Abstract group 532400.d: $C_{110}.D_{22}:F_{11}$

• Label: 532400.d
• Order: \( 2^{4} \cdot 5^{2} \cdot 11^{3} \)
• Exponent: \( 2^{2} \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 5^{2} \)
• $\card{Z(G)}$: \( 2 \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3 \cdot 5^{4} \cdot 11^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3 \cdot 5^{3} \)
• Perm deg.: $46$
• Trans deg.: $440$
• Rank: $3$

Group information

Description:	$C_{110}.D_{22}:F_{11}$
Order:	\(532400\)\(\medspace = 2^{4} \cdot 5^{2} \cdot 11^{3} \)
Exponent:	\(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \)
Automorphism group:	$C_{11}^3.C_5^2.C_2^4.C_5.C_{30}.C_2^4$, of order \(1277760000\)\(\medspace = 2^{9} \cdot 3 \cdot 5^{4} \cdot 11^{3} \)
Composition factors:	$C_2$ x 4, $C_5$ x 2, $C_{11}$ x 3
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	2663	792	26624	90512	1330	322608	1330	15180	5320	5320	60720	532400
Conjugacy classes	1	3	6	24	72	43	144	43	39	172	172	156	875
Divisions	1	3	6	6	18	43	36	43	24	43	43	24	290
Autjugacy classes	1	2	2	5	10	3	10	3	3	3	3	3	48

Dimension	1	2	4	8	10	20	40	80	160
Irr. complex chars.	200	50	0	0	60	315	250	0	0	875
Irr. rational chars.	8	2	48	12	12	33	77	33	65	290

Minimal presentations

Permutation degree:	$46$
Transitive degree:	$440$
Rank:	$3$
Inequivalent generating triples:	$35997696$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d \mid a^{10}=c^{220}=d^{11}=1, b^{22}=c^{66}, b^{a}=b^{13}c^{144}d^{9}, c^{a}= \!\cdots\! \rangle}$
Permutation group:	Degree $46$ $\langle(20,21,22,23,24), (1,2,5,3,4,9,7,8,11,6,10)(20,22,24,21,23)(26,32,35,43,34,40,37,44,36,46,45) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 5 & 7 & 4 & 0 \\ 10 & 10 & 10 & 0 \\ 8 & 10 & 6 & 4 \end{array}\right), \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 5 & 9 & 4 & 0 \\ 2 & 2 & 10 & 0 \\ 2 & 2 & 6 & 4 \end{array}\right), \left(\begin{array}{rrrr} 8 & 2 & 9 & 5 \\ 10 & 5 & 10 & 8 \\ 8 & 5 & 4 & 0 \\ 3 & 5 & 7 & 5 \end{array}\right), \left(\begin{array}{rrrr} 3 & 2 & 9 & 8 \\ 10 & 6 & 4 & 9 \\ 6 & 4 & 1 & 9 \\ 0 & 6 & 1 & 4 \end{array}\right), \left(\begin{array}{rrrr} 9 & 0 & 0 & 0 \\ 7 & 5 & 10 & 0 \\ 6 & 6 & 5 & 0 \\ 3 & 6 & 4 & 1 \end{array}\right), \left(\begin{array}{rrrr} 2 & 9 & 1 & 9 \\ 8 & 8 & 9 & 5 \\ 10 & 5 & 2 & 0 \\ 3 & 6 & 0 & 10 \end{array}\right), \left(\begin{array}{rrrr} 3 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 3 \end{array}\right), \left(\begin{array}{rrrr} 8 & 0 & 0 & 0 \\ 0 & 8 & 0 & 0 \\ 0 & 0 & 8 & 0 \\ 0 & 0 & 0 & 8 \end{array}\right), \left(\begin{array}{rrrr} 0 & 10 & 3 & 5 \\ 10 & 8 & 3 & 3 \\ 6 & 7 & 4 & 1 \\ 0 & 6 & 1 & 1 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{11})$
Direct product:	$C_5$ $\, \times\, $ $(C_{11}^3:(Q_8\times C_{10}))$
Semidirect product:	$(C_{110}.D_{22})$ $\,\rtimes\,$ $F_{11}$	$(C_2.D_{11}^3)$ $\,\rtimes\,$ $C_5^2$	$(C_{110}.D_{11}^2)$ $\,\rtimes\,$ $C_{10}$	$(C_{110}.D_{11}^2)$ $\,\rtimes\,$ $C_{10}$	all 24
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{10}$ . $(D_{11}^3:C_5)$	$C_{110}$ . $(D_{22}:F_{11})$	$(C_{110}.D_{11}^2)$ . $C_{10}$	$(C_{11}^2:D_{22})$ . $C_{10}^2$	all 32

