Abstract group 145200.l: $C_{220}:F_{11}:S_3$

• Label: 145200.l
• Order: \( 2^{4} \cdot 3 \cdot 5^{2} \cdot 11^{2} \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 5^{2} \)
• $\card{Z(G)}$: \( 2 \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3 \cdot 5^{2} \cdot 11^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 5 \)
• Perm deg.: $42$
• Trans deg.: $660$
• Rank: $3$

Group information

Description:	$C_{220}:F_{11}:S_3$
Order:	\(145200\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \cdot 11^{2} \)
Exponent:	\(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \)
Automorphism group:	$C_{11}^2.C_{30}.C_{10}.C_2^6.C_2$, of order \(4646400\)\(\medspace = 2^{9} \cdot 3 \cdot 5^{2} \cdot 11^{2} \)
Composition factors:	$C_2$ x 4, $C_3$, $C_5$ x 2, $C_{11}$ x 2
Derived length:	$3$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	5	6	10	11	12	15	20	22	30	44	55	60	110	220
Elements	1	507	242	244	2424	726	67368	120	968	5808	10656	2760	17424	240	480	23232	11040	960	145200
Conjugacy classes	1	7	1	2	24	3	168	3	4	24	48	7	72	4	12	96	28	16	520
Divisions	1	7	1	2	6	3	42	3	2	6	12	7	18	3	3	12	7	3	138
Autjugacy classes	1	3	1	2	5	2	15	2	2	5	10	3	10	2	2	10	3	2	80

Dimension	1	2	4	8	16	30	60	120	240	480
Irr. complex chars.	200	250	0	0	0	40	30	0	0	0	520
Irr. rational chars.	8	6	50	36	12	8	4	9	4	1	138

Minimal presentations

Permutation degree:	$42$
Transitive degree:	$660$
Rank:	$3$
Inequivalent generating triples:	$182988288$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	60	not computed	240
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d \mid a^{10}=b^{30}=c^{11}=d^{44}=[c,d]=1, b^{a}=b^{11}c^{8}d^{24}, c^{a}= \!\cdots\! \rangle}$
Permutation group:	Degree $42$ $\langle(38,39,40,41,42), (2,7,10,29,6,21,19,14,16,9)(3,12,17,11,5,20,4,15,13,18) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 3 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 3 \end{array}\right), \left(\begin{array}{rrrr} 7 & 8 & 10 & 9 \\ 5 & 8 & 7 & 2 \\ 6 & 9 & 0 & 8 \\ 10 & 4 & 3 & 7 \end{array}\right), \left(\begin{array}{rrrr} 0 & 8 & 9 & 4 \\ 2 & 7 & 2 & 9 \\ 9 & 1 & 7 & 3 \\ 2 & 9 & 9 & 3 \end{array}\right), \left(\begin{array}{rrrr} 8 & 2 & 9 & 8 \\ 5 & 6 & 0 & 9 \\ 8 & 6 & 1 & 9 \\ 1 & 8 & 6 & 10 \end{array}\right), \left(\begin{array}{rrrr} 5 & 3 & 8 & 0 \\ 0 & 0 & 1 & 6 \\ 5 & 1 & 4 & 8 \\ 8 & 7 & 5 & 2 \end{array}\right), \left(\begin{array}{rrrr} 9 & 6 & 5 & 2 \\ 9 & 8 & 4 & 5 \\ 3 & 8 & 2 & 5 \\ 7 & 3 & 2 & 1 \end{array}\right), \left(\begin{array}{rrrr} 9 & 9 & 2 & 3 \\ 3 & 8 & 2 & 2 \\ 9 & 0 & 1 & 2 \\ 1 & 9 & 8 & 0 \end{array}\right), \left(\begin{array}{rrrr} 8 & 0 & 0 & 0 \\ 0 & 8 & 0 & 0 \\ 0 & 0 & 8 & 0 \\ 0 & 0 & 0 & 8 \end{array}\right), \left(\begin{array}{rrrr} 9 & 7 & 7 & 9 \\ 10 & 0 & 1 & 8 \\ 8 & 5 & 8 & 2 \\ 4 & 0 & 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{11})$
Direct product:	$C_5$ $\, \times\, $ $(C_{11}^2:(C_{10}\times D_{12}))$
Semidirect product:	$(C_{220}:F_{11})$ $\,\rtimes\,$ $S_3$	$(C_{110}.F_{11})$ $\,\rtimes\,$ $D_6$	$(C_{55}:F_{11})$ $\,\rtimes\,$ $D_{12}$	$(C_{11}^2:D_6)$ $\,\rtimes\,$ $C_{10}^2$	all 60
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_{110}:F_{11})$ . $D_6$	$(C_{22}:F_{11})$ . $(S_3\times C_{10})$	$(C_{110}:D_{11})$ . $(S_3\times C_{10})$	$(C_{11}:D_{22})$ . $(D_6\times C_5^2)$	all 18

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{2} \times C_{10}^{2} \simeq C_{2}^{3} \times C_{5}^{2}$
Schur multiplier:	$C_{2}^{2} \times C_{10}$
Commutator length:	$1$

