Abstract group 48400.j: $C_{55}:(Q_8\times F_{11})$

• Label: 48400.j
• Order: \( 2^{4} \cdot 5^{2} \cdot 11^{2} \)
• Exponent: \( 2^{2} \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 5^{2} \)
• $\card{Z(G)}$: \( 2 \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 5^{3} \cdot 11^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 5^{2} \)
• Perm deg.: $35$
• Trans deg.: $440$
• Rank: $3$

Group information

Description:	$C_{55}:(Q_8\times F_{11})$
Order:	\(48400\)\(\medspace = 2^{4} \cdot 5^{2} \cdot 11^{2} \)
Exponent:	\(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \)
Automorphism group:	$C_{11}^2.C_{10}^2.C_{10}.C_2^6$, of order \(7744000\)\(\medspace = 2^{9} \cdot 5^{3} \cdot 11^{2} \)
Composition factors:	$C_2$ x 4, $C_5$ x 2, $C_{11}$ x 2
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.

Group statistics

Order	1	2	4	5	10	11	20	22	44	55	110	220
Elements	1	243	332	2424	8232	120	30368	120	1120	480	480	4480	48400
Conjugacy classes	1	3	6	24	72	7	144	7	16	28	28	64	400
Divisions	1	3	6	6	18	7	36	7	11	7	7	11	120
Autjugacy classes	1	2	3	5	10	2	15	2	3	2	2	3	50

Dimension	1	2	4	8	10	20	40	80	160
Irr. complex chars.	200	50	0	0	40	110	0	0	0	400
Irr. rational chars.	8	2	48	12	8	12	13	12	5	120

Minimal presentations

Permutation degree:	$35$
Transitive degree:	$440$
Rank:	$3$
Inequivalent generating triples:	$4499712$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	20	40	160
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c \mid a^{10}=c^{220}=1, b^{22}=c^{22}, b^{a}=b^{13}c^{188}, c^{a}=c^{101}, c^{b}=c^{131} \rangle$
Permutation group:	Degree $35$ $\langle(31,32,33,34,35), (2,7,3,5,15)(4,8,9,13,12)(6,11,22,19,18)(10,21,20,17,16) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 3 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 3 \end{array}\right), \left(\begin{array}{rrrr} 0 & 9 & 2 & 2 \\ 3 & 8 & 0 & 4 \\ 0 & 1 & 10 & 0 \\ 4 & 5 & 6 & 4 \end{array}\right), \left(\begin{array}{rrrr} 0 & 4 & 10 & 2 \\ 2 & 10 & 4 & 10 \\ 6 & 2 & 8 & 7 \\ 8 & 6 & 9 & 7 \end{array}\right), \left(\begin{array}{rrrr} 3 & 2 & 9 & 8 \\ 5 & 1 & 0 & 9 \\ 7 & 5 & 4 & 9 \\ 3 & 7 & 6 & 2 \end{array}\right), \left(\begin{array}{rrrr} 8 & 7 & 4 & 6 \\ 5 & 5 & 1 & 4 \\ 0 & 4 & 4 & 4 \\ 6 & 0 & 6 & 1 \end{array}\right), \left(\begin{array}{rrrr} 8 & 0 & 0 & 0 \\ 0 & 8 & 0 & 0 \\ 0 & 0 & 8 & 0 \\ 0 & 0 & 0 & 8 \end{array}\right), \left(\begin{array}{rrrr} 4 & 10 & 1 & 7 \\ 1 & 3 & 5 & 1 \\ 6 & 7 & 10 & 1 \\ 9 & 6 & 10 & 9 \end{array}\right), \left(\begin{array}{rrrr} 8 & 2 & 9 & 5 \\ 10 & 5 & 10 & 8 \\ 8 & 5 & 4 & 0 \\ 3 & 5 & 7 & 5 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{11})$
Direct product:	$C_5$ $\, \times\, $ $(C_{11}:(Q_8\times F_{11}))$
Semidirect product:	$(C_{55}:F_{11})$ $\,\rtimes\,$ $Q_8$	$(C_{55}:Q_8)$ $\,\rtimes\,$ $F_{11}$	$C_{55}$ $\,\rtimes\,$ $(Q_8\times F_{11})$	$(C_{220}.D_{22})$ $\,\rtimes\,$ $C_5$	all 22
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{20}$ . $(D_{11}:F_{11})$	$C_{10}$ . $(D_{22}:F_{11})$	$(C_{220}:F_{11})$ . $C_2$	$(C_{44}:F_{11})$ . $C_{10}$	all 38