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 928456 subgroups in 5136 conjugacy classes, 210 normal (19 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{10}$	$G/Z \simeq$ $D_{11}^3:C_5$
Commutator:	$G' \simeq$ $C_{11}^2\times C_{22}$	$G/G' \simeq$ $C_2\times C_{10}^2$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{11}.C_{110}.C_{22}.C_{10}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{11}^2\times C_{110}$	$G/\operatorname{Fit} \simeq$ $C_2^2\times C_{10}$
Radical:	$R \simeq$ $C_{110}.D_{22}:F_{11}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{11}^2\times C_{110}$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_{10}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times Q_8$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}^3$

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 532400: $C_{110}.D_{22}:F_{11}$

Order 266200: $C_{11}^3:(C_{10}\times C_{20})$ x 3, $C_{11}^3:(Q_8\times C_5^2)$ x 3, $C_{11}^3:(Q_8\times C_5^2)$

Order 133100: $C_5\times C_{11}^3:C_{20}$ x 3, $C_5\times C_{11}^3:C_{20}$ x 3, $C_{11}^3:C_{10}^2$

Order 106480: $C_{11}^3:(Q_8\times C_{10})$ x 5, $C_5\times C_2.D_{11}^3$

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

Order 53240: $C_{11}^3:(C_2\times C_{20})$ x 15, $C_{11}^3:(C_5\times Q_8)$ x 15, $C_{11}^3:(C_5\times Q_8)$ x 5, $C_{110}.D_{11}^2$ x 3, $C_{110}.D_{11}^2$ x 3, $C_{110}.D_{11}^2$

Order 48400: $C_{55}:(Q_8\times F_{11})$ x 3

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

Order 26620: $C_{11}^3:C_{20}$ x 15, $C_{11}^3:C_{20}$ x 15, $C_2\times C_{11}^3:C_{10}$ x 5, $C_{11}^2:C_{220}$ x 3, $C_{11}^2:C_{220}$ x 3, $C_2\times C_{11}^3:C_{10}$

Order 24200: $C_{11}:(C_{20}\times F_{11})$ x 21, $C_{220}.F_{11}$ x 6, $C_{11}^2:(Q_8\times C_5^2)$ x 6, $C_{220}:F_{11}$ x 3

Order 21296: $C_2.D_{11}^3$

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

Order 12100: $C_{110}:F_{11}$ x 43, $C_{110}.F_{11}$ x 42, $C_5\times C_{11}^2:C_{20}$ x 3, $C_{110}.F_{11}$ x 3

Order 10648: $C_{22}.D_{11}^2$ x 3, $C_{11}^3:Q_8$ x 3, $C_{11}^3:Q_8$

Order 9680: $C_{11}:(Q_8\times F_{11})$ x 15, $C_{220}.D_{22}$ x 3

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

Order 6050: $C_{55}:F_{11}$ x 86, $C_5\times C_{11}^2:C_{10}$ x 43

Order 5324: $C_{11}^2:C_{44}$ x 3, $C_{11}^2:C_{44}$ x 3, $C_{11}^2:D_{22}$

Order 4840: $C_{11}:(C_4\times F_{11})$ x 105, $C_{44}.F_{11}$ x 30, $C_{11}^2:(C_5\times Q_8)$ x 30, $C_{110}.D_{22}$ x 21, $C_{44}:F_{11}$ x 15, $C_{44}.C_{110}$ x 6, $C_{110}.D_{22}$ x 6, $C_{220}:D_{11}$ x 3

