Abstract group 798600.i: $C_{11}^3:(S_3\times C_{10}^2)$

• Label: 798600.i
• Order: \( 2^{3} \cdot 3 \cdot 5^{2} \cdot 11^{3} \)
• Exponent: \( 2 \cdot 3 \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 5^{2} \)
• $\card{Z(G)}$: \( 2 \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3 \cdot 5^{3} \cdot 11^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 5^{2} \)
• Perm deg.: $40$
• Trans deg.: not computed
• Rank: $3$

Group information

Description:	$C_{11}^3:(S_3\times C_{10}^2)$
Order:	\(798600\)\(\medspace = 2^{3} \cdot 3 \cdot 5^{2} \cdot 11^{3} \)
Exponent:	\(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \)
Automorphism group:	Group of order \(63888000\)\(\medspace = 2^{7} \cdot 3 \cdot 5^{3} \cdot 11^{3} \)
Composition factors:	$C_2$ x 3, $C_3$, $C_5$ x 2, $C_{11}$ x 3
Derived length:	$3$

This group is nonabelian, solvable, and an A-group. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	5	6	10	11	15	22	30	33	55	66	110	165	330
Elements	1	3455	242	26624	5566	413120	1330	54208	16510	181984	2420	5320	2420	66040	9680	9680	798600
Conjugacy classes	1	7	1	24	3	168	29	24	55	72	1	116	1	220	4	4	730
Divisions	1	7	1	6	3	42	29	6	55	18	1	29	1	55	1	1	256
Autjugacy classes	1	4	1	5	2	20	7	5	11	10	1	7	1	11	1	1	88

Minimal presentations

Permutation degree:	$40$
Transitive degree:	not computed
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	30	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{10}=b^{6}=c^{11}=d^{11}=e^{110}=[c,d]=[c,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $40$ $\langle(34,35,36,38,37), (1,2,6,19,5,18,17,4,7,9,16)(3,10,13,24,12,14,8,20,25,15,23) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 3 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 3 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 5 & 7 & 4 & 0 \\ 10 & 10 & 10 & 0 \\ 8 & 10 & 6 & 4 \end{array}\right), \left(\begin{array}{rrrr} 5 & 3 & 8 & 0 \\ 0 & 0 & 1 & 6 \\ 5 & 1 & 4 & 8 \\ 8 & 7 & 5 & 2 \end{array}\right), \left(\begin{array}{rrrr} 2 & 6 & 9 & 4 \\ 4 & 8 & 0 & 0 \\ 5 & 10 & 4 & 10 \\ 5 & 4 & 9 & 4 \end{array}\right), \left(\begin{array}{rrrr} 8 & 0 & 0 & 0 \\ 0 & 8 & 0 & 0 \\ 0 & 0 & 8 & 0 \\ 0 & 0 & 0 & 8 \end{array}\right), \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 5 & 9 & 4 & 0 \\ 2 & 2 & 10 & 0 \\ 2 & 2 & 6 & 4 \end{array}\right), \left(\begin{array}{rrrr} 0 & 10 & 3 & 5 \\ 10 & 8 & 3 & 3 \\ 6 & 7 & 4 & 1 \\ 0 & 6 & 1 & 1 \end{array}\right), \left(\begin{array}{rrrr} 3 & 2 & 9 & 8 \\ 10 & 6 & 4 & 9 \\ 6 & 4 & 1 & 9 \\ 0 & 6 & 1 & 4 \end{array}\right), \left(\begin{array}{rrrr} 9 & 0 & 0 & 0 \\ 7 & 5 & 10 & 0 \\ 6 & 6 & 5 & 0 \\ 3 & 6 & 4 & 1 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{11})$
Direct product:	$C_2$ $\, \times\, $ $C_5$ $\, \times\, $ $(C_{11}^3:(S_3\times C_{10}))$
Semidirect product:	$C_{11}^3$ $\,\rtimes\,$ $(S_3\times C_{10}^2)$	$(C_5\times C_{11}^3:C_{15})$ $\,\rtimes\,$ $C_2^3$	$(C_{11}^3:C_5^2)$ $\,\rtimes\,$ $(C_2\times D_6)$	$(C_{11}^3:(C_2\times D_6))$ $\,\rtimes\,$ $C_5^2$	all 5
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_{11}\wr S_3)$ . $C_{10}^2$	$(C_{11}^3:C_{10}^2)$ . $S_3$	$(C_{11}^3:C_6)$ . $C_{10}^2$	$(C_{11}^2:D_{33})$ . $C_{10}^2$	all 69

Elements of the group are displayed as matrices in $\GL_{4}(\F_{11})$.