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:	$C_{2} \times C_{10}$
Commutator length:	$1$

Subgroups

There are 26048 subgroups in 928 conjugacy classes, 182 normal (27 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{10}$	$G/Z \simeq$ $D_{22}:F_{11}$
Commutator:	$G' \simeq$ $C_{11}\times C_{22}$	$G/G' \simeq$ $C_2\times C_{10}^2$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_5\times D_{22}:F_{11}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{11}\times C_{220}$	$G/\operatorname{Fit} \simeq$ $C_2\times C_{10}$
Radical:	$R \simeq$ $C_{55}:(Q_8\times F_{11})$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{11}\times C_{110}$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_{10}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times Q_8$
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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Classes of subgroups up to conjugation

Order 48400: $C_{55}:(Q_8\times F_{11})$

Order 24200: $C_{11}:(C_{20}\times F_{11})$ x 2, $C_{220}.F_{11}$ x 2, $C_{11}^2:(Q_8\times C_5^2)$ x 2, $C_{220}:F_{11}$

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

Order 9680: $C_{11}:(Q_8\times F_{11})$ x 5, $C_{220}.D_{22}$

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

Order 4840: $C_{44}.F_{11}$ x 10, $C_{11}^2:(C_5\times Q_8)$ x 10, $C_{11}:(C_4\times F_{11})$ x 10, $C_{44}:F_{11}$ x 5, $C_{44}.C_{110}$ x 2, $C_{110}.D_{22}$ x 2, $C_{110}.D_{22}$ x 2, $C_{220}:D_{11}$

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

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

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

Order 2200: $C_{20}\times F_{11}$ x 11, $C_{220}.C_{10}$ x 6, $C_{220}.C_{10}$ x 2

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

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

Order 1100: $C_{220}:C_5$ x 11, $C_{55}:C_{20}$ x 11, $C_{10}\times F_{11}$ x 7

Order 968: $C_{11}^2:Q_8$ x 2, $C_{11}^2:Q_8$ x 2, $C_{22}.D_{22}$ x 2, $C_{44}:D_{11}$

Order 880: $Q_8\times F_{11}$ x 10, $C_{20}.D_{22}$ x 2

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

Order 550: $C_5\times F_{11}$ x 14, $C_{110}:C_5$ x 7

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

Order 440: $C_4\times F_{11}$ x 55, $C_{44}.C_{10}$ x 30, $C_{20}\times D_{11}$ x 11, $C_{44}.C_{10}$ x 10, $C_{55}:Q_8$ x 6, $Q_8\times C_{55}$ x 2

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

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

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

Order 220: $C_{11}:C_{20}$ x 55, $C_{11}:C_{20}$ x 55, $C_2\times F_{11}$ x 35, $C_{220}$ x 11, $C_{11}:C_{20}$ x 11, $C_5\times D_{22}$ x 7

Order 200: $Q_8\times C_5^2$ x 4, $C_{10}\times C_{20}$ x 3

Order 176: $Q_8\times D_{11}$ x 2

Order 121: $C_{11}^2$

Order 110: $F_{11}$ x 70, $C_{11}:C_{10}$ x 35, $C_5\times D_{11}$ x 14, $C_{110}$ x 7

Order 100: $C_5\times C_{20}$ x 6, $C_{10}^2$

Order 88: $C_4\times D_{11}$ x 11, $C_{11}:Q_8$ x 6, $Q_8\times C_{11}$ x 2

Order 80: $Q_8\times C_{10}$ x 6

Order 55: $C_{11}:C_5$ x 35, $C_{55}$ x 7

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

Order 44: $C_{44}$ x 11, $C_{11}:C_4$ x 11, $D_{22}$ x 7

Order 40: $C_5\times Q_8$ x 24, $C_2\times C_{20}$ x 18

Order 25: $C_5^2$

Order 22: $D_{11}$ x 14, $C_{22}$ x 7

Order 20: $C_{20}$ x 36, $C_2\times C_{10}$ x 6

Order 16: $C_2\times Q_8$

Order 11: $C_{11}$ x 7

Order 10: $C_{10}$ x 18

Order 8: $Q_8$ x 4, $C_2\times C_4$ x 3

Order 5: $C_5$ x 6

Order 4: $C_4$ x 6, $C_2^2$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 48400: $C_{55}:(Q_8\times F_{11})$

Order 24200: $C_{11}:(C_{20}\times F_{11})$, $C_{220}.F_{11}$, $C_{11}^2:(Q_8\times C_5^2)$, $C_{220}:F_{11}$