Order 4400: $C_{44}.C_{10}^2$ x 3

Order 3025: $C_{11}^2:C_5^2$ x 43

Order 2662: $C_{11}^3:C_2$ x 2, $C_{11}^2\times C_{22}$

Order 2420: $C_{22}:F_{11}$ x 215, $C_{11}^2:C_{20}$ x 210, $C_{110}:D_{11}$ x 43, $C_{11}:C_{220}$ x 42, $C_{11}^2:C_{20}$ x 15, $C_{11}^2:C_{20}$ x 15, $C_{11}^2:C_{20}$ x 3, $C_{11}\times C_{220}$ x 3

Order 2200: $C_{20}\times F_{11}$ x 24, $C_{220}.C_{10}$ x 9, $C_{220}.C_{10}$ x 3

Order 1936: $C_{44}.D_{22}$ x 3

Order 1331: $C_{11}^3$

Order 1210: $C_{11}:F_{11}$ x 430, $C_{11}^2:C_{10}$ x 215, $C_{55}:D_{11}$ x 86, $C_{11}\times C_{110}$ x 43

Order 1100: $C_{10}\times F_{11}$ x 43, $C_{220}:C_5$ x 24, $C_{55}:C_{20}$ x 24

Order 968: $C_{22}.D_{22}$ x 21, $C_{11}^2:Q_8$ x 6, $C_{11}^2:Q_8$ x 6, $C_{44}:D_{11}$ x 3

Order 880: $Q_8\times F_{11}$ x 15, $C_{20}.D_{22}$ x 3

Order 605: $C_{11}^2:C_5$ x 215, $C_{11}\times C_{55}$ x 43

Order 550: $C_5\times F_{11}$ x 86, $C_{110}:C_5$ x 43

Order 484: $C_{11}:D_{22}$ x 43, $C_{11}:C_{44}$ x 42, $C_{11}\times C_{44}$ x 3, $C_{11}^2:C_4$ x 3

Order 440: $C_4\times F_{11}$ x 120, $C_{44}.C_{10}$ x 45, $C_{20}\times D_{11}$ x 24, $C_{44}.C_{10}$ x 15, $C_{55}:Q_8$ x 9, $Q_8\times C_{55}$ x 3

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

Order 275: $C_{55}:C_5$ x 43

Order 242: $C_{11}:D_{11}$ x 86, $C_{11}\times C_{22}$ x 43

Order 220: $C_2\times F_{11}$ x 215, $C_{11}:C_{20}$ x 120, $C_{11}:C_{20}$ x 120, $C_5\times D_{22}$ x 43, $C_{220}$ x 24, $C_{11}:C_{20}$ x 24

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 3

Order 121: $C_{11}^2$ x 43

Order 110: $F_{11}$ x 430, $C_{11}:C_{10}$ x 215, $C_5\times D_{11}$ x 86, $C_{110}$ x 43

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

Order 88: $C_4\times D_{11}$ x 24, $C_{11}:Q_8$ x 9, $Q_8\times C_{11}$ x 3

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

Order 55: $C_{11}:C_5$ x 215, $C_{55}$ x 43

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

Order 44: $D_{22}$ x 43, $C_{44}$ x 24, $C_{11}:C_4$ x 24

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 86, $C_{22}$ x 43

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 43

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 532400: $C_{110}.D_{22}:F_{11}$

Order 266200: $C_{11}^3:(C_{10}\times C_{20})$, $C_{11}^3:(Q_8\times C_5^2)$, $C_{11}^3:(Q_8\times C_5^2)$

Order 133100: $C_5\times C_{11}^3:C_{20}$, $C_5\times C_{11}^3:C_{20}$, $C_{11}^3:C_{10}^2$

Order 106480: $C_5\times C_2.D_{11}^3$, $C_{11}^3:(Q_8\times C_{10})$

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

Order 53240: $C_{110}.D_{11}^2$, $C_{11}^3:(C_5\times Q_8)$, $C_{11}^3:(C_2\times C_{20})$, $C_{110}.D_{11}^2$, $C_{11}^3:(C_5\times Q_8)$, $C_{110}.D_{11}^2$

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

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

Order 26620: $C_{11}^2:C_{220}$, $C_2\times C_{11}^3:C_{10}$, $C_2\times C_{11}^3:C_{10}$, $C_{11}^3:C_{20}$, $C_{11}^3:C_{20}$, $C_{11}^2:C_{220}$