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 190 normal subgroups (45 characteristic).

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

Special subgroups

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

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 1 (order at least 798600) are shown, as well as all normal subgroups of any index.

Order 798600: $C_{11}^3:(S_3\times C_{10}^2)$

Order 399300: $C_{11}\wr C_3:C_{10}^2$ x 4, $C_{10}\times C_{11}^3:C_{30}$, $C_{11}\wr C_3:C_{10}^2$, $C_{11}\wr C_3:C_{10}^2$

Order 266200: $C_2\times C_{11}^3:C_{10}^2$

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

Order 159720: $C_{11}^3:(C_{10}\times D_6)$ x 5, $C_{11}^3:(C_{10}\times D_6)$

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 20, $C_2\times C_{11}^3:C_{30}$ x 5, $C_{11}^3:(S_3\times C_{10})$ x 5, $C_{11}^3:(S_3\times C_{10})$ x 5, $C_5\times C_{11}^3:D_6$ x 4, $C_2\times C_{11}^3:C_{30}$, $C_{10}\times C_{11}\wr S_3$, $C_5\times C_{11}^2:D_{66}$

Order 72600: $C_{11}^2:(S_3\times C_{10}^2)$

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, $C_{11}^3:(C_5\times S_3)$ x 10, $C_{11}^3:C_{30}$ x 10, $C_{11}^3:C_{30}$ x 5, $C_{11}^3:C_{30}$ x 2, $C_5\times C_{11}\wr S_3$ x 2, $C_5\times C_{11}^2:D_{33}$ x 2, $C_{11}^2:C_{330}$

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

Order 31944: $C_{11}^3:(C_2\times D_6)$

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

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

Order 15972: $C_{11}^3:D_6$ x 4, $C_2\times C_{11}^3:C_6$, $C_{11}^2:D_{66}$, $C_{11}^3:D_6$

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 7986: $C_{11}^3:C_6$ x 2, $C_{11}^2:D_{33}$ x 2, $C_{11}\wr S_3$ x 2, $C_{11}^2:C_{66}$

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

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

Order 6600: $C_{66}:C_{10}^2$

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

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

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

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

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

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

Order 1331: $C_{11}^3$

Order 1210: $C_{11}\times C_{110}$

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

Order 605: $C_{11}\times C_{55}$

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

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

Order 121: $C_{11}^2$

Order 110: $C_{110}$

Order 55: $C_{55}$

Order 25: $C_5^2$

Order 22: $C_{22}$

Order 11: $C_{11}$

Order 10: $C_{10}$

Order 8: $C_2^3$

Order 5: $C_5$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 1 (order at least 798600) are shown, as well as all normal subgroups of any index.

Order 798600: $C_{11}^3:(S_3\times C_{10}^2)$

Order 399300: $C_{10}\times C_{11}^3:C_{30}$, $C_{11}\wr C_3:C_{10}^2$, $C_{11}\wr C_3:C_{10}^2$, $C_{11}\wr C_3:C_{10}^2$

Order 266200: $C_2\times C_{11}^3:C_{10}^2$

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

Order 159720: $C_{11}^3:(C_{10}\times D_6)$, $C_{11}^3:(C_{10}\times D_6)$

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

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

Order 79860: $C_2\times C_{11}^3:C_{30}$, $C_2\times C_{11}^3:C_{30}$, $C_5\times C_{11}^3:D_6$, $C_{11}^3:(S_3\times C_{10})$, $C_{11}^3:(S_3\times C_{10})$, $C_{11}^3:(S_3\times C_{10})$, $C_{10}\times C_{11}\wr S_3$, $C_5\times C_{11}^2:D_{66}$

Order 72600: $C_{11}^2:(S_3\times C_{10}^2)$

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

Order 39930: $C_{11}^3:C_{30}$, $C_{11}^3:(C_5\times S_3)$, $C_{11}^3:(C_5\times S_3)$, $C_{11}^3:C_{30}$, $C_{11}^3:C_{30}$, $C_5\times C_{11}\wr S_3$, $C_{11}^2:C_{330}$, $C_5\times C_{11}^2:D_{33}$