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

Order 9680: $C_{220}.D_{22}$, $C_{11}:(Q_8\times F_{11})$

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

Order 4840: $C_{44}.C_{110}$, $C_{110}.D_{22}$, $C_{44}:F_{11}$, $C_{220}:D_{11}$, $C_{110}.D_{22}$, $C_{44}.F_{11}$, $C_{11}^2:(C_5\times Q_8)$, $C_{11}:(C_4\times F_{11})$

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

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

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

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

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

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

Order 1100: $C_{220}:C_5$ x 3, $C_{55}:C_{20}$ x 3, $C_{10}\times F_{11}$ x 2

Order 968: $C_{44}:D_{11}$, $C_{11}^2:Q_8$, $C_{11}^2:Q_8$, $C_{22}.D_{22}$

Order 880: $C_{20}.D_{22}$, $Q_8\times F_{11}$

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

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

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

Order 440: $C_4\times F_{11}$ x 3, $C_{20}\times D_{11}$ x 3, $C_{44}.C_{10}$ x 2, $C_{55}:Q_8$ x 2, $Q_8\times C_{55}$, $C_{44}.C_{10}$

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

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

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

Order 220: $C_{220}$ x 3, $C_{11}:C_{20}$ x 3, $C_{11}:C_{20}$ x 3, $C_{11}:C_{20}$ x 3, $C_2\times F_{11}$ x 2, $C_5\times D_{22}$ x 2

Order 200: $Q_8\times C_5^2$ x 2, $C_{10}\times C_{20}$ x 2

Order 176: $Q_8\times D_{11}$

Order 121: $C_{11}^2$

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

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

Order 88: $C_4\times D_{11}$ x 3, $C_{11}:Q_8$ x 2, $Q_8\times C_{11}$

Order 80: $Q_8\times C_{10}$ x 2

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

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

Order 44: $C_{44}$ x 3, $C_{11}:C_4$ x 3, $D_{22}$ x 2

Order 40: $C_2\times C_{20}$ x 4, $C_5\times Q_8$ x 4

Order 25: $C_5^2$

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

Order 20: $C_{20}$ x 6, $C_2\times C_{10}$ x 2

Order 16: $C_2\times Q_8$

Order 11: $C_{11}$ x 2

Order 10: $C_{10}$ x 4

Order 8: $Q_8$ x 2, $C_2\times C_4$ x 2

Order 5: $C_5$ x 2

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 48400: $C_{55}:(Q_8\times F_{11})$ ($C_1$)

Order 24200: $C_{11}:(C_{20}\times F_{11})$ ($C_2$) x 2, $C_{220}.F_{11}$ ($C_2$) x 2, $C_{11}^2:(Q_8\times C_5^2)$ ($C_2$) x 2, $C_{220}:F_{11}$ ($C_2$)

Order 12100: $C_{110}.F_{11}$ ($C_2^2$) x 4, $C_5\times C_{11}^2:C_{20}$ ($C_2^2$), $C_{110}.F_{11}$ ($C_2^2$), $C_{110}:F_{11}$ ($C_2^2$)

Order 9680: $C_{11}:(Q_8\times F_{11})$ ($C_5$) x 5, $C_{220}.D_{22}$ ($C_5$)

Order 6050: $C_{55}:F_{11}$ ($Q_8$) x 2, $C_5\times C_{11}^2:C_{10}$ ($C_2^3$)

Order 4840: $C_{44}.F_{11}$ ($C_{10}$) x 10, $C_{11}^2:(C_5\times Q_8)$ ($C_{10}$) x 10, $C_{11}:(C_4\times F_{11})$ ($C_{10}$) x 10, $C_{44}:F_{11}$ ($C_{10}$) x 5, $C_{44}.C_{110}$ ($C_{10}$) x 2, $C_{110}.D_{22}$ ($C_{10}$) x 2, $C_{110}.D_{22}$ ($C_{10}$) x 2, $C_{220}:D_{11}$ ($C_{10}$)

Order 3025: $C_{11}^2:C_5^2$ ($C_2\times Q_8$)

Order 2420: $C_{11}^2:C_{20}$ ($C_2\times C_{10}$) x 20, $C_{11}^2:C_{20}$ ($C_2\times C_{10}$) x 5, $C_{22}:F_{11}$ ($C_2\times C_{10}$) x 5, $C_{11}^2:C_{20}$ ($C_2\times C_{10}$) x 5, $C_{11}:C_{220}$ ($C_2\times C_{10}$) x 4, $C_{11}^2:C_{20}$ ($C_2\times C_{10}$), $C_{11}\times C_{220}$ ($C_2\times C_{10}$), $C_{110}:D_{11}$ ($C_2\times C_{10}$)

