Abstract group 319440.v: $\He_{11}:C_{120}:C_2$

• Label: 319440.v
• Order: \( 2^{4} \cdot 3 \cdot 5 \cdot 11^{3} \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 5 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3 \cdot 5 \cdot 11^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $1331$
• Trans deg.: $1331$
• Rank: $2$

Group information

Description:	$\He_{11}:C_{120}:C_2$
Order:	\(319440\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 11^{3} \)
Exponent:	\(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
Automorphism group:	$\He_{11}:C_{120}:C_2$, of order \(319440\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 11^{3} \)
Composition factors:	$C_2$ x 4, $C_3$, $C_5$, $C_{11}$ x 3
Derived length:	$4$

This group is nonabelian and solvable.

Group statistics

Order	1	2	3	4	5	6	8	10	11	12	15	20	22	24	30	33	40	44	60	66	120	132
Elements	1	1573	242	1694	5324	242	5324	69212	1330	484	10648	74536	15730	10648	10648	2420	21296	16940	21296	2420	42592	4840	319440
Conjugacy classes	1	2	1	2	4	1	2	8	2	2	4	8	2	4	4	1	8	3	8	1	16	2	86
Divisions	1	2	1	2	1	1	1	2	2	1	1	2	2	1	1	1	1	2	1	1	1	1	29
Autjugacy classes	1	2	1	2	4	1	2	8	2	2	4	8	2	4	4	1	8	3	8	1	16	2	86

Dimension	1	2	4	8	16	32	110	120	220	440
Irr. complex chars.	20	55	0	0	0	0	4	2	5	0	86
Irr. rational chars.	4	3	6	4	2	1	2	2	4	1	29

Minimal presentations

Permutation degree:	$1331$
Transitive degree:	$1331$
Rank:	$2$
Inequivalent generating pairs:	$76032$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	110	110	110
Arbitrary	110	110	110

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{10}=b^{24}=c^{11}=d^{11}=e^{11}=[c,e]=[d,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $1331$ $\langle(2,16,31,82,163,154,90,303,23,25)(3,33,48,416,560,849,488,490,40,42)(4,50,64,580,709,528,398,641,56,58) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 6 & 9 & 1 & 4 \\ 4 & 6 & 3 & 1 \\ 8 & 10 & 7 & 2 \\ 2 & 8 & 7 & 7 \end{array}\right), \left(\begin{array}{rrrr} 2 & 2 & 0 & 8 \\ 0 & 8 & 8 & 3 \\ 4 & 10 & 4 & 1 \\ 1 & 6 & 5 & 2 \end{array}\right), \left(\begin{array}{rrrr} 3 & 0 & 10 & 10 \\ 3 & 0 & 3 & 10 \\ 3 & 8 & 2 & 0 \\ 1 & 3 & 8 & 10 \end{array}\right), \left(\begin{array}{rrrr} 3 & 6 & 1 & 10 \\ 10 & 0 & 0 & 4 \\ 8 & 2 & 7 & 6 \\ 1 & 10 & 10 & 7 \end{array}\right), \left(\begin{array}{rrrr} 6 & 0 & 8 & 5 \\ 2 & 9 & 6 & 10 \\ 4 & 2 & 7 & 6 \\ 2 & 9 & 8 & 1 \end{array}\right), \left(\begin{array}{rrrr} 10 & 10 & 10 & 0 \\ 1 & 1 & 8 & 4 \\ 0 & 3 & 2 & 1 \\ 3 & 6 & 5 & 0 \end{array}\right), \left(\begin{array}{rrrr} 0 & 3 & 6 & 0 \\ 10 & 7 & 3 & 6 \\ 4 & 3 & 6 & 8 \\ 8 & 4 & 1 & 2 \end{array}\right), \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 8 & 5 & 0 & 0 \\ 5 & 0 & 5 & 0 \\ 9 & 9 & 1 & 9 \end{array}\right), \left(\begin{array}{rrrr} 5 & 0 & 0 & 0 \\ 2 & 8 & 0 & 0 \\ 4 & 0 & 8 & 0 \\ 7 & 2 & 10 & 4 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{11})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(\He_{11}:C_{40})$ $\,\rtimes\,$ $S_3$	$(\He_{11}:C_{120})$ $\,\rtimes\,$ $C_2$	$(\He_{11}:C_{24})$ $\,\rtimes\,$ $C_{10}$	$\He_{11}$ $\,\rtimes\,$ $(C_{120}:C_2)$	all 11
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{11}$ . $(F_{121}:C_2)$	$(\He_{11}:C_{30})$ . $D_4$	$(\He_{11}:C_{20})$ . $D_6$	$(\He_{11}:D_{12})$ . $C_{10}$	all 11
Aut. group:	$\Aut(\He_{11}:C_8)$	$\Aut(\He_{11}:C_{15})$	$\Aut(\He_{11}:C_{20})$	$\Aut(\He_{11}:D_{12})$	all 13