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

Order 21296: $C_2.D_{11}^3$

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

Order 12100: $C_{110}.F_{11}$ x 4, $C_{110}:F_{11}$ x 3, $C_5\times C_{11}^2:C_{20}$, $C_{110}.F_{11}$

Order 10648: $C_{22}.D_{11}^2$, $C_{11}^3:Q_8$, $C_{11}^3:Q_8$

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

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

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

Order 5324: $C_{11}^2:D_{22}$, $C_{11}^2:C_{44}$, $C_{11}^2:C_{44}$

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

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

Order 3025: $C_{11}^2:C_5^2$ x 3

Order 2662: $C_{11}^2\times C_{22}$, $C_{11}^3:C_2$

Order 2420: $C_{11}:C_{220}$ x 4, $C_{11}^2:C_{20}$ x 4, $C_{110}:D_{11}$ x 3, $C_{22}:F_{11}$ x 3, $C_{11}^2:C_{20}$, $C_{11}^2:C_{20}$, $C_{11}\times C_{220}$, $C_{11}^2:C_{20}$

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 1331: $C_{11}^3$

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

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

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

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

Order 605: $C_{11}\times C_{55}$ x 3, $C_{11}^2:C_5$ x 3

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

Order 484: $C_{11}:C_{44}$ x 4, $C_{11}:D_{22}$ x 3, $C_{11}\times C_{44}$, $C_{11}^2:C_4$

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 3

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

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

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

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

Order 121: $C_{11}^2$ x 3

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

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 3, $C_{11}:C_5$ x 3

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

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

Order 40: $C_5\times Q_8$ x 4, $C_2\times C_{20}$ x 2

Order 25: $C_5^2$

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

Order 20: $C_{20}$ x 4, $C_2\times C_{10}$ x 2

Order 16: $C_2\times Q_8$

Order 11: $C_{11}$ x 3

Order 10: $C_{10}$ x 4

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

Order 5: $C_5$ x 2

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 532400: $C_{110}.D_{22}:F_{11}$ ($C_1$)

Order 266200: $C_{11}^3:(C_{10}\times C_{20})$ ($C_2$) x 3, $C_{11}^3:(Q_8\times C_5^2)$ ($C_2$) x 3, $C_{11}^3:(Q_8\times C_5^2)$ ($C_2$)

Order 133100: $C_5\times C_{11}^3:C_{20}$ ($C_2^2$) x 3, $C_5\times C_{11}^3:C_{20}$ ($C_2^2$) x 3, $C_{11}^3:C_{10}^2$ ($C_2^2$)

Order 106480: $C_{11}^3:(Q_8\times C_{10})$ ($C_5$) x 5, $C_5\times C_2.D_{11}^3$ ($C_5$)

Order 66550: $C_5\times C_{11}^3:C_{10}$ ($Q_8$) x 2, $C_5\times C_{11}^3:C_{10}$ ($C_2^3$)

Order 53240: $C_{11}^3:(C_2\times C_{20})$ ($C_{10}$) x 15, $C_{11}^3:(C_5\times Q_8)$ ($C_{10}$) x 15, $C_{11}^3:(C_5\times Q_8)$ ($C_{10}$) x 5, $C_{110}.D_{11}^2$ ($C_{10}$) x 3, $C_{110}.D_{11}^2$ ($C_{10}$) x 3, $C_{110}.D_{11}^2$ ($C_{10}$)

Order 33275: $C_{11}^3:C_5^2$ ($C_2\times Q_8$)

Order 26620: $C_{11}^3:C_{20}$ ($C_2\times C_{10}$) x 15, $C_{11}^3:C_{20}$ ($C_2\times C_{10}$) x 15, $C_2\times C_{11}^3:C_{10}$ ($C_2\times C_{10}$) x 5, $C_{11}^2:C_{220}$ ($C_2\times C_{10}$) x 3, $C_{11}^2:C_{220}$ ($C_2\times C_{10}$) x 3, $C_2\times C_{11}^3:C_{10}$ ($C_2\times C_{10}$)

Order 21296: $C_2.D_{11}^3$ ($C_5^2$)