Subgroups

There are 132208 subgroups in 1360 conjugacy classes, 222 normal (45 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{10}$	$G/Z \simeq$ $C_{11}^2:(C_{10}\times D_6)$
Commutator:	$G' \simeq$ $C_{11}^2:C_6$	$G/G' \simeq$ $C_2\times C_{10}^2$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{11}^2:(S_3\times C_{10}^2)$
Fitting:	$\operatorname{Fit} \simeq$ $C_{11}\times C_{220}$	$G/\operatorname{Fit} \simeq$ $S_3\times C_{10}$
Radical:	$R \simeq$ $C_{220}:F_{11}:S_3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{11}\times C_{110}$	$G/\operatorname{soc} \simeq$ $C_{10}\times D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 145200: $C_{220}:F_{11}:S_3$

Order 72600: $C_{11}^2:(S_3\times C_{10}^2)$ x 2, $C_{11}^2:(D_{12}\times C_5^2)$ x 2, $C_{11}^2:(D_{12}\times C_5^2)$ x 2, $C_{20}\times C_{11}^2:C_{30}$

Order 48400: $C_5\times D_{44}:F_{11}$

Order 36300: $C_{11}^2:(D_6\times C_5^2)$ x 4, $C_{11}^2:(D_6\times C_5^2)$ x 4, $C_5\times C_{11}^2:C_{60}$, $C_{10}\times C_{11}^2:C_{30}$, $C_5\times C_{11}^2:C_{60}$

Order 29040: $C_{11}^2:(C_{10}\times D_{12})$ x 5, $(C_{11}\times C_{220}):D_6$

Order 24200: $C_{220}:F_{11}$ x 2, $C_5\times D_{22}:F_{11}$ x 2, $C_5\times D_{22}:F_{11}$ x 2, $C_{220}:F_{11}$

Order 18150: $C_{11}^2:(S_3\times C_5^2)$ x 4, $C_5\times C_{11}^2:C_{30}$ x 2, $C_5\times C_{11}^2:C_{30}$

Order 14520: $C_{11}^2:(C_5\times D_{12})$ x 10, $C_{11}^2:(C_5\times D_{12})$ x 10, $C_{11}^2:(C_{10}\times D_6)$ x 10, $C_4\times C_{11}^2:C_{30}$ x 5, $C_5\times C_{11}^2:D_{12}$ x 2, $C_{10}\times C_{11}^2:D_6$ x 2, $C_5\times C_{11}^2:D_{12}$ x 2, $C_4\times C_{11}^2:C_{30}$

Order 12100: $C_{110}:F_{11}$ x 4, $C_{11}^2:C_{10}^2$ x 4, $C_5\times C_{11}^2:C_{20}$, $C_{110}.F_{11}$, $C_{110}:F_{11}$

Order 9680: $D_{44}:F_{11}$ x 5, $C_{220}:D_{22}$

Order 9075: $C_5\times C_{11}^2:C_{15}$

Order 7260: $C_{11}^2:(S_3\times C_{10})$ x 20, $C_{11}^2:(S_3\times C_{10})$ x 20, $C_{11}^2:C_{60}$ x 5, $C_{11}^2:C_{60}$ x 5, $C_2\times C_{11}^2:C_{30}$ x 5, $C_{10}\times C_{11}^2:S_3$ x 4, $C_5\times C_{11}^2:D_6$ x 4, $C_2\times C_{11}^2:C_{30}$, $C_{11}^2:C_{60}$, $C_{11}^2:C_{60}$

Order 6050: $C_{55}:F_{11}$ x 4, $C_{55}:F_{11}$ x 2, $C_5\times C_{11}^2:C_{10}$

Order 5808: $(C_{11}\times C_{44}):D_6$

Order 4840: $C_{44}:F_{11}$ x 10, $D_{22}:F_{11}$ x 10, $D_{22}:F_{11}$ x 10, $C_{44}:F_{11}$ x 5, $C_{55}\times D_{44}$ x 2, $C_{110}.D_{22}$ x 2, $C_{10}\times D_{11}^2$ x 2, $C_{220}:D_{11}$

Order 4400: $C_{44}:C_{10}^2$ x 2

Order 3630: $C_{11}^2:(C_5\times S_3)$ x 20, $C_{11}^2:C_{30}$ x 10, $C_{11}^2:C_{30}$ x 5, $C_5\times C_{11}^2:S_3$ x 4, $C_{11}^2:C_{30}$ x 2, $C_{11}^2:C_{30}$

Order 3025: $C_{11}^2:C_5^2$

Order 2904: $C_{11}^2:D_{12}$ x 2, $C_{11}^2:D_{12}$ x 2, $C_2\times C_{11}^2:D_6$ x 2, $C_4\times C_{11}^2:C_6$

Order 2420: $C_{22}:F_{11}$ x 20, $D_{11}:F_{11}$ x 20, $C_{11}^2:C_{20}$ x 5, $C_{22}:F_{11}$ x 5, $C_{11}^2:C_{20}$ x 5, $D_{11}\times C_{110}$ x 4, $C_5\times D_{11}^2$ x 4, $C_{11}^2:C_{20}$, $C_{11}\times C_{220}$, $C_{110}:D_{11}$

