Abstract group 319440.x: $C_{11}^3:(C_{10}\times D_{12})$

• Label: 319440.x
• Order: \( 2^{4} \cdot 3 \cdot 5 \cdot 11^{3} \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 5 \)
• $\card{Z(G)}$: 10
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3 \cdot 5^{2} \cdot 11^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 5^{2} \)
• Perm deg.: $53$
• Trans deg.: $660$
• Rank: $3$

Group information

Description:	$C_{11}^3:(C_{10}\times D_{12})$
Order:	\(319440\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 11^{3} \)
Exponent:	\(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \)
Automorphism group:	Group of order \(51110400\)\(\medspace = 2^{9} \cdot 3 \cdot 5^{2} \cdot 11^{3} \)
Composition factors:	$C_2$ x 4, $C_3$, $C_5$, $C_{11}$ x 3
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	33	44	55	60	66	110	132	165	220	330	660
Elements	1	4247	242	264	4	5566	16988	1330	5808	968	1056	31690	22264	2420	5060	5320	23232	2420	126760	4840	9680	20240	9680	19360	319440
Conjugacy classes	1	7	1	2	4	3	28	95	4	4	8	225	12	5	25	380	16	5	900	10	20	100	20	40	1915
Divisions	1	7	1	2	1	3	7	19	2	1	2	45	3	1	4	19	2	1	45	1	1	4	1	1	174
Autjugacy classes	1	4	1	2	1	2	4	5	2	1	2	9	2	1	3	5	2	1	9	1	1	3	1	1	64

Minimal presentations

Permutation degree:	$53$
Transitive degree:	$660$
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{10}=b^{22}=c^{12}=d^{11}=e^{11}=[b,d]=[d,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $53$ $\langle(1,2,8,12,23,32)(3,9,21)(4,14,22,6,11,26)(5,15,28,17,10,25)(7,19,20,18,27,31) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 3 & 0 & 0 & 0 \\ 5 & 8 & 4 & 0 \\ 0 & 0 & 3 & 0 \\ 6 & 0 & 6 & 8 \end{array}\right), \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 5 & 9 & 4 & 0 \\ 2 & 2 & 10 & 0 \\ 2 & 2 & 6 & 4 \end{array}\right), \left(\begin{array}{rrrr} 8 & 2 & 4 & 4 \\ 7 & 2 & 10 & 2 \\ 5 & 2 & 1 & 6 \\ 10 & 2 & 9 & 0 \end{array}\right), \left(\begin{array}{rrrr} 3 & 2 & 9 & 8 \\ 10 & 6 & 4 & 9 \\ 6 & 4 & 1 & 9 \\ 0 & 6 & 1 & 4 \end{array}\right), \left(\begin{array}{rrrr} 10 & 2 & 2 & 0 \\ 8 & 3 & 7 & 8 \\ 9 & 1 & 10 & 7 \\ 9 & 7 & 10 & 10 \end{array}\right), \left(\begin{array}{rrrr} 3 & 1 & 0 & 9 \\ 9 & 2 & 4 & 0 \\ 9 & 10 & 8 & 6 \\ 2 & 6 & 10 & 9 \end{array}\right), \left(\begin{array}{rrrr} 3 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 3 \end{array}\right), \left(\begin{array}{rrrr} 8 & 0 & 0 & 0 \\ 0 & 8 & 0 & 0 \\ 0 & 0 & 8 & 0 \\ 0 & 0 & 0 & 8 \end{array}\right), \left(\begin{array}{rrrr} 0 & 10 & 3 & 5 \\ 10 & 8 & 3 & 3 \\ 6 & 7 & 4 & 1 \\ 0 & 6 & 1 & 1 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{11})$
Direct product:	not computed
Semidirect product:	$C_{11}^3$ $\,\rtimes\,$ $(C_{10}\times D_{12})$	$(C_{11}\wr C_3)$ $\,\rtimes\,$ $(D_4\times C_{10})$	$(C_{11}^2:C_{165})$ $\,\rtimes\,$ $(C_2\times D_4)$	$(C_{11}^2\times C_{55})$ $\,\rtimes\,$ $(C_2\times D_{12})$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_{11}^3:C_{30})$ . $D_4$ (2)	$(C_{110}.D_{11}^2)$ . $S_3$	$(C_{11}^3:D_{12})$ . $C_{10}$ (2)	$(C_{11}^3:D_{12})$ . $C_{10}$	all 60

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

Homology

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

Subgroups

There are 82 normal subgroups (54 characteristic).

