Abstract group 399300.d: $C_{11}\wr C_3:C_{10}^2$

• Label: 399300.d
• Order: \( 2^{2} \cdot 3 \cdot 5^{2} \cdot 11^{3} \)
• Exponent: \( 2 \cdot 3 \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 5^{2} \)
• $\card{Z(G)}$: \( 2 \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3 \cdot 5^{3} \cdot 11^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 5^{2} \)
• Perm deg.: $40$
• Trans deg.: $330$
• Rank: $2$

Group information

Description:	$C_{11}\wr C_3:C_{10}^2$
Order:	\(399300\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \cdot 11^{3} \)
Exponent:	\(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \)
Automorphism group:	$C_{11}^3.C_{15}.C_{10}^2.C_2^4$, of order \(31944000\)\(\medspace = 2^{6} \cdot 3 \cdot 5^{3} \cdot 11^{3} \)
Composition factors:	$C_2$ x 2, $C_3$, $C_5$ x 2, $C_{11}$ x 3
Derived length:	$3$

This group is nonabelian, monomial (hence solvable), and an A-group.

Group statistics

Order	1	2	3	5	6	10	11	15	22	30	33	55	66	110	165	330
Elements	1	67	242	26624	242	186608	1330	54208	9250	54208	2420	5320	2420	37000	9680	9680	399300
Conjugacy classes	1	3	1	24	1	72	57	24	105	24	2	228	2	420	8	8	980
Divisions	1	3	1	6	1	18	29	6	53	6	1	29	1	53	1	1	210
Autjugacy classes	1	2	1	5	1	10	7	5	10	5	1	7	1	10	1	1	68

Dimension	1	2	4	5	8	10	15	20	30	40	60	80	120	240
Irr. complex chars.	100	50	0	40	0	20	440	0	330	0	0	0	0	0	980
Irr. rational chars.	4	2	24	0	12	4	0	2	46	4	32	2	46	32	210

Minimal presentations

Permutation degree:	$40$
Transitive degree:	$330$
Rank:	$2$
Inequivalent generating pairs:	$864$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	15	not computed	120
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d \mid a^{10}=b^{33}=c^{11}=d^{110}=[c,d]=1, b^{a}=b^{26}c^{2}d^{80}, c^{a}= \!\cdots\! \rangle}$
Permutation group:	Degree $40$ $\langle(34,35,36,38,37), (1,2,5,16,15,6,19,13,4,12,17)(3,7,18,33,10,11,22,32,14,29,30) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 10 & 2 & 8 & 3 \\ 9 & 5 & 6 & 0 \\ 9 & 5 & 7 & 2 \\ 1 & 9 & 1 & 3 \end{array}\right), \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 5 & 9 & 4 & 0 \\ 2 & 2 & 10 & 0 \\ 2 & 2 & 6 & 4 \end{array}\right), \left(\begin{array}{rrrr} 2 & 8 & 5 & 0 \\ 1 & 5 & 8 & 8 \\ 3 & 2 & 5 & 7 \\ 6 & 3 & 8 & 10 \end{array}\right), \left(\begin{array}{rrrr} 3 & 2 & 9 & 8 \\ 10 & 6 & 4 & 9 \\ 6 & 4 & 1 & 9 \\ 0 & 6 & 1 & 4 \end{array}\right), \left(\begin{array}{rrrr} 9 & 0 & 0 & 0 \\ 7 & 5 & 10 & 0 \\ 6 & 6 & 5 & 0 \\ 3 & 6 & 4 & 1 \end{array}\right), \left(\begin{array}{rrrr} 3 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 3 \end{array}\right), \left(\begin{array}{rrrr} 8 & 0 & 0 & 0 \\ 0 & 8 & 0 & 0 \\ 0 & 0 & 8 & 0 \\ 0 & 0 & 0 & 8 \end{array}\right), \left(\begin{array}{rrrr} 0 & 10 & 3 & 5 \\ 10 & 8 & 3 & 3 \\ 6 & 7 & 4 & 1 \\ 0 & 6 & 1 & 1 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{11})$
Direct product:	$C_2$ $\, \times\, $ $C_5$ $\, \times\, $ $(C_{11}^3:(C_5\times S_3))$
Semidirect product:	$(C_{11}^3:C_5^2)$ $\,\rtimes\,$ $D_6$	$(C_{11}^3:D_6)$ $\,\rtimes\,$ $C_5^2$	$C_{11}^3$ $\,\rtimes\,$ $(D_6\times C_5^2)$	$(C_{11}^3:C_{30})$ $\,\rtimes\,$ $C_{10}$	all 36
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

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

Subgroups

There are 441704 subgroups in 1792 conjugacy classes, 78 normal (33 characteristic).