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

Homology

Abelianization:	$C_{2} \times C_{10} \simeq C_{2}^{2} \times C_{5}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 126212 subgroups in 140 conjugacy classes, 24 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $\He_{11}:C_{120}:C_2$
Commutator:	$G' \simeq$ $\He_{11}:C_{12}$	$G/G' \simeq$ $C_2\times C_{10}$
Frattini:	$\Phi \simeq$ $C_{11}$	$G/\Phi \simeq$ $F_{121}:C_2$
Fitting:	$\operatorname{Fit} \simeq$ $\He_{11}$	$G/\operatorname{Fit} \simeq$ $C_{120}:C_2$
Radical:	$R \simeq$ $\He_{11}:C_{120}:C_2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{11}$	$G/\operatorname{soc} \simeq$ $F_{121}:C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\SD_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $\He_{11}$

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

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 319440: $\He_{11}:C_{120}:C_2$

Order 159720: $\He_{11}:C_{120}$, $\He_{11}:C_{15}:Q_8$, $\He_{11}:(C_5\times D_{12})$

Order 106480: $\He_{11}:(C_5\times \SD_{16})$

Order 79860: $\He_{11}:(S_3\times C_{10})$, $\He_{11}:C_{60}$, $\He_{11}:C_3:C_{20}$

Order 63888: $\He_{11}:C_{24}:C_2$

Order 53240: $\He_{11}:(C_5\times Q_8)$, $\He_{11}:C_{40}$, $\He_{11}:(C_5\times D_4)$

Order 39930: $\He_{11}:(C_5\times S_3)$, $\He_{11}:C_{30}$

Order 31944: $\He_{11}:C_{24}$, $\He_{11}:C_3:Q_8$, $\He_{11}:D_{12}$

Order 26620: $\He_{11}:C_{20}$ x 2, $C_{11}^2:(C_2\times F_{11})$

Order 21296: $\He_{11}:\SD_{16}$

Order 19965: $\He_{11}:C_{15}$

Order 15972: $\He_{11}:D_6$, $\He_{11}:C_{12}$, $\He_{11}:C_3:C_4$

Order 13310: $C_{11}^2:F_{11}$, $C_{11}^2:F_{11}$

Order 10648: $\He_{11}:Q_8$, $\He_{11}:C_8$, $\He_{11}:D_4$

Order 7986: $\He_{11}:C_6$, $\He_{11}:S_3$

Order 6655: $\He_{11}:C_5$

Order 5324: $\He_{11}:C_4$ x 2, $C_{11}^2:D_{22}$

Order 3993: $\He_{11}:C_3$

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

Order 2640: $D_{132}.C_{10}$

Order 2420: $D_{11}:F_{11}$

Order 1331: $\He_{11}$

Order 1320: $C_{132}:C_{10}$, $C_{132}.C_{10}$, $C_{11}:C_{120}$

Order 1210: $C_{11}:F_{11}$, $C_{11}:F_{11}$, $C_{11}:F_{11}$

Order 880: $Q_8:F_{11}$

Order 660: $C_{11}:C_{60}$, $C_{33}:C_{20}$, $C_{66}:C_{10}$

Order 605: $C_{11}^2:C_5$

Order 528: $C_{33}:\SD_{16}$

Order 484: $D_{11}^2$

Order 440: $C_{44}:C_{10}$, $C_{44}.C_{10}$, $C_{11}:C_{40}$

Order 330: $C_{11}:C_{30}$, $C_{33}:C_{10}$

Order 264: $D_{132}$, $C_{11}:C_{24}$, $C_{33}:Q_8$

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

Order 240: $C_{120}:C_2$

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

Order 176: $Q_8:D_{11}$

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

Order 132: $D_{66}$, $C_{132}$, $C_3:C_{44}$

Order 121: $C_{11}^2$

Order 120: $C_{120}$, $C_5\times D_{12}$, $C_{15}:Q_8$

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

Order 88: $D_{44}$, $Q_8\times C_{11}$, $C_{11}:C_8$

Order 80: $C_5\times \SD_{16}$

Order 66: $C_{66}$, $D_{33}$

Order 60: $C_{60}$, $S_3\times C_{10}$, $C_3:C_{20}$

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

Order 48: $C_{24}:C_2$

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

Order 40: $C_{40}$, $C_5\times Q_8$, $C_5\times D_4$

Order 33: $C_{33}$

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

Order 24: $D_{12}$, $C_3:Q_8$, $C_{24}$

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

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

Order 16: $\SD_{16}$

Order 15: $C_{15}$

Order 12: $D_6$, $C_{12}$, $C_3:C_4$

Order 11: $C_{11}$ x 2

Order 10: $C_{10}$ x 2

Order 8: $Q_8$, $D_4$, $C_8$

Order 6: $C_6$, $S_3$

Order 5: $C_5$

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 319440: $\He_{11}:C_{120}:C_2$