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

Special subgroups

Center:	a subgroup isomorphic to $C_{10}$
Commutator:	a subgroup isomorphic to $C_{11}^2:C_{66}$
Frattini:	a subgroup isomorphic to $C_2$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
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 2904 (order at least 110) are shown, as well as all normal subgroups of any index.

Order 319440: $C_{11}^3:(C_{10}\times D_{12})$

Order 159720: $C_5\times C_{11}^3:D_{12}$ x 2, $C_{11}^3:(C_{10}\times D_6)$ x 2, $C_5\times C_{11}^3:D_{12}$, $C_5\times C_{11}^2:D_{132}$, $C_{11}^3:(C_2\times C_{60})$

Order 106480: $C_5\times D_{22}:D_{11}^2$

Order 79860: $C_5\times C_{11}^3:D_6$ x 4, $C_{10}\times C_{11}\wr S_3$ x 2, $C_5\times C_{11}^2:D_{66}$ x 2, $C_2\times C_{11}^3:C_{30}$, $C_{11}^3:C_{60}$, $C_{11}^2:C_{660}$

Order 63888: $C_{11}^3:(C_2\times D_{12})$

Order 53240: $C_{110}.D_{11}^2$ x 2, $C_{110}:D_{11}^2$ x 2, $C_{110}.D_{11}^2$, $C_{110}.D_{11}^2$, $C_{110}.D_{11}^2$

Order 39930: $C_{11}^3:C_{30}$ 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 31944: $C_{11}^3:(C_2\times D_6)$ x 2, $C_{11}^3:D_{12}$ x 2, $C_{11}^2:(C_{12}\times D_{11})$, $C_{11}^2:D_{132}$, $C_{11}^3:D_{12}$

Order 29040: $(C_{11}\times C_{220}):D_6$

Order 26620: $C_{55}:D_{11}^2$ x 4, $C_{55}.C_{22}^2$ x 2, $C_{11}:D_{11}\times C_{110}$ x 2, $C_{11}^2:C_{220}$, $C_2\times C_{11}^3:C_{10}$, $C_{11}^2:C_{220}$

Order 21296: $D_{22}:D_{11}^2$

Order 19965: $C_{11}^2:C_{165}$

Order 15972: $C_{11}^3:D_6$ x 4, $C_{11}^2:D_{66}$ x 2, $C_{11}^3:D_6$ x 2, $C_{11}^3:C_{12}$, $C_2\times C_{11}^3:C_6$, $C_{11}^2:C_{132}$

Order 14520: $C_5\times C_{11}^2:D_{12}$ 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 13310: $C_{11}^2:C_{110}$ x 2, $C_{11}^3:C_{10}$ x 2, $C_{11}^2:C_{110}$ x 2, $C_{11}^2\times C_{110}$

Order 10648: $C_{22}:D_{11}^2$ x 2, $C_{11}^2:D_{44}$ x 2, $C_{22}.D_{11}^2$, $C_{11}^2:D_{44}$, $C_{11}^3:D_4$

Order 9680: $C_5\times D_{22}:D_{22}$ x 2, $C_{220}:D_{22}$

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: $C_{10}\times C_{11}^2:S_3$ 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 6655: $C_{11}^2\times C_{55}$

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

Order 5324: $C_{11}:D_{11}^2$ x 4, $C_{11}^2:D_{22}$ x 2, $D_{22}\times C_{11}^2$ x 2, $C_{11}^2:D_{22}$, $C_{11}^2:C_{44}$, $C_{11}^2:C_{44}$

Order 4840: $C_{10}\times D_{11}^2$ x 14, $D_{22}:C_{110}$ x 4, $C_{55}:D_{44}$ x 4, $C_{110}.D_{22}$ x 3, $C_{55}\times D_{44}$ x 2, $C_{110}.D_{22}$ x 2, $C_{110}:D_{22}$ x 2, $C_{220}:D_{11}$

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

Order 3630: $C_5\times C_{11}^2:S_3$ x 4, $C_{11}^2:C_{30}$ x 2, $C_{11}^2:C_{30}$

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

Order 2640: $C_{220}:D_6$

Order 2420: $C_5\times D_{11}^2$ x 48, $D_{11}\times C_{110}$ x 28, $C_{110}:D_{11}$ x 25, $C_{11}:C_{220}$ x 6, $C_{22}\times C_{110}$ x 2, $C_{11}^2:C_{20}$, $C_{11}\times C_{220}$