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

Special subgroups

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

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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Classes of subgroups up to conjugation

Order 399300: $C_{11}\wr C_3:C_{10}^2$

Order 199650: $C_{11}^3:(S_3\times C_5^2)$ x 2, $C_5\times C_{11}^3:C_{30}$

Order 133100: $C_{11}^3:C_{10}^2$

Order 99825: $C_5\times C_{11}^3:C_{15}$

Order 79860: $C_{11}^3:(S_3\times C_{10})$ x 5, $C_{10}\times C_{11}\wr S_3$

Order 66550: $C_5\times C_{11}^3:C_{10}$ x 2, $C_5\times C_{11}^3:C_{10}$

Order 39930: $C_{11}^3:(C_5\times S_3)$ x 10, $C_{11}^3:C_{30}$ x 5, $C_5\times C_{11}\wr S_3$ x 2, $C_{11}^2:C_{330}$

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

Order 33275: $C_{11}^3:C_5^2$

Order 26620: $C_2\times C_{11}^3:C_{10}$ x 5, $C_{55}.C_{22}^2$

Order 19965: $C_{11}^3:C_{15}$ x 5, $C_{11}^2:C_{165}$

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

Order 15972: $C_{11}^3:D_6$

Order 13310: $C_{11}^3:C_{10}$ x 10, $C_{11}^3:C_{10}$ x 5, $C_{11}^2:C_{110}$ x 2, $C_{11}^2\times C_{110}$

Order 12100: $C_{110}:F_{11}$ x 12, $C_{11}^2:C_{10}^2$

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

Order 7986: $C_{11}\wr S_3$ x 2, $C_{11}^2:C_{66}$

Order 7260: $C_{11}^2:(S_3\times C_{10})$ x 5, $C_{10}\times C_{11}^2:S_3$

Order 6655: $C_{11}^3:C_5$ x 5, $C_{11}^2\times C_{55}$

Order 6050: $C_5\times C_{11}^2:C_{10}$ x 31, $C_{55}:F_{11}$ x 24

Order 5324: $D_{22}\times C_{11}^2$

Order 3993: $C_{11}\wr C_3$

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

Order 3300: $C_{330}:C_{10}$

Order 3025: $C_{11}^2:C_5^2$ x 29

Order 2662: $D_{11}\times C_{11}^2$ x 2, $C_{11}^2\times C_{22}$

Order 2420: $C_{22}:F_{11}$ x 60, $D_{11}\times C_{110}$ x 12, $C_{22}^2:C_5$ x 5, $C_{22}\times C_{110}$

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

Order 1650: $C_{165}:C_{10}$ x 2, $C_{110}:C_{15}$

Order 1452: $C_{11}^2:D_6$

Order 1331: $C_{11}^3$

Order 1210: $C_{11}^2:C_{10}$ x 155, $C_{11}:F_{11}$ x 120, $C_{11}\times C_{110}$ x 31, $D_{11}\times C_{55}$ x 24

Order 1100: $C_{110}:C_{10}$ x 12, $C_{10}\times F_{11}$

Order 825: $C_{165}:C_5$

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

Order 660: $C_{66}:C_{10}$ x 5, $S_3\times C_{110}$

Order 605: $C_{11}^2:C_5$ x 145, $C_{11}\times C_{55}$ x 29

Order 550: $C_{110}:C_5$ x 53, $C_5\times F_{11}$ x 2

Order 484: $C_{11}\times D_{22}$ x 12, $C_{22}^2$

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

Order 330: $C_{33}:C_{10}$ x 10, $C_{11}:C_{30}$ x 5, $S_3\times C_{55}$ x 2, $C_{330}$

Order 300: $D_6\times C_5^2$

Order 275: $C_{55}:C_5$ x 29

Order 242: $C_{11}\times C_{22}$ x 31, $C_{11}\times D_{11}$ x 24

Order 220: $C_{22}:C_{10}$ x 60, $C_2\times C_{110}$ x 12, $C_2\times F_{11}$ x 5, $C_5\times D_{22}$

Order 165: $C_{11}:C_{15}$ x 5, $C_{165}$

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

Order 132: $S_3\times C_{22}$

Order 121: $C_{11}^2$ x 29

Order 110: $C_{11}:C_{10}$ x 265, $C_{110}$ x 53, $F_{11}$ x 10, $C_5\times D_{11}$ x 2