Order 2200: $C_{22}.C_{10}^2$ x 4, $C_{22}:C_{10}^2$ x 4, $C_{20}\times F_{11}$ x 3, $C_{220}:C_{10}$ x 2, $C_{220}:C_{10}$ x 2

Order 1936: $C_{44}:D_{22}$

Order 1815: $C_{11}^2:C_{15}$ x 5, $C_{11}^2:C_{15}$

Order 1452: $C_{11}^2:D_6$ x 4, $C_{11}^2:D_6$ x 4, $C_2\times C_{11}^2:C_6$, $C_{11}^2:C_{12}$, $C_{11}^2:C_{12}$

Order 1210: $C_{11}:F_{11}$ x 20, $C_{11}:F_{11}$ x 10, $C_{11}^2:C_{10}$ x 5, $D_{11}\times C_{55}$ x 4, $C_{55}:D_{11}$ x 2, $C_{11}\times C_{110}$

Order 1200: $C_{12}:C_{10}^2$

Order 1100: $C_{10}\times F_{11}$ x 15, $C_{110}:C_{10}$ x 4, $C_{220}:C_5$ x 3, $C_{55}:C_{20}$ x 3

Order 968: $D_{11}\times D_{22}$ x 2, $C_{11}\times D_{44}$ x 2, $D_{22}:D_{11}$ x 2, $C_{44}:D_{11}$

Order 880: $D_4\times F_{11}$ x 10, $C_{20}:D_{22}$ x 2

Order 726: $C_{11}^2:S_3$ x 4, $C_{11}^2:C_6$ x 2, $C_{11}^2:C_6$

Order 605: $C_{11}^2:C_5$ x 5, $C_{11}\times C_{55}$

Order 600: $D_{12}\times C_5^2$ x 4, $S_3\times C_{10}^2$ x 2, $C_{10}\times C_{60}$

Order 550: $C_5\times F_{11}$ x 10, $C_{110}:C_5$ x 7

Order 484: $D_{11}^2$ x 4, $C_{11}\times D_{22}$ x 4, $C_{11}\times C_{44}$, $C_{11}^2:C_4$, $C_{11}:D_{22}$

Order 440: $C_2^2\times F_{11}$ x 20, $D_{22}:C_{10}$ x 20, $C_4\times F_{11}$ x 15, $C_{44}:C_{10}$ x 10, $C_{44}:C_{10}$ x 10, $C_{10}\times D_{22}$ x 4, $C_{55}:D_4$ x 4, $C_{20}\times D_{11}$ x 3, $D_4\times C_{55}$ x 2, $C_5\times D_{44}$ x 2

Order 400: $C_4:C_{10}^2$

Order 363: $C_{11}^2:C_3$

Order 300: $D_6\times C_5^2$ x 8, $C_5\times C_{60}$ x 2, $C_{10}\times C_{30}$

Order 275: $C_{55}:C_5$ x 3

Order 242: $C_{11}\times D_{11}$ x 4, $C_{11}:D_{11}$ x 2, $C_{11}\times C_{22}$

Order 240: $C_{10}\times D_{12}$ x 6

Order 220: $C_2\times F_{11}$ x 75, $C_{22}:C_{10}$ x 20, $C_{11}:C_{20}$ x 15, $C_5\times D_{22}$ x 15, $C_{11}:C_{20}$ x 15, $C_2\times C_{110}$ x 4, $C_{220}$ x 3, $C_{11}:C_{20}$ x 3

Order 200: $D_4\times C_5^2$ x 4, $C_2\times C_{10}^2$ x 2, $C_{10}\times C_{20}$

Order 176: $D_4\times D_{11}$ x 2

Order 150: $S_3\times C_5^2$ x 4, $C_5\times C_{30}$ x 3

Order 121: $C_{11}^2$

Order 120: $C_5\times D_{12}$ x 24, $C_{10}\times D_6$ x 12, $C_2\times C_{60}$ x 6

Order 110: $F_{11}$ x 50, $C_{11}:C_{10}$ x 35, $C_5\times D_{11}$ x 10, $C_{110}$ x 7

Order 100: $C_{10}^2$ x 9, $C_5\times C_{20}$ x 2

Order 88: $C_{11}:D_4$ x 4, $C_2\times D_{22}$ x 4, $C_4\times D_{11}$ x 3, $D_4\times C_{11}$ x 2, $D_{44}$ x 2

Order 80: $D_4\times C_{10}$ x 6

Order 75: $C_5\times C_{15}$

Order 60: $S_3\times C_{10}$ x 48, $C_{60}$ x 12, $C_2\times C_{30}$ x 6

Order 55: $C_{11}:C_5$ x 15, $C_{55}$ x 3

Order 50: $C_5\times C_{10}$ x 7

Order 48: $C_2\times D_{12}$

Order 44: $D_{22}$ x 15, $C_2\times C_{22}$ x 4, $C_{44}$ x 3, $C_{11}:C_4$ x 3

Order 40: $C_5\times D_4$ x 24, $C_2^2\times C_{10}$ x 12, $C_2\times C_{20}$ x 6