Order 1936: $D_{22}:D_{22}$ x 2, $C_{44}:D_{22}$

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

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 1331: $C_{11}^3$

Order 1320: $C_{165}:D_4$ x 2, $C_{30}:D_{22}$ x 2, $D_{11}\times C_{60}$, $C_5\times D_{132}$, $D_{12}\times C_{55}$

Order 1210: $D_{11}\times C_{55}$ x 48, $C_{55}:D_{11}$ x 40, $C_{11}\times C_{110}$ x 21

Order 968: $D_{11}\times D_{22}$ x 14, $C_{22}\wr C_2$ x 4, $C_{11}:D_{44}$ x 4, $C_{22}.D_{22}$ x 3, $C_{22}:D_{22}$ x 2, $C_{11}\times D_{44}$ x 2, $D_{22}:D_{11}$ x 2, $C_{44}:D_{11}$

Order 880: $C_{20}:D_{22}$ x 3

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

Order 660: $C_{15}:D_{22}$ x 4, $C_5\times D_{66}$ x 2, $S_3\times C_{110}$ x 2, $C_{11}:C_{60}$, $C_{15}\times D_{22}$, $C_{660}$

Order 605: $C_{11}\times C_{55}$ x 19

Order 528: $D_{11}\times D_{12}$

Order 484: $D_{11}^2$ x 48, $C_{11}\times D_{22}$ x 28, $C_{11}:D_{22}$ x 25, $C_{11}:C_{44}$ x 6, $C_{22}^2$ x 2, $C_{11}\times C_{44}$, $C_{11}^2:C_4$

Order 440: $C_{10}\times D_{22}$ x 16, $C_{55}:D_4$ x 6, $C_{20}\times D_{11}$ x 4, $D_4\times C_{55}$ x 3, $C_5\times D_{44}$ x 3

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

Order 330: $C_5\times D_{33}$ x 2, $S_3\times C_{55}$ x 2, $C_{15}\times D_{11}$ x 2, $C_{330}$

Order 264: $S_3\times D_{22}$ x 2, $C_{11}:D_{12}$ x 2, $D_{132}$, $C_{11}\times D_{12}$, $C_{12}\times D_{11}$

Order 242: $C_{11}\times D_{11}$ x 48, $C_{11}:D_{11}$ x 40, $C_{11}\times C_{22}$ x 21

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

Order 220: $C_5\times D_{22}$ x 87, $C_2\times C_{110}$ x 16, $C_{220}$ x 4, $C_{11}:C_{20}$ x 4

Order 176: $D_4\times D_{11}$ x 3

Order 165: $C_{165}$

Order 132: $S_3\times D_{11}$ x 4, $D_{66}$ x 2, $S_3\times C_{22}$ x 2, $C_3\times D_{22}$, $C_{132}$, $C_{11}:C_{12}$

Order 121: $C_{11}^2$ x 19

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

Order 110: $C_5\times D_{11}$ x 64, $C_{110}$ x 45

Order 55: $C_{55}$

Order 44: $C_{11}:C_4$

Order 22: $C_{22}$

Order 11: $C_{11}$

Order 10: $C_{10}$

Order 5: $C_5$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 319440: $C_{11}^3:(C_{10}\times D_{12})$ ($C_1$)

Order 159720: $C_5\times C_{11}^3:D_{12}$ ($C_2$) x 2, $C_{11}^3:(C_{10}\times D_6)$ ($C_2$) x 2, $C_5\times C_{11}^3:D_{12}$ ($C_2$), $C_5\times C_{11}^2:D_{132}$ ($C_2$), $C_{11}^3:(C_2\times C_{60})$ ($C_2$)

Order 79860: $C_{10}\times C_{11}\wr S_3$ ($C_2^2$) x 2, $C_5\times C_{11}^2:D_{66}$ ($C_2^2$) x 2, $C_2\times C_{11}^3:C_{30}$ ($C_2^2$), $C_{11}^3:C_{60}$ ($C_2^2$), $C_{11}^2:C_{660}$ ($C_2^2$)

Order 63888: $C_{11}^3:(C_2\times D_{12})$ ($C_5$)

Order 53240: $C_{110}.D_{11}^2$ ($S_3$)

Order 39930: $C_{11}^3:C_{30}$ ($D_4$) x 2, $C_{11}^2:C_{330}$ ($C_2^3$)