Order 1936: $C_{44}.D_{22}$ ($C_5^2$)

Order 1210: $C_{11}:F_{11}$ ($C_5\times Q_8$) x 10, $C_{11}^2:C_{10}$ ($C_2^2\times C_{10}$) x 5, $C_{55}:D_{11}$ ($C_5\times Q_8$) x 2, $C_{11}\times C_{110}$ ($C_2^2\times C_{10}$)

Order 968: $C_{11}^2:Q_8$ ($C_5\times C_{10}$) x 2, $C_{11}^2:Q_8$ ($C_5\times C_{10}$) x 2, $C_{22}.D_{22}$ ($C_5\times C_{10}$) x 2, $C_{44}:D_{11}$ ($C_5\times C_{10}$)

Order 605: $C_{11}^2:C_5$ ($Q_8\times C_{10}$) x 5, $C_{11}\times C_{55}$ ($Q_8\times C_{10}$)

Order 484: $C_{11}:C_{44}$ ($C_{10}^2$) x 4, $C_{11}\times C_{44}$ ($C_{10}^2$), $C_{11}^2:C_4$ ($C_{10}^2$), $C_{11}:D_{22}$ ($C_{10}^2$)

Order 440: $C_{55}:Q_8$ ($F_{11}$) x 2

Order 242: $C_{11}:D_{11}$ ($Q_8\times C_5^2$) x 2, $C_{11}\times C_{22}$ ($C_2\times C_{10}^2$)

Order 220: $C_{11}:C_{20}$ ($C_2\times F_{11}$) x 4, $C_{220}$ ($C_2\times F_{11}$) x 2

Order 121: $C_{11}^2$ ($C_4.C_{10}^2$)

Order 110: $C_{110}$ ($C_2^2\times F_{11}$) x 2

Order 88: $C_{11}:Q_8$ ($C_5\times F_{11}$) x 2

Order 55: $C_{55}$ ($Q_8\times F_{11}$) x 2

Order 44: $C_{11}:C_4$ ($C_{10}\times F_{11}$) x 4, $C_{44}$ ($C_{10}\times F_{11}$) x 2

Order 22: $C_{22}$ ($C_{22}:C_{10}^2$) x 2

Order 20: $C_{20}$ ($D_{11}:F_{11}$)

Order 11: $C_{11}$ ($C_{44}.C_{10}^2$) x 2

Order 10: $C_{10}$ ($D_{22}:F_{11}$)

Order 5: $C_5$ ($C_{11}:(Q_8\times F_{11})$)

Order 4: $C_4$ ($C_{11}^2:C_{10}^2$)

Order 2: $C_2$ ($C_5\times D_{22}:F_{11}$)

Order 1: $C_1$ ($C_{55}:(Q_8\times F_{11})$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 48400: $C_{55}:(Q_8\times F_{11})$ ($C_1$)

Order 24200: $C_{11}:(C_{20}\times F_{11})$ ($C_2$), $C_{220}.F_{11}$ ($C_2$), $C_{11}^2:(Q_8\times C_5^2)$ ($C_2$), $C_{220}:F_{11}$ ($C_2$)

Order 12100: $C_{110}.F_{11}$ ($C_2^2$), $C_5\times C_{11}^2:C_{20}$ ($C_2^2$), $C_{110}.F_{11}$ ($C_2^2$), $C_{110}:F_{11}$ ($C_2^2$)

Order 9680: $C_{220}.D_{22}$ ($C_5$), $C_{11}:(Q_8\times F_{11})$ ($C_5$)

Order 6050: $C_5\times C_{11}^2:C_{10}$ ($C_2^3$), $C_{55}:F_{11}$ ($Q_8$)

Order 4840: $C_{44}.C_{110}$ ($C_{10}$), $C_{110}.D_{22}$ ($C_{10}$), $C_{44}:F_{11}$ ($C_{10}$), $C_{220}:D_{11}$ ($C_{10}$), $C_{110}.D_{22}$ ($C_{10}$), $C_{44}.F_{11}$ ($C_{10}$), $C_{11}^2:(C_5\times Q_8)$ ($C_{10}$), $C_{11}:(C_4\times F_{11})$ ($C_{10}$)

Order 3025: $C_{11}^2:C_5^2$ ($C_2\times Q_8$)