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

Order 31944: $C_{11}^3:(C_2\times D_6)$

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

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

Order 15972: $C_2\times C_{11}^3:C_6$, $C_{11}^3:D_6$, $C_{11}^2:D_{66}$, $C_{11}^3:D_6$

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

Order 7986: $C_{11}^3:C_6$, $C_{11}^2:D_{33}$, $C_{11}^2:C_{66}$, $C_{11}\wr S_3$

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

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

Order 6600: $C_{66}:C_{10}^2$

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

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

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

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

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

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

Order 1331: $C_{11}^3$

Order 1210: $C_{11}\times C_{110}$

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

Order 605: $C_{11}\times C_{55}$

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

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

Order 121: $C_{11}^2$

Order 110: $C_{110}$

Order 55: $C_{55}$

Order 25: $C_5^2$

Order 22: $C_{22}$

Order 11: $C_{11}$

Order 10: $C_{10}$

Order 8: $C_2^3$

Order 5: $C_5$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 798600: $C_{11}^3:(S_3\times C_{10}^2)$ ($C_1$)

Order 399300: $C_{11}\wr C_3:C_{10}^2$ ($C_2$) x 4, $C_{10}\times C_{11}^3:C_{30}$ ($C_2$), $C_{11}\wr C_3:C_{10}^2$ ($C_2$), $C_{11}\wr C_3:C_{10}^2$ ($C_2$)

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

Order 159720: $C_{11}^3:(C_{10}\times D_6)$ ($C_5$) x 5, $C_{11}^3:(C_{10}\times D_6)$ ($C_5$)

Order 133100: $C_{11}^3:C_{10}^2$ ($S_3$)

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

Order 79860: $C_{11}^3:(S_3\times C_{10})$ ($C_{10}$) x 20, $C_2\times C_{11}^3:C_{30}$ ($C_{10}$) x 5, $C_{11}^3:(S_3\times C_{10})$ ($C_{10}$) x 5, $C_{11}^3:(S_3\times C_{10})$ ($C_{10}$) x 5, $C_5\times C_{11}^3:D_6$ ($C_{10}$) x 4, $C_2\times C_{11}^3:C_{30}$ ($C_{10}$), $C_{10}\times C_{11}\wr S_3$ ($C_{10}$), $C_5\times C_{11}^2:D_{66}$ ($C_{10}$)

Order 66550: $C_5\times C_{11}^3:C_{10}$ ($D_6$) x 2, $C_5\times C_{11}^3:C_{10}$ ($D_6$)

Order 39930: $C_{11}^3:(C_5\times S_3)$ ($C_2\times C_{10}$) x 10, $C_{11}^3:(C_5\times S_3)$ ($C_2\times C_{10}$) x 10, $C_{11}^3:C_{30}$ ($C_2\times C_{10}$) x 10, $C_{11}^3:C_{30}$ ($C_2\times C_{10}$) x 5, $C_{11}^3:C_{30}$ ($C_2\times C_{10}$) x 2, $C_5\times C_{11}\wr S_3$ ($C_2\times C_{10}$) x 2, $C_5\times C_{11}^2:D_{33}$ ($C_2\times C_{10}$) x 2, $C_{11}^2:C_{330}$ ($C_2\times C_{10}$)

Order 33275: $C_{11}^3:C_5^2$ ($C_2\times D_6$)

Order 31944: $C_{11}^3:(C_2\times D_6)$ ($C_5^2$)

Order 26620: $C_2\times C_{11}^3:C_{10}$ ($C_5\times S_3$) x 5, $C_2\times C_{11}^3:C_{10}$ ($C_5\times S_3$)

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

Order 15972: $C_{11}^3:D_6$ ($C_5\times C_{10}$) x 4, $C_2\times C_{11}^3:C_6$ ($C_5\times C_{10}$), $C_{11}^2:D_{66}$ ($C_5\times C_{10}$), $C_{11}^3:D_6$ ($C_5\times C_{10}$)