Order 30: $C_5\times S_3$ x 24, $C_{30}$ x 18

Order 25: $C_5^2$

Order 24: $D_{12}$ x 4, $C_2\times D_6$ x 2, $C_2\times C_{12}$

Order 22: $D_{11}$ x 10, $C_{22}$ x 7

Order 20: $C_2\times C_{10}$ x 54, $C_{20}$ x 12

Order 16: $C_2\times D_4$

Order 15: $C_{15}$ x 6

Order 12: $D_6$ x 8, $C_{12}$ x 2, $C_2\times C_6$

Order 11: $C_{11}$ x 3

Order 10: $C_{10}$ x 42

Order 8: $D_4$ x 4, $C_2^3$ x 2, $C_2\times C_4$

Order 6: $S_3$ x 4, $C_6$ x 3

Order 5: $C_5$ x 6

Order 4: $C_2^2$ x 9, $C_4$ x 2

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 145200: $C_{220}:F_{11}:S_3$

Order 72600: $C_{11}^2:(S_3\times C_{10}^2)$, $C_{11}^2:(D_{12}\times C_5^2)$, $C_{11}^2:(D_{12}\times C_5^2)$, $C_{20}\times C_{11}^2:C_{30}$

Order 48400: $C_5\times D_{44}:F_{11}$

Order 36300: $C_{11}^2:(D_6\times C_5^2)$, $C_{11}^2:(D_6\times C_5^2)$, $C_5\times C_{11}^2:C_{60}$, $C_{10}\times C_{11}^2:C_{30}$, $C_5\times C_{11}^2:C_{60}$

Order 29040: $C_{11}^2:(C_{10}\times D_{12})$, $(C_{11}\times C_{220}):D_6$

Order 24200: $C_{220}:F_{11}$, $C_{220}:F_{11}$, $C_5\times D_{22}:F_{11}$, $C_5\times D_{22}:F_{11}$

Order 18150: $C_{11}^2:(S_3\times C_5^2)$, $C_5\times C_{11}^2:C_{30}$, $C_5\times C_{11}^2:C_{30}$

Order 14520: $C_4\times C_{11}^2:C_{30}$, $C_4\times C_{11}^2:C_{30}$, $C_5\times C_{11}^2:D_{12}$, $C_{10}\times C_{11}^2:D_6$, $C_{11}^2:(C_5\times D_{12})$, $C_{11}^2:(C_5\times D_{12})$, $C_{11}^2:(C_{10}\times D_6)$, $C_5\times C_{11}^2:D_{12}$

Order 12100: $C_{110}:F_{11}$, $C_{11}^2:C_{10}^2$, $C_5\times C_{11}^2:C_{20}$, $C_{110}.F_{11}$, $C_{110}:F_{11}$

Order 9680: $C_{220}:D_{22}$, $D_{44}:F_{11}$

Order 9075: $C_5\times C_{11}^2:C_{15}$

Order 7260: $C_{10}\times C_{11}^2:S_3$, $C_{11}^2:C_{60}$, $C_2\times C_{11}^2:C_{30}$, $C_{11}^2:C_{60}$, $C_{11}^2:C_{60}$, $C_{11}^2:C_{60}$, $C_5\times C_{11}^2:D_6$, $C_{11}^2:(S_3\times C_{10})$, $C_{11}^2:(S_3\times C_{10})$, $C_2\times C_{11}^2:C_{30}$

Order 6050: $C_{55}:F_{11}$, $C_5\times C_{11}^2:C_{10}$, $C_{55}:F_{11}$

Order 5808: $(C_{11}\times C_{44}):D_6$

Order 4840: $C_{44}:F_{11}$, $D_{22}:F_{11}$, $C_{55}\times D_{44}$, $C_{110}.D_{22}$, $C_{44}:F_{11}$, $C_{220}:D_{11}$, $C_{10}\times D_{11}^2$, $D_{22}:F_{11}$

Order 4400: $C_{44}:C_{10}^2$

Order 3630: $C_{11}^2:(C_5\times S_3)$, $C_5\times C_{11}^2:S_3$, $C_{11}^2:C_{30}$, $C_{11}^2:C_{30}$, $C_{11}^2:C_{30}$, $C_{11}^2:C_{30}$

Order 3025: $C_{11}^2:C_5^2$

Order 2904: $C_{11}^2:D_{12}$, $C_4\times C_{11}^2:C_6$, $C_{11}^2:D_{12}$, $C_2\times C_{11}^2:D_6$

Order 2420: $C_{22}:F_{11}$, $D_{11}:F_{11}$, $D_{11}\times C_{110}$, $C_{11}^2:C_{20}$, $C_{11}^2:C_{20}$, $C_{11}\times C_{220}$, $C_{110}:D_{11}$, $C_{22}:F_{11}$, $C_5\times D_{11}^2$, $C_{11}^2:C_{20}$

Order 2200: $C_{20}\times F_{11}$ x 2, $C_{220}:C_{10}$, $C_{22}.C_{10}^2$, $C_{220}:C_{10}$, $C_{22}:C_{10}^2$