Order 2420: $C_{11}:C_{220}$ ($C_2\times C_{10}$), $C_{11}^2:C_{20}$ ($C_2\times C_{10}$), $C_{11}^2:C_{20}$ ($C_2\times C_{10}$), $C_{11}\times C_{220}$ ($C_2\times C_{10}$), $C_{110}:D_{11}$ ($C_2\times C_{10}$), $C_{22}:F_{11}$ ($C_2\times C_{10}$), $C_{11}^2:C_{20}$ ($C_2\times C_{10}$), $C_{11}^2:C_{20}$ ($C_2\times C_{10}$)

Order 1936: $C_{44}.D_{22}$ ($C_5^2$)

Order 1210: $C_{11}:F_{11}$ ($C_5\times Q_8$), $C_{11}\times C_{110}$ ($C_2^2\times C_{10}$), $C_{55}:D_{11}$ ($C_5\times Q_8$), $C_{11}^2:C_{10}$ ($C_2^2\times C_{10}$)

Order 968: $C_{44}:D_{11}$ ($C_5\times C_{10}$), $C_{11}^2:Q_8$ ($C_5\times C_{10}$), $C_{11}^2:Q_8$ ($C_5\times C_{10}$), $C_{22}.D_{22}$ ($C_5\times C_{10}$)

Order 605: $C_{11}\times C_{55}$ ($Q_8\times C_{10}$), $C_{11}^2:C_5$ ($Q_8\times C_{10}$)

Order 484: $C_{11}\times C_{44}$ ($C_{10}^2$), $C_{11}^2:C_4$ ($C_{10}^2$), $C_{11}:C_{44}$ ($C_{10}^2$), $C_{11}:D_{22}$ ($C_{10}^2$)

Order 440: $C_{55}:Q_8$ ($F_{11}$)

Order 242: $C_{11}\times C_{22}$ ($C_2\times C_{10}^2$), $C_{11}:D_{11}$ ($Q_8\times C_5^2$)

Order 220: $C_{220}$ ($C_2\times F_{11}$), $C_{11}:C_{20}$ ($C_2\times F_{11}$)

Order 121: $C_{11}^2$ ($C_4.C_{10}^2$)

Order 110: $C_{110}$ ($C_2^2\times F_{11}$)

Order 88: $C_{11}:Q_8$ ($C_5\times F_{11}$)

Order 55: $C_{55}$ ($Q_8\times F_{11}$)

Order 44: $C_{44}$ ($C_{10}\times F_{11}$), $C_{11}:C_4$ ($C_{10}\times F_{11}$)

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

Order 20: $C_{20}$ ($D_{11}:F_{11}$)

Order 11: $C_{11}$ ($C_{44}.C_{10}^2$)

Order 10: $C_{10}$ ($D_{22}:F_{11}$)

Order 5: $C_5$ ($C_{11}:(Q_8\times F_{11})$)

Order 4: $C_4$ ($C_{11}^2:C_{10}^2$)

Order 2: $C_2$ ($C_5\times D_{22}:F_{11}$)

Order 1: $C_1$ ($C_{55}:(Q_8\times F_{11})$)

Series

Derived series

$C_{55}:(Q_8\times F_{11})$

$\rhd$

$C_{55}:(Q_8\times F_{11})$

$\rhd$

$C_{11}\times C_{22}$

$\rhd$

$C_{11}\times C_{22}$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_{55}:(Q_8\times F_{11})$

$\rhd$

$C_{55}:(Q_8\times F_{11})$

$\rhd$

$C_{220}.F_{11}$

$\rhd$

$C_{220}.F_{11}$

$\rhd$

$C_{44}.C_{110}$

$\rhd$

$C_{44}.C_{110}$

$\rhd$

$C_{11}\times C_{220}$

$\rhd$

$C_{11}\times C_{220}$

$\rhd$

$C_{220}$

$\rhd$

$C_{220}$

$\rhd$

$C_{110}$

$\rhd$

$C_{110}$

$\rhd$

$C_{55}$

$\rhd$

$C_{55}$

$\rhd$

$C_{11}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_{55}:(Q_8\times F_{11})$

$\rhd$

$C_{55}:(Q_8\times F_{11})$

$\rhd$

$C_{11}\times C_{22}$

$\rhd$

$C_{11}\times C_{22}$

$\rhd$

$C_{11}^2$

$\rhd$

$C_{11}^2$

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 3 larger groups in the database.

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

Character theory

Complex character table

See the $400 \times 400$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $120 \times 120$ rational character table.