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

Order 7986: $C_{11}^3:C_6$ ($C_{10}^2$) x 2, $C_{11}^2:D_{33}$ ($C_{10}^2$) x 2, $C_{11}\wr S_3$ ($C_{10}^2$) x 2, $C_{11}^2:C_{66}$ ($C_{10}^2$)

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

Order 6655: $C_{11}^3:C_5$ ($C_{10}\times D_6$) x 5, $C_{11}^2\times C_{55}$ ($C_{10}\times D_6$)

Order 5324: $C_{11}^2:D_{22}$ ($S_3\times C_5^2$)

Order 3993: $C_{11}\wr C_3$ ($C_2\times C_{10}^2$)

Order 3630: $C_5\times C_{11}^2:S_3$ ($C_2\times F_{11}$) x 2, $C_{11}^2:C_{30}$ ($C_2\times F_{11}$)

Order 2662: $C_{11}^3:C_2$ ($D_6\times C_5^2$) x 2, $C_{11}^2\times C_{22}$ ($D_6\times C_5^2$)

Order 1815: $C_{11}^2:C_{15}$ ($C_2^2\times F_{11}$)

Order 1452: $C_{11}^2:D_6$ ($C_5\times F_{11}$)

Order 1331: $C_{11}^3$ ($S_3\times C_{10}^2$)

Order 1210: $C_{11}\times C_{110}$ ($S_3\times F_{11}$)

Order 726: $C_{11}^2:S_3$ ($C_{10}\times F_{11}$) x 2, $C_{11}^2:C_6$ ($C_{10}\times F_{11}$)

Order 605: $C_{11}\times C_{55}$ ($D_6\times F_{11}$)

Order 363: $C_{11}^2:C_3$ ($C_{22}:C_{10}^2$)

Order 242: $C_{11}\times C_{22}$ ($C_{33}:C_{10}^2$)

Order 121: $C_{11}^2$ ($C_{66}:C_{10}^2$)

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

Order 55: $C_{55}$ ($C_{11}^2:(C_{10}\times D_6)$)

Order 22: $C_{22}$ ($C_{11}^2:(D_6\times C_5^2)$)

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

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

Order 5: $C_5$ ($C_{11}^3:(C_{10}\times D_6)$)

Order 2: $C_2$ ($C_{11}\wr C_3:C_{10}^2$)

Order 1: $C_1$ ($C_{11}^3:(S_3\times C_{10}^2)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 798600: $C_{11}^3:(S_3\times C_{10}^2)$ ($C_1$)

Order 399300: $C_{10}\times C_{11}^3:C_{30}$ ($C_2$), $C_{11}\wr C_3:C_{10}^2$ ($C_2$), $C_{11}\wr C_3:C_{10}^2$ ($C_2$), $C_{11}\wr C_3:C_{10}^2$ ($C_2$)

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

Order 159720: $C_{11}^3:(C_{10}\times D_6)$ ($C_5$), $C_{11}^3:(C_{10}\times D_6)$ ($C_5$)

Order 133100: $C_{11}^3:C_{10}^2$ ($S_3$)

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

Order 79860: $C_2\times C_{11}^3:C_{30}$ ($C_{10}$), $C_2\times C_{11}^3:C_{30}$ ($C_{10}$), $C_5\times C_{11}^3:D_6$ ($C_{10}$), $C_{11}^3:(S_3\times C_{10})$ ($C_{10}$), $C_{11}^3:(S_3\times C_{10})$ ($C_{10}$), $C_{11}^3:(S_3\times C_{10})$ ($C_{10}$), $C_{10}\times C_{11}\wr S_3$ ($C_{10}$), $C_5\times C_{11}^2:D_{66}$ ($C_{10}$)

Order 66550: $C_5\times C_{11}^3:C_{10}$ ($D_6$), $C_5\times C_{11}^3:C_{10}$ ($D_6$)

Order 39930: $C_{11}^3:C_{30}$ ($C_2\times C_{10}$), $C_{11}^3:(C_5\times S_3)$ ($C_2\times C_{10}$), $C_{11}^3:(C_5\times S_3)$ ($C_2\times C_{10}$), $C_{11}^3:C_{30}$ ($C_2\times C_{10}$), $C_{11}^3:C_{30}$ ($C_2\times C_{10}$), $C_5\times C_{11}\wr S_3$ ($C_2\times C_{10}$), $C_{11}^2:C_{330}$ ($C_2\times C_{10}$), $C_5\times C_{11}^2:D_{33}$ ($C_2\times C_{10}$)