Order 1936: $C_{44}:D_{22}$

Order 1815: $C_{11}^2:C_{15}$, $C_{11}^2:C_{15}$

Order 1452: $C_2\times C_{11}^2:C_6$, $C_{11}^2:D_6$, $C_{11}^2:D_6$, $C_{11}^2:C_{12}$, $C_{11}^2:C_{12}$

Order 1210: $C_{11}:F_{11}$, $C_{11}\times C_{110}$, $C_{55}:D_{11}$, $D_{11}\times C_{55}$, $C_{11}^2:C_{10}$, $C_{11}:F_{11}$

Order 1200: $C_{12}:C_{10}^2$

Order 1100: $C_{10}\times F_{11}$ x 4, $C_{220}:C_5$ x 2, $C_{55}:C_{20}$ x 2, $C_{110}:C_{10}$

Order 968: $D_{11}\times D_{22}$, $C_{44}:D_{11}$, $C_{11}\times D_{44}$, $D_{22}:D_{11}$

Order 880: $C_{20}:D_{22}$, $D_4\times F_{11}$

Order 726: $C_{11}^2:C_6$, $C_{11}^2:C_6$, $C_{11}^2:S_3$

Order 605: $C_{11}\times C_{55}$, $C_{11}^2:C_5$

Order 600: $D_{12}\times C_5^2$ x 2, $S_3\times C_{10}^2$, $C_{10}\times C_{60}$

Order 550: $C_5\times F_{11}$ x 3, $C_{110}:C_5$ x 3

Order 484: $D_{11}^2$, $C_{11}\times C_{44}$, $C_{11}^2:C_4$, $C_{11}:D_{22}$, $C_{11}\times D_{22}$

Order 440: $C_4\times F_{11}$ x 2, $C_{20}\times D_{11}$ x 2, $C_{44}:C_{10}$, $C_{10}\times D_{22}$, $C_2^2\times F_{11}$, $D_4\times C_{55}$, $C_{55}:D_4$, $C_5\times D_{44}$, $C_{44}:C_{10}$, $D_{22}:C_{10}$

Order 400: $C_4:C_{10}^2$

Order 363: $C_{11}^2:C_3$

Order 300: $D_6\times C_5^2$ x 2, $C_5\times C_{60}$ x 2, $C_{10}\times C_{30}$

Order 275: $C_{55}:C_5$ x 2

Order 242: $C_{11}\times C_{22}$, $C_{11}:D_{11}$, $C_{11}\times D_{11}$

Order 240: $C_{10}\times D_{12}$ x 2

Order 220: $C_2\times F_{11}$ x 4, $C_5\times D_{22}$ x 4, $C_{220}$ x 2, $C_{11}:C_{20}$ x 2, $C_{11}:C_{20}$ x 2, $C_{11}:C_{20}$ x 2, $C_{22}:C_{10}$, $C_2\times C_{110}$

Order 200: $D_4\times C_5^2$ x 2, $C_2\times C_{10}^2$, $C_{10}\times C_{20}$

Order 176: $D_4\times D_{11}$

Order 150: $C_5\times C_{30}$ x 2, $S_3\times C_5^2$

Order 121: $C_{11}^2$

Order 120: $C_5\times D_{12}$ x 4, $C_{10}\times D_6$ x 2, $C_2\times C_{60}$ x 2

Order 110: $C_{110}$ x 3, $C_5\times D_{11}$ x 3, $C_{11}:C_{10}$ x 3, $F_{11}$ x 3

Order 100: $C_{10}^2$ x 3, $C_5\times C_{20}$ x 2

Order 88: $C_4\times D_{11}$ x 2, $D_4\times C_{11}$, $C_{11}:D_4$, $D_{44}$, $C_2\times D_{22}$

Order 80: $D_4\times C_{10}$ x 2

Order 75: $C_5\times C_{15}$

Order 60: $C_{60}$ x 4, $S_3\times C_{10}$ x 4, $C_2\times C_{30}$ x 2

Order 55: $C_{55}$ x 2, $C_{11}:C_5$ x 2

Order 50: $C_5\times C_{10}$ x 3

Order 48: $C_2\times D_{12}$

Order 44: $D_{22}$ x 4, $C_{44}$ x 2, $C_{11}:C_4$ x 2, $C_2\times C_{22}$

Order 40: $C_5\times D_4$ x 4, $C_2\times C_{20}$ x 2, $C_2^2\times C_{10}$ x 2

Order 30: $C_{30}$ x 4, $C_5\times S_3$ x 2

Order 25: $C_5^2$

Order 24: $D_{12}$ x 2, $C_2\times C_{12}$, $C_2\times D_6$

Order 22: $C_{22}$ x 3, $D_{11}$ x 3

Order 20: $C_2\times C_{10}$ x 6, $C_{20}$ x 4

Order 16: $C_2\times D_4$

Order 15: $C_{15}$ x 2

Order 12: $D_6$ x 2, $C_{12}$ x 2, $C_2\times C_6$

Order 11: $C_{11}$ x 2

Order 10: $C_{10}$ x 6

Order 8: $D_4$ x 2, $C_2^3$, $C_2\times C_4$

Order 6: $C_6$ x 2, $S_3$

Order 5: $C_5$ x 2

Order 4: $C_2^2$ x 3, $C_4$ x 2

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 145200: $C_{220}:F_{11}:S_3$ ($C_1$)