Order 13310: $C_{11}^3:C_{10}$ ($C_5\times Q_8$) x 10, $C_{11}^3:C_{10}$ ($C_2^2\times C_{10}$) x 5, $C_{11}^3:C_{10}$ ($C_5\times Q_8$) x 2, $C_{11}^2\times C_{110}$ ($C_2^2\times C_{10}$)

Order 10648: $C_{22}.D_{11}^2$ ($C_5\times C_{10}$) x 3, $C_{11}^3:Q_8$ ($C_5\times C_{10}$) x 3, $C_{11}^3:Q_8$ ($C_5\times C_{10}$)

Order 6655: $C_{11}^3:C_5$ ($Q_8\times C_{10}$) x 5, $C_{11}^2\times C_{55}$ ($Q_8\times C_{10}$)

Order 5324: $C_{11}^2:C_{44}$ ($C_{10}^2$) x 3, $C_{11}^2:C_{44}$ ($C_{10}^2$) x 3, $C_{11}^2:D_{22}$ ($C_{10}^2$)

Order 4840: $C_{110}.D_{22}$ ($F_{11}$) x 3

Order 2662: $C_{11}^3:C_2$ ($Q_8\times C_5^2$) x 2, $C_{11}^2\times C_{22}$ ($C_2\times C_{10}^2$)

Order 2420: $C_{11}:C_{220}$ ($C_2\times F_{11}$) x 6, $C_{11}^2:C_{20}$ ($C_2\times F_{11}$) x 3

Order 1331: $C_{11}^3$ ($C_4.C_{10}^2$)

Order 1210: $C_{11}\times C_{110}$ ($C_2^2\times F_{11}$) x 3

Order 968: $C_{11}^2:Q_8$ ($C_5\times F_{11}$) x 3

Order 605: $C_{11}\times C_{55}$ ($Q_8\times F_{11}$) x 3

Order 484: $C_{11}:C_{44}$ ($C_{10}\times F_{11}$) x 6, $C_{11}^2:C_4$ ($C_{10}\times F_{11}$) x 3

Order 242: $C_{11}\times C_{22}$ ($C_{22}:C_{10}^2$) x 3

Order 220: $C_{11}:C_{20}$ ($D_{11}:F_{11}$) x 3

Order 121: $C_{11}^2$ ($C_{44}.C_{10}^2$) x 3

Order 110: $C_{110}$ ($D_{22}:F_{11}$) x 3

Order 55: $C_{55}$ ($C_{11}:(Q_8\times F_{11})$) x 3

Order 44: $C_{11}:C_4$ ($C_{11}^2:C_{10}^2$) x 3

Order 22: $C_{22}$ ($C_5\times D_{22}:F_{11}$) x 3

Order 11: $C_{11}$ ($C_{55}:(Q_8\times F_{11})$) x 3

Order 10: $C_{10}$ ($D_{11}^3:C_5$)

Order 5: $C_5$ ($C_{11}^3:(Q_8\times C_{10})$)

Order 2: $C_2$ ($C_{11}.C_{110}.C_{22}.C_{10}$)

Order 1: $C_1$ ($C_{110}.D_{22}:F_{11}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 532400: $C_{110}.D_{22}:F_{11}$ ($C_1$)

Order 266200: $C_{11}^3:(C_{10}\times C_{20})$ ($C_2$), $C_{11}^3:(Q_8\times C_5^2)$ ($C_2$), $C_{11}^3:(Q_8\times C_5^2)$ ($C_2$)

Order 133100: $C_5\times C_{11}^3:C_{20}$ ($C_2^2$), $C_5\times C_{11}^3:C_{20}$ ($C_2^2$), $C_{11}^3:C_{10}^2$ ($C_2^2$)

Order 106480: $C_5\times C_2.D_{11}^3$ ($C_5$), $C_{11}^3:(Q_8\times C_{10})$ ($C_5$)

Order 66550: $C_5\times C_{11}^3:C_{10}$ ($Q_8$), $C_5\times C_{11}^3:C_{10}$ ($C_2^3$)

Order 53240: $C_{110}.D_{11}^2$ ($C_{10}$), $C_{11}^3:(C_5\times Q_8)$ ($C_{10}$), $C_{11}^3:(C_2\times C_{20})$ ($C_{10}$), $C_{110}.D_{11}^2$ ($C_{10}$), $C_{11}^3:(C_5\times Q_8)$ ($C_{10}$), $C_{110}.D_{11}^2$ ($C_{10}$)