Order 33275: $C_{11}^3:C_5^2$ ($C_2\times D_6$)

Order 31944: $C_{11}^3:(C_2\times D_6)$ ($C_5^2$)

Order 26620: $C_2\times C_{11}^3:C_{10}$ ($C_5\times S_3$), $C_2\times C_{11}^3:C_{10}$ ($C_5\times S_3$)

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

Order 15972: $C_2\times C_{11}^3:C_6$ ($C_5\times C_{10}$), $C_{11}^3:D_6$ ($C_5\times C_{10}$), $C_{11}^2:D_{66}$ ($C_5\times C_{10}$), $C_{11}^3:D_6$ ($C_5\times C_{10}$)

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

Order 7986: $C_{11}^3:C_6$ ($C_{10}^2$), $C_{11}^2:D_{33}$ ($C_{10}^2$), $C_{11}^2:C_{66}$ ($C_{10}^2$), $C_{11}\wr S_3$ ($C_{10}^2$)

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

Order 6655: $C_{11}^2\times C_{55}$ ($C_{10}\times D_6$), $C_{11}^3:C_5$ ($C_{10}\times D_6$)

Order 5324: $C_{11}^2:D_{22}$ ($S_3\times C_5^2$)

Order 3993: $C_{11}\wr C_3$ ($C_2\times C_{10}^2$)

Order 3630: $C_5\times C_{11}^2:S_3$ ($C_2\times F_{11}$), $C_{11}^2:C_{30}$ ($C_2\times F_{11}$)

Order 2662: $C_{11}^2\times C_{22}$ ($D_6\times C_5^2$), $C_{11}^3:C_2$ ($D_6\times C_5^2$)

Order 1815: $C_{11}^2:C_{15}$ ($C_2^2\times F_{11}$)

Order 1452: $C_{11}^2:D_6$ ($C_5\times F_{11}$)

Order 1331: $C_{11}^3$ ($S_3\times C_{10}^2$)

Order 1210: $C_{11}\times C_{110}$ ($S_3\times F_{11}$)

Order 726: $C_{11}^2:C_6$ ($C_{10}\times F_{11}$), $C_{11}^2:S_3$ ($C_{10}\times F_{11}$)

Order 605: $C_{11}\times C_{55}$ ($D_6\times F_{11}$)

Order 363: $C_{11}^2:C_3$ ($C_{22}:C_{10}^2$)

Order 242: $C_{11}\times C_{22}$ ($C_{33}:C_{10}^2$)

Order 121: $C_{11}^2$ ($C_{66}:C_{10}^2$)

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

Order 55: $C_{55}$ ($C_{11}^2:(C_{10}\times D_6)$)

Order 22: $C_{22}$ ($C_{11}^2:(D_6\times C_5^2)$)

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

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

Order 5: $C_5$ ($C_{11}^3:(C_{10}\times D_6)$)

Order 2: $C_2$ ($C_{11}\wr C_3:C_{10}^2$)

Order 1: $C_1$ ($C_{11}^3:(S_3\times C_{10}^2)$)

Series

Derived series

$C_{11}^3:(S_3\times C_{10}^2)$

$\rhd$

$C_{11}\wr C_3$

$\rhd$

$C_{11}^2$

$\rhd$

$C_1$

Chief series

$C_{11}^3:(S_3\times C_{10}^2)$

$\rhd$

$C_{10}\times C_{11}^3:C_{30}$

$\rhd$

$C_5\times C_{11}^3:C_{30}$

$\rhd$

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

$\rhd$

$C_{11}^2:C_{165}$

$\rhd$

$C_{11}\wr C_3$

$\rhd$

$C_{11}^3$

$\rhd$

$C_{11}^2$

$\rhd$

$C_1$

Lower central series

$C_{11}^3:(S_3\times C_{10}^2)$

$\rhd$

$C_{11}\wr C_3$

Upper central series

$C_1$

$\lhd$

$C_{10}$

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

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

Character theory

Complex character table

The $730 \times 730$ character table is not available for this group.

Rational character table

The $256 \times 256$ rational character table is not available for this group.