Order 31944: $C_{11}^3:(C_2\times D_6)$ ($C_{10}$) x 2, $C_{11}^3:D_{12}$ ($C_{10}$) x 2, $C_{11}^2:(C_{12}\times D_{11})$ ($C_{10}$), $C_{11}^2:D_{132}$ ($C_{10}$), $C_{11}^3:D_{12}$ ($C_{10}$)

Order 26620: $C_{11}^2:C_{220}$ ($D_6$), $C_2\times C_{11}^3:C_{10}$ ($D_6$), $C_{11}^2:C_{220}$ ($D_6$)

Order 19965: $C_{11}^2:C_{165}$ ($C_2\times D_4$)

Order 15972: $C_{11}^2:D_{66}$ ($C_2\times C_{10}$) x 2, $C_{11}^3:D_6$ ($C_2\times C_{10}$) x 2, $C_{11}^3:C_{12}$ ($C_2\times C_{10}$), $C_2\times C_{11}^3:C_6$ ($C_2\times C_{10}$), $C_{11}^2:C_{132}$ ($C_2\times C_{10}$)

Order 14520: $C_5\times C_{11}^2:D_{12}$ ($D_{11}$)

Order 13310: $C_{11}^3:C_{10}$ ($D_{12}$) x 2, $C_{11}^2\times C_{110}$ ($C_2\times D_6$)

Order 10648: $C_{22}.D_{11}^2$ ($C_5\times S_3$)

Order 7986: $C_{11}^3:C_6$ ($C_5\times D_4$) x 2, $C_{11}^2:C_{66}$ ($C_2^2\times C_{10}$)

Order 7260: $C_{10}\times C_{11}^2:S_3$ ($D_{22}$) x 2, $C_{11}^2:C_{60}$ ($D_{22}$)

Order 6655: $C_{11}^2\times C_{55}$ ($C_2\times D_{12}$)

Order 5324: $C_{11}^2:D_{22}$ ($S_3\times C_{10}$), $C_{11}^2:C_{44}$ ($S_3\times C_{10}$), $C_{11}^2:C_{44}$ ($S_3\times C_{10}$)

Order 3993: $C_{11}\wr C_3$ ($D_4\times C_{10}$)

Order 3630: $C_{11}^2:C_{30}$ ($C_2\times D_{22}$)

Order 2904: $C_{11}^2:D_{12}$ ($C_5\times D_{11}$)

Order 2662: $C_{11}^3:C_2$ ($C_5\times D_{12}$) x 2, $C_{11}^2\times C_{22}$ ($C_{10}\times D_6$)

Order 2420: $C_{11}^2:C_{20}$ ($S_3\times D_{11}$)

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

Order 1452: $C_{11}^2:D_6$ ($C_5\times D_{22}$) x 2, $C_{11}^2:C_{12}$ ($C_5\times D_{22}$)

Order 1331: $C_{11}^3$ ($C_{10}\times D_{12}$)

Order 1210: $C_{11}\times C_{110}$ ($S_3\times D_{22}$)

Order 726: $C_{11}^2:C_6$ ($C_{10}\times D_{22}$)

Order 605: $C_{11}\times C_{55}$ ($D_{11}\times D_{12}$)

Order 484: $C_{11}^2:C_4$ ($C_{15}:D_{22}$)

Order 363: $C_{11}^2:C_3$ ($C_{20}:D_{22}$)

Order 242: $C_{11}\times C_{22}$ ($C_{30}:D_{22}$)

Order 220: $C_{11}:C_{20}$ ($C_{11}^2:D_6$)

Order 121: $C_{11}^2$ ($C_{220}:D_6$)

Order 110: $C_{110}$ ($C_2\times C_{11}^2:D_6$)

Order 55: $C_{55}$ ($(C_{11}\times C_{44}):D_6$)

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

Order 22: $C_{22}$ ($C_{10}\times C_{11}^2:D_6$)

Order 11: $C_{11}$ ($(C_{11}\times C_{220}):D_6$)

Order 10: $C_{10}$ ($C_{11}^3:(C_2^2\times S_3)$)

Order 5: $C_5$ ($C_{11}^3:(C_2\times D_{12})$)

Order 2: $C_2$ ($C_{11}^3.C_{30}.C_2^2$)

Order 1: $C_1$ ($C_{11}^3:(C_{10}\times D_{12})$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

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 $1915 \times 1915$ character table is not available for this group.

Rational character table

The $174 \times 174$ rational character table is not available for this group.