Order 33275: $C_{11}^3:C_5^2$ ($C_2\times Q_8$)

Order 26620: $C_{11}^2:C_{220}$ ($C_2\times C_{10}$), $C_2\times C_{11}^3:C_{10}$ ($C_2\times C_{10}$), $C_2\times C_{11}^3:C_{10}$ ($C_2\times C_{10}$), $C_{11}^3:C_{20}$ ($C_2\times C_{10}$), $C_{11}^3:C_{20}$ ($C_2\times C_{10}$), $C_{11}^2:C_{220}$ ($C_2\times C_{10}$)

Order 21296: $C_2.D_{11}^3$ ($C_5^2$)

Order 13310: $C_{11}^3:C_{10}$ ($C_2^2\times C_{10}$), $C_{11}^2\times C_{110}$ ($C_2^2\times C_{10}$), $C_{11}^3:C_{10}$ ($C_5\times Q_8$), $C_{11}^3:C_{10}$ ($C_5\times Q_8$)

Order 10648: $C_{22}.D_{11}^2$ ($C_5\times C_{10}$), $C_{11}^3:Q_8$ ($C_5\times C_{10}$), $C_{11}^3:Q_8$ ($C_5\times C_{10}$)

Order 6655: $C_{11}^2\times C_{55}$ ($Q_8\times C_{10}$), $C_{11}^3:C_5$ ($Q_8\times C_{10}$)

Order 5324: $C_{11}^2:D_{22}$ ($C_{10}^2$), $C_{11}^2:C_{44}$ ($C_{10}^2$), $C_{11}^2:C_{44}$ ($C_{10}^2$)

Order 4840: $C_{110}.D_{22}$ ($F_{11}$)

Order 2662: $C_{11}^2\times C_{22}$ ($C_2\times C_{10}^2$), $C_{11}^3:C_2$ ($Q_8\times C_5^2$)

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

Order 1331: $C_{11}^3$ ($C_4.C_{10}^2$)

Order 1210: $C_{11}\times C_{110}$ ($C_2^2\times F_{11}$)

Order 968: $C_{11}^2:Q_8$ ($C_5\times F_{11}$)

Order 605: $C_{11}\times C_{55}$ ($Q_8\times F_{11}$)

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

Order 242: $C_{11}\times C_{22}$ ($C_{22}:C_{10}^2$)

Order 220: $C_{11}:C_{20}$ ($D_{11}:F_{11}$)

Order 121: $C_{11}^2$ ($C_{44}.C_{10}^2$)

Order 110: $C_{110}$ ($D_{22}:F_{11}$)

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

Order 44: $C_{11}:C_4$ ($C_{11}^2:C_{10}^2$)

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

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

Order 10: $C_{10}$ ($D_{11}^3:C_5$)

Order 5: $C_5$ ($C_{11}^3:(Q_8\times C_{10})$)

Order 2: $C_2$ ($C_{11}.C_{110}.C_{22}.C_{10}$)

Order 1: $C_1$ ($C_{110}.D_{22}:F_{11}$)

Series

Derived series

$C_{110}.D_{22}:F_{11}$

$\rhd$

$C_{11}^2\times C_{22}$

$\rhd$

$C_1$

Chief series

$C_{110}.D_{22}:F_{11}$

$\rhd$

$C_{11}^3:(Q_8\times C_5^2)$

$\rhd$

$C_{110}.D_{11}^2$

$\rhd$

$C_{11}^2:C_{220}$

$\rhd$

$C_{11}:C_{220}$

$\rhd$

$C_{11}\times C_{110}$

$\rhd$

$C_{11}\times C_{55}$

$\rhd$

$C_{11}^2$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Lower central series

$C_{110}.D_{22}:F_{11}$

$\rhd$

$C_{11}^2\times C_{22}$

$\rhd$

$C_{11}^3$

Upper central series

$C_1$

$\lhd$

$C_{10}$

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 $875 \times 875$ character table is not available for this group.

Rational character table

See the $290 \times 290$ rational character table (warning: may be slow to load).