Order 72600: $C_{11}^2:(S_3\times C_{10}^2)$ ($C_2$) x 2, $C_{11}^2:(D_{12}\times C_5^2)$ ($C_2$) x 2, $C_{11}^2:(D_{12}\times C_5^2)$ ($C_2$) x 2, $C_{20}\times C_{11}^2:C_{30}$ ($C_2$)

Order 36300: $C_{11}^2:(D_6\times C_5^2)$ ($C_2^2$) x 4, $C_5\times C_{11}^2:C_{60}$ ($C_2^2$), $C_{10}\times C_{11}^2:C_{30}$ ($C_2^2$), $C_5\times C_{11}^2:C_{60}$ ($C_2^2$)

Order 29040: $C_{11}^2:(C_{10}\times D_{12})$ ($C_5$) x 5, $(C_{11}\times C_{220}):D_6$ ($C_5$)

Order 24200: $C_{220}:F_{11}$ ($S_3$)

Order 18150: $C_5\times C_{11}^2:C_{30}$ ($D_4$) x 2, $C_5\times C_{11}^2:C_{30}$ ($C_2^3$)

Order 14520: $C_{11}^2:(C_5\times D_{12})$ ($C_{10}$) x 10, $C_{11}^2:(C_5\times D_{12})$ ($C_{10}$) x 10, $C_{11}^2:(C_{10}\times D_6)$ ($C_{10}$) x 10, $C_4\times C_{11}^2:C_{30}$ ($C_{10}$) x 5, $C_5\times C_{11}^2:D_{12}$ ($C_{10}$) x 2, $C_{10}\times C_{11}^2:D_6$ ($C_{10}$) x 2, $C_5\times C_{11}^2:D_{12}$ ($C_{10}$) x 2, $C_4\times C_{11}^2:C_{30}$ ($C_{10}$)

Order 12100: $C_5\times C_{11}^2:C_{20}$ ($D_6$), $C_{110}.F_{11}$ ($D_6$), $C_{110}:F_{11}$ ($D_6$)

Order 9075: $C_5\times C_{11}^2:C_{15}$ ($C_2\times D_4$)

Order 7260: $C_{11}^2:(S_3\times C_{10})$ ($C_2\times C_{10}$) x 20, $C_{11}^2:C_{60}$ ($C_2\times C_{10}$) x 5, $C_{11}^2:C_{60}$ ($C_2\times C_{10}$) x 5, $C_2\times C_{11}^2:C_{30}$ ($C_2\times C_{10}$) x 5, $C_{10}\times C_{11}^2:S_3$ ($C_2\times C_{10}$) x 4, $C_2\times C_{11}^2:C_{30}$ ($C_2\times C_{10}$), $C_{11}^2:C_{60}$ ($C_2\times C_{10}$), $C_{11}^2:C_{60}$ ($C_2\times C_{10}$)

Order 6050: $C_{55}:F_{11}$ ($D_{12}$) x 2, $C_5\times C_{11}^2:C_{10}$ ($C_2\times D_6$)

Order 5808: $(C_{11}\times C_{44}):D_6$ ($C_5^2$)

Order 4840: $C_{44}:F_{11}$ ($C_5\times S_3$) x 5, $C_{220}:D_{11}$ ($C_5\times S_3$)

Order 3630: $C_{11}^2:C_{30}$ ($C_5\times D_4$) x 10, $C_{11}^2:C_{30}$ ($C_2^2\times C_{10}$) x 5, $C_{11}^2:C_{30}$ ($C_5\times D_4$) x 2, $C_{11}^2:C_{30}$ ($C_2^2\times C_{10}$)

Order 3025: $C_{11}^2:C_5^2$ ($C_2\times D_{12}$)

Order 2904: $C_{11}^2:D_{12}$ ($C_5\times C_{10}$) x 2, $C_{11}^2:D_{12}$ ($C_5\times C_{10}$) x 2, $C_2\times C_{11}^2:D_6$ ($C_5\times C_{10}$) x 2, $C_4\times C_{11}^2:C_6$ ($C_5\times C_{10}$)

Order 2420: $C_{11}^2:C_{20}$ ($S_3\times C_{10}$) x 5, $C_{22}:F_{11}$ ($S_3\times C_{10}$) x 5, $C_{11}^2:C_{20}$ ($S_3\times C_{10}$) x 5, $C_{11}^2:C_{20}$ ($S_3\times C_{10}$), $C_{11}\times C_{220}$ ($S_3\times C_{10}$), $C_{110}:D_{11}$ ($S_3\times C_{10}$)

Order 1815: $C_{11}^2:C_{15}$ ($D_4\times C_{10}$) x 5, $C_{11}^2:C_{15}$ ($D_4\times C_{10}$)

Order 1452: $C_{11}^2:D_6$ ($C_{10}^2$) x 4, $C_2\times C_{11}^2:C_6$ ($C_{10}^2$), $C_{11}^2:C_{12}$ ($C_{10}^2$), $C_{11}^2:C_{12}$ ($C_{10}^2$)