Order 100: $C_{10}^2$

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

Order 66: $S_3\times C_{11}$ x 2, $C_{66}$

Order 60: $S_3\times C_{10}$ x 6

Order 55: $C_{11}:C_5$ x 145, $C_{55}$ x 29

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

Order 44: $C_2\times C_{22}$ x 12, $D_{22}$

Order 33: $C_{33}$

Order 30: $C_5\times S_3$ x 12, $C_{30}$ x 6

Order 25: $C_5^2$

Order 22: $C_{22}$ x 53, $D_{11}$ x 2

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

Order 15: $C_{15}$ x 6

Order 12: $D_6$

Order 11: $C_{11}$ x 29

Order 10: $C_{10}$ x 18

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

Order 5: $C_5$ x 6

Order 4: $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 399300: $C_{11}\wr C_3:C_{10}^2$

Order 199650: $C_5\times C_{11}^3:C_{30}$, $C_{11}^3:(S_3\times C_5^2)$

Order 133100: $C_{11}^3:C_{10}^2$

Order 99825: $C_5\times C_{11}^3:C_{15}$

Order 79860: $C_{11}^3:(S_3\times C_{10})$, $C_{10}\times C_{11}\wr S_3$

Order 66550: $C_5\times C_{11}^3:C_{10}$, $C_5\times C_{11}^3:C_{10}$

Order 39930: $C_{11}^3:(C_5\times S_3)$, $C_{11}^3:C_{30}$, $C_5\times C_{11}\wr S_3$, $C_{11}^2:C_{330}$

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

Order 33275: $C_{11}^3:C_5^2$

Order 26620: $C_2\times C_{11}^3:C_{10}$, $C_{55}.C_{22}^2$

Order 19965: $C_{11}^3:C_{15}$, $C_{11}^2:C_{165}$

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

Order 15972: $C_{11}^3:D_6$

Order 13310: $C_{11}^3:C_{10}$, $C_{11}^3:C_{10}$, $C_{11}^2:C_{110}$, $C_{11}^2\times C_{110}$

Order 12100: $C_{110}:F_{11}$ x 3, $C_{11}^2:C_{10}^2$

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

Order 7986: $C_{11}^2:C_{66}$, $C_{11}\wr S_3$

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

Order 6655: $C_{11}^2\times C_{55}$, $C_{11}^3:C_5$

Order 6050: $C_5\times C_{11}^2:C_{10}$ x 8, $C_{55}:F_{11}$ x 3

Order 5324: $D_{22}\times C_{11}^2$

Order 3993: $C_{11}\wr C_3$

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

Order 3300: $C_{330}:C_{10}$

Order 3025: $C_{11}^2:C_5^2$ x 7

Order 2662: $C_{11}^2\times C_{22}$, $D_{11}\times C_{11}^2$

Order 2420: $C_{22}:F_{11}$ x 3, $D_{11}\times C_{110}$ x 3, $C_{22}\times C_{110}$, $C_{22}^2:C_5$

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

Order 1650: $C_{110}:C_{15}$, $C_{165}:C_{10}$

Order 1452: $C_{11}^2:D_6$

Order 1331: $C_{11}^3$

Order 1210: $C_{11}\times C_{110}$ x 8, $C_{11}^2:C_{10}$ x 8, $D_{11}\times C_{55}$ x 3, $C_{11}:F_{11}$ x 3

Order 1100: $C_{110}:C_{10}$ x 3, $C_{10}\times F_{11}$

Order 825: $C_{165}:C_5$

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

Order 660: $S_3\times C_{110}$, $C_{66}:C_{10}$

Order 605: $C_{11}\times C_{55}$ x 7, $C_{11}^2:C_5$ x 7

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

Order 484: $C_{11}\times D_{22}$ x 3, $C_{22}^2$

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

Order 330: $S_3\times C_{55}$, $C_{11}:C_{30}$, $C_{33}:C_{10}$, $C_{330}$

Order 300: $D_6\times C_5^2$

Order 275: $C_{55}:C_5$ x 7

Order 242: $C_{11}\times C_{22}$ x 8, $C_{11}\times D_{11}$ x 3

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

Order 165: $C_{165}$, $C_{11}:C_{15}$

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

Order 132: $S_3\times C_{22}$

Order 121: $C_{11}^2$ x 7

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

Order 100: $C_{10}^2$

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

Order 66: $C_{66}$, $S_3\times C_{11}$

Order 60: $S_3\times C_{10}$ x 2

Order 55: $C_{55}$ x 7, $C_{11}:C_5$ x 7

Order 50: $C_5\times C_{10}$ x 2

Order 44: $C_2\times C_{22}$ x 3, $D_{22}$

Order 33: $C_{33}$

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

Order 25: $C_5^2$

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

Order 20: $C_2\times C_{10}$ x 2

Order 15: $C_{15}$ x 2

Order 12: $D_6$

Order 11: $C_{11}$ x 7

Order 10: $C_{10}$ x 4

Order 6: $C_6$, $S_3$

Order 5: $C_5$ x 2

Order 4: $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 399300: $C_{11}\wr C_3:C_{10}^2$ ($C_1$)

Order 199650: $C_{11}^3:(S_3\times C_5^2)$ ($C_2$) x 2, $C_5\times C_{11}^3:C_{30}$ ($C_2$)

Order 99825: $C_5\times C_{11}^3:C_{15}$ ($C_2^2$)

Order 79860: $C_{11}^3:(S_3\times C_{10})$ ($C_5$) x 5, $C_{10}\times C_{11}\wr S_3$ ($C_5$)

Order 66550: $C_5\times C_{11}^3:C_{10}$ ($S_3$)

Order 39930: $C_{11}^3:(C_5\times S_3)$ ($C_{10}$) x 10, $C_{11}^3:C_{30}$ ($C_{10}$) x 5, $C_5\times C_{11}\wr S_3$ ($C_{10}$) x 2, $C_{11}^2:C_{330}$ ($C_{10}$)

Order 33275: $C_{11}^3:C_5^2$ ($D_6$)

Order 19965: $C_{11}^3:C_{15}$ ($C_2\times C_{10}$) x 5, $C_{11}^2:C_{165}$ ($C_2\times C_{10}$)

Order 15972: $C_{11}^3:D_6$ ($C_5^2$)

Order 13310: $C_{11}^3:C_{10}$ ($C_5\times S_3$) x 5, $C_{11}^2\times C_{110}$ ($C_5\times S_3$)

Order 7986: $C_{11}\wr S_3$ ($C_5\times C_{10}$) x 2, $C_{11}^2:C_{66}$ ($C_5\times C_{10}$)

Order 7260: $C_{10}\times C_{11}^2:S_3$ ($C_{11}:C_5$)

Order 6655: $C_{11}^3:C_5$ ($S_3\times C_{10}$) x 5, $C_{11}^2\times C_{55}$ ($S_3\times C_{10}$)

Order 3993: $C_{11}\wr C_3$ ($C_{10}^2$)

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

Order 2662: $C_{11}^2\times C_{22}$ ($S_3\times C_5^2$)

Order 1815: $C_{11}^2:C_{15}$ ($C_{22}:C_{10}$)

Order 1452: $C_{11}^2:D_6$ ($C_{55}:C_5$)

Order 1331: $C_{11}^3$ ($D_6\times C_5^2$)

Order 1210: $C_{11}\times C_{110}$ ($C_{33}:C_{10}$)

Order 726: $C_{11}^2:S_3$ ($C_{110}:C_5$) x 2, $C_{11}^2:C_6$ ($C_{110}:C_5$)

Order 605: $C_{11}\times C_{55}$ ($C_{66}:C_{10}$)

Order 363: $C_{11}^2:C_3$ ($C_{110}:C_{10}$)

Order 242: $C_{11}\times C_{22}$ ($C_{165}:C_{10}$)

Order 121: $C_{11}^2$ ($C_{330}:C_{10}$)

Order 110: $C_{110}$ ($C_{11}^2:(C_5\times S_3)$)

Order 55: $C_{55}$ ($C_{11}^2:(S_3\times C_{10})$)

Order 22: $C_{22}$ ($C_{11}^2:(S_3\times C_5^2)$)

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

Order 10: $C_{10}$ ($C_{11}^3:(C_5\times S_3)$)

Order 5: $C_5$ ($C_{11}^3:(S_3\times C_{10})$)

Order 2: $C_2$ ($C_{11}^3:(S_3\times C_5^2)$)

Order 1: $C_1$ ($C_{11}\wr C_3:C_{10}^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 399300: $C_{11}\wr C_3:C_{10}^2$ ($C_1$)

Order 199650: $C_5\times C_{11}^3:C_{30}$ ($C_2$), $C_{11}^3:(S_3\times C_5^2)$ ($C_2$)

Order 99825: $C_5\times C_{11}^3:C_{15}$ ($C_2^2$)

Order 79860: $C_{11}^3:(S_3\times C_{10})$ ($C_5$), $C_{10}\times C_{11}\wr S_3$ ($C_5$)