Order 1210: $C_{11}:F_{11}$ ($C_5\times D_{12}$) x 10, $C_{11}^2:C_{10}$ ($C_{10}\times D_6$) x 5, $C_{55}:D_{11}$ ($C_5\times D_{12}$) x 2, $C_{11}\times C_{110}$ ($C_{10}\times D_6$)

Order 968: $C_{44}:D_{11}$ ($S_3\times C_5^2$)

Order 726: $C_{11}^2:C_6$ ($D_4\times C_5^2$) x 2, $C_{11}^2:C_6$ ($C_2\times C_{10}^2$)

Order 605: $C_{11}^2:C_5$ ($C_{10}\times D_{12}$) x 5, $C_{11}\times C_{55}$ ($C_{10}\times D_{12}$)

Order 484: $C_{11}\times C_{44}$ ($D_6\times C_5^2$), $C_{11}^2:C_4$ ($D_6\times C_5^2$), $C_{11}:D_{22}$ ($D_6\times C_5^2$)

Order 363: $C_{11}^2:C_3$ ($C_4:C_{10}^2$)

Order 242: $C_{11}:D_{11}$ ($D_{12}\times C_5^2$) x 2, $C_{11}\times C_{22}$ ($S_3\times C_{10}^2$)

Order 121: $C_{11}^2$ ($C_{12}:C_{10}^2$)

Order 20: $C_{20}$ ($C_{11}^2:(S_3\times C_{10})$)

Order 10: $C_{10}$ ($C_{11}^2:(C_{10}\times D_6)$)

Order 5: $C_5$ ($C_{11}^2:(C_{10}\times D_{12})$)

Order 4: $C_4$ ($C_{11}^2:(D_6\times C_5^2)$)

Order 2: $C_2$ ($C_{11}^2:(S_3\times C_{10}^2)$)

Order 1: $C_1$ ($C_{220}:F_{11}:S_3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 145200: $C_{220}:F_{11}:S_3$ ($C_1$)

Order 72600: $C_{11}^2:(S_3\times C_{10}^2)$ ($C_2$), $C_{11}^2:(D_{12}\times C_5^2)$ ($C_2$), $C_{11}^2:(D_{12}\times C_5^2)$ ($C_2$), $C_{20}\times C_{11}^2:C_{30}$ ($C_2$)

Order 36300: $C_{11}^2:(D_6\times C_5^2)$ ($C_2^2$), $C_5\times C_{11}^2:C_{60}$ ($C_2^2$), $C_{10}\times C_{11}^2:C_{30}$ ($C_2^2$), $C_5\times C_{11}^2:C_{60}$ ($C_2^2$)

Order 29040: $C_{11}^2:(C_{10}\times D_{12})$ ($C_5$), $(C_{11}\times C_{220}):D_6$ ($C_5$)

Order 24200: $C_{220}:F_{11}$ ($S_3$)

Order 18150: $C_5\times C_{11}^2:C_{30}$ ($C_2^3$), $C_5\times C_{11}^2:C_{30}$ ($D_4$)

Order 14520: $C_4\times C_{11}^2:C_{30}$ ($C_{10}$), $C_4\times C_{11}^2:C_{30}$ ($C_{10}$), $C_5\times C_{11}^2:D_{12}$ ($C_{10}$), $C_{10}\times C_{11}^2:D_6$ ($C_{10}$), $C_{11}^2:(C_5\times D_{12})$ ($C_{10}$), $C_{11}^2:(C_5\times D_{12})$ ($C_{10}$), $C_{11}^2:(C_{10}\times D_6)$ ($C_{10}$), $C_5\times C_{11}^2:D_{12}$ ($C_{10}$)

Order 12100: $C_5\times C_{11}^2:C_{20}$ ($D_6$), $C_{110}.F_{11}$ ($D_6$), $C_{110}:F_{11}$ ($D_6$)

Order 9075: $C_5\times C_{11}^2:C_{15}$ ($C_2\times D_4$)

Order 7260: $C_{10}\times C_{11}^2:S_3$ ($C_2\times C_{10}$), $C_{11}^2:C_{60}$ ($C_2\times C_{10}$), $C_2\times C_{11}^2:C_{30}$ ($C_2\times C_{10}$), $C_{11}^2:C_{60}$ ($C_2\times C_{10}$), $C_{11}^2:C_{60}$ ($C_2\times C_{10}$), $C_{11}^2:C_{60}$ ($C_2\times C_{10}$), $C_{11}^2:(S_3\times C_{10})$ ($C_2\times C_{10}$), $C_2\times C_{11}^2:C_{30}$ ($C_2\times C_{10}$)

Order 6050: $C_5\times C_{11}^2:C_{10}$ ($C_2\times D_6$), $C_{55}:F_{11}$ ($D_{12}$)

Order 5808: $(C_{11}\times C_{44}):D_6$ ($C_5^2$)

Order 4840: $C_{44}:F_{11}$ ($C_5\times S_3$), $C_{220}:D_{11}$ ($C_5\times S_3$)

Order 3630: $C_{11}^2:C_{30}$ ($C_2^2\times C_{10}$), $C_{11}^2:C_{30}$ ($C_5\times D_4$), $C_{11}^2:C_{30}$ ($C_5\times D_4$), $C_{11}^2:C_{30}$ ($C_2^2\times C_{10}$)