Order 66550: $C_5\times C_{11}^3:C_{10}$ ($S_3$)

Order 39930: $C_{11}^3:(C_5\times S_3)$ ($C_{10}$), $C_{11}^3:C_{30}$ ($C_{10}$), $C_5\times C_{11}\wr S_3$ ($C_{10}$), $C_{11}^2:C_{330}$ ($C_{10}$)

Order 33275: $C_{11}^3:C_5^2$ ($D_6$)

Order 19965: $C_{11}^3:C_{15}$ ($C_2\times C_{10}$), $C_{11}^2:C_{165}$ ($C_2\times C_{10}$)

Order 15972: $C_{11}^3:D_6$ ($C_5^2$)

Order 13310: $C_{11}^3:C_{10}$ ($C_5\times S_3$), $C_{11}^2\times C_{110}$ ($C_5\times S_3$)

Order 7986: $C_{11}^2:C_{66}$ ($C_5\times C_{10}$), $C_{11}\wr S_3$ ($C_5\times C_{10}$)

Order 7260: $C_{10}\times C_{11}^2:S_3$ ($C_{11}:C_5$)

Order 6655: $C_{11}^2\times C_{55}$ ($S_3\times C_{10}$), $C_{11}^3:C_5$ ($S_3\times C_{10}$)

Order 3993: $C_{11}\wr C_3$ ($C_{10}^2$)

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

Order 2662: $C_{11}^2\times C_{22}$ ($S_3\times C_5^2$)

Order 1815: $C_{11}^2:C_{15}$ ($C_{22}:C_{10}$)

Order 1452: $C_{11}^2:D_6$ ($C_{55}:C_5$)

Order 1331: $C_{11}^3$ ($D_6\times C_5^2$)

Order 1210: $C_{11}\times C_{110}$ ($C_{33}:C_{10}$)

Order 726: $C_{11}^2:C_6$ ($C_{110}:C_5$), $C_{11}^2:S_3$ ($C_{110}:C_5$)

Order 605: $C_{11}\times C_{55}$ ($C_{66}:C_{10}$)

Order 363: $C_{11}^2:C_3$ ($C_{110}:C_{10}$)

Order 242: $C_{11}\times C_{22}$ ($C_{165}:C_{10}$)

Order 121: $C_{11}^2$ ($C_{330}:C_{10}$)

Order 110: $C_{110}$ ($C_{11}^2:(C_5\times S_3)$)

Order 55: $C_{55}$ ($C_{11}^2:(S_3\times C_{10})$)

Order 22: $C_{22}$ ($C_{11}^2:(S_3\times C_5^2)$)

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

Order 10: $C_{10}$ ($C_{11}^3:(C_5\times S_3)$)

Order 5: $C_5$ ($C_{11}^3:(S_3\times C_{10})$)

Order 2: $C_2$ ($C_{11}^3:(S_3\times C_5^2)$)

Order 1: $C_1$ ($C_{11}\wr C_3:C_{10}^2$)

Series

Derived series

$C_{11}\wr C_3:C_{10}^2$

$\rhd$

$C_{11}\wr C_3:C_{10}^2$

$\rhd$

$C_{11}\wr C_3$

$\rhd$

$C_{11}\wr C_3$

$\rhd$

$C_{11}^2$

$\rhd$

$C_{11}^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_{11}\wr C_3:C_{10}^2$

$\rhd$

$C_{11}\wr C_3:C_{10}^2$

$\rhd$

$C_{11}^3:(S_3\times C_5^2)$

$\rhd$

$C_{11}^3:(S_3\times C_5^2)$

$\rhd$

$C_5\times C_{11}^3:C_{15}$

$\rhd$

$C_5\times C_{11}^3:C_{15}$

$\rhd$

$C_{11}^3:C_{15}$

$\rhd$

$C_{11}^3:C_{15}$

$\rhd$

$C_{11}\wr C_3$

$\rhd$

$C_{11}\wr C_3$

$\rhd$

$C_{11}^3$

$\rhd$

$C_{11}^3$

$\rhd$

$C_{11}^2$

$\rhd$

$C_{11}^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_{11}\wr C_3:C_{10}^2$

$\rhd$

$C_{11}\wr C_3:C_{10}^2$

$\rhd$

$C_{11}\wr C_3$

$\rhd$

$C_{11}\wr C_3$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_{10}$

$\lhd$

$C_{10}$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

The $980 \times 980$ character table is not available for this group.

Rational character table

See the $210 \times 210$ rational character table (warning: may be slow to load).