Order 3025: $C_{11}^2:C_5^2$ ($C_2\times D_{12}$)

Order 2904: $C_{11}^2:D_{12}$ ($C_5\times C_{10}$), $C_4\times C_{11}^2:C_6$ ($C_5\times C_{10}$), $C_{11}^2:D_{12}$ ($C_5\times C_{10}$), $C_2\times C_{11}^2:D_6$ ($C_5\times C_{10}$)

Order 2420: $C_{11}^2:C_{20}$ ($S_3\times C_{10}$), $C_{11}^2:C_{20}$ ($S_3\times C_{10}$), $C_{11}\times C_{220}$ ($S_3\times C_{10}$), $C_{110}:D_{11}$ ($S_3\times C_{10}$), $C_{22}:F_{11}$ ($S_3\times C_{10}$), $C_{11}^2:C_{20}$ ($S_3\times C_{10}$)

Order 1815: $C_{11}^2:C_{15}$ ($D_4\times C_{10}$), $C_{11}^2:C_{15}$ ($D_4\times C_{10}$)

Order 1452: $C_2\times C_{11}^2:C_6$ ($C_{10}^2$), $C_{11}^2:D_6$ ($C_{10}^2$), $C_{11}^2:C_{12}$ ($C_{10}^2$), $C_{11}^2:C_{12}$ ($C_{10}^2$)

Order 1210: $C_{11}:F_{11}$ ($C_5\times D_{12}$), $C_{11}\times C_{110}$ ($C_{10}\times D_6$), $C_{55}:D_{11}$ ($C_5\times D_{12}$), $C_{11}^2:C_{10}$ ($C_{10}\times D_6$)

Order 968: $C_{44}:D_{11}$ ($S_3\times C_5^2$)

Order 726: $C_{11}^2:C_6$ ($C_2\times C_{10}^2$), $C_{11}^2:C_6$ ($D_4\times C_5^2$)

Order 605: $C_{11}\times C_{55}$ ($C_{10}\times D_{12}$), $C_{11}^2:C_5$ ($C_{10}\times D_{12}$)

Order 484: $C_{11}\times C_{44}$ ($D_6\times C_5^2$), $C_{11}^2:C_4$ ($D_6\times C_5^2$), $C_{11}:D_{22}$ ($D_6\times C_5^2$)

Order 363: $C_{11}^2:C_3$ ($C_4:C_{10}^2$)

Order 242: $C_{11}\times C_{22}$ ($S_3\times C_{10}^2$), $C_{11}:D_{11}$ ($D_{12}\times C_5^2$)

Order 121: $C_{11}^2$ ($C_{12}:C_{10}^2$)

Order 20: $C_{20}$ ($C_{11}^2:(S_3\times C_{10})$)

Order 10: $C_{10}$ ($C_{11}^2:(C_{10}\times D_6)$)

Order 5: $C_5$ ($C_{11}^2:(C_{10}\times D_{12})$)

Order 4: $C_4$ ($C_{11}^2:(D_6\times C_5^2)$)

Order 2: $C_2$ ($C_{11}^2:(S_3\times C_{10}^2)$)

Order 1: $C_1$ ($C_{220}:F_{11}:S_3$)

Series

Derived series

$C_{220}:F_{11}:S_3$

$\rhd$

$C_{220}:F_{11}:S_3$

$\rhd$

$C_{11}^2:C_6$

$\rhd$

$C_{11}^2:C_6$

$\rhd$

$C_{11}^2$

$\rhd$

$C_{11}^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_{220}:F_{11}:S_3$

$\rhd$

$C_{220}:F_{11}:S_3$

$\rhd$

$C_{11}^2:(S_3\times C_{10}^2)$

$\rhd$

$C_{11}^2:(S_3\times C_{10}^2)$

$\rhd$

$C_{11}^2:(D_6\times C_5^2)$

$\rhd$

$C_{11}^2:(D_6\times C_5^2)$

$\rhd$

$C_5\times C_{11}^2:C_{30}$

$\rhd$

$C_5\times C_{11}^2:C_{30}$

$\rhd$

$C_{11}^2:C_{30}$

$\rhd$

$C_{11}^2:C_{30}$

$\rhd$

$C_{11}^2:C_6$

$\rhd$

$C_{11}^2:C_6$

$\rhd$

$C_{11}^2:C_3$

$\rhd$

$C_{11}^2:C_3$

$\rhd$

$C_{11}^2$

$\rhd$

$C_{11}^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_{220}:F_{11}:S_3$

$\rhd$

$C_{220}:F_{11}:S_3$

$\rhd$

$C_{11}^2:C_6$

$\rhd$

$C_{11}^2:C_6$

$\rhd$

$C_{11}^2:C_3$

$\rhd$

$C_{11}^2:C_3$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_{10}$

$\lhd$

$C_{10}$

$\lhd$

$C_{20}$

$\lhd$

$C_{20}$

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

The $520 \times 520$ character table is not available for this group.

Rational character table

See the $138 \times 138$ rational character table (warning: may be slow to load).