Order 159720: $\He_{11}:C_{120}$, $\He_{11}:C_{15}:Q_8$, $\He_{11}:(C_5\times D_{12})$

Order 106480: $\He_{11}:(C_5\times \SD_{16})$

Order 79860: $\He_{11}:(S_3\times C_{10})$, $\He_{11}:C_{60}$, $\He_{11}:C_3:C_{20}$

Order 63888: $\He_{11}:C_{24}:C_2$

Order 53240: $\He_{11}:(C_5\times Q_8)$, $\He_{11}:C_{40}$, $\He_{11}:(C_5\times D_4)$

Order 39930: $\He_{11}:(C_5\times S_3)$, $\He_{11}:C_{30}$

Order 31944: $\He_{11}:C_{24}$, $\He_{11}:C_3:Q_8$, $\He_{11}:D_{12}$

Order 26620: $\He_{11}:C_{20}$ x 2, $C_{11}^2:(C_2\times F_{11})$

Order 21296: $\He_{11}:\SD_{16}$

Order 19965: $\He_{11}:C_{15}$

Order 15972: $\He_{11}:D_6$, $\He_{11}:C_{12}$, $\He_{11}:C_3:C_4$

Order 13310: $C_{11}^2:F_{11}$, $C_{11}^2:F_{11}$

Order 10648: $\He_{11}:Q_8$, $\He_{11}:C_8$, $\He_{11}:D_4$

Order 7986: $\He_{11}:C_6$, $\He_{11}:S_3$

Order 6655: $\He_{11}:C_5$

Order 5324: $\He_{11}:C_4$ x 2, $C_{11}^2:D_{22}$

Order 3993: $\He_{11}:C_3$

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

Order 2640: $D_{132}.C_{10}$

Order 2420: $D_{11}:F_{11}$

Order 1331: $\He_{11}$

Order 1320: $C_{132}:C_{10}$, $C_{132}.C_{10}$, $C_{11}:C_{120}$

Order 1210: $C_{11}:F_{11}$, $C_{11}:F_{11}$, $C_{11}:F_{11}$

Order 880: $Q_8:F_{11}$

Order 660: $C_{11}:C_{60}$, $C_{33}:C_{20}$, $C_{66}:C_{10}$

Order 605: $C_{11}^2:C_5$

Order 528: $C_{33}:\SD_{16}$

Order 484: $D_{11}^2$

Order 440: $C_{44}:C_{10}$, $C_{44}.C_{10}$, $C_{11}:C_{40}$

Order 330: $C_{11}:C_{30}$, $C_{33}:C_{10}$

Order 264: $D_{132}$, $C_{11}:C_{24}$, $C_{33}:Q_8$

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

Order 240: $C_{120}:C_2$

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

Order 176: $Q_8:D_{11}$

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

Order 132: $D_{66}$, $C_{132}$, $C_3:C_{44}$

Order 121: $C_{11}^2$

Order 120: $C_{120}$, $C_5\times D_{12}$, $C_{15}:Q_8$

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

Order 88: $D_{44}$, $Q_8\times C_{11}$, $C_{11}:C_8$

Order 80: $C_5\times \SD_{16}$

Order 66: $C_{66}$, $D_{33}$

Order 60: $C_{60}$, $S_3\times C_{10}$, $C_3:C_{20}$

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

Order 48: $C_{24}:C_2$

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

Order 40: $C_{40}$, $C_5\times Q_8$, $C_5\times D_4$

Order 33: $C_{33}$

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

Order 24: $D_{12}$, $C_3:Q_8$, $C_{24}$

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

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

Order 16: $\SD_{16}$

Order 15: $C_{15}$

Order 12: $D_6$, $C_{12}$, $C_3:C_4$

Order 11: $C_{11}$ x 2

Order 10: $C_{10}$ x 2

Order 8: $Q_8$, $D_4$, $C_8$

Order 6: $C_6$, $S_3$

Order 5: $C_5$

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 319440: $\He_{11}:C_{120}:C_2$ ($C_1$)

Order 159720: $\He_{11}:C_{120}$ ($C_2$), $\He_{11}:C_{15}:Q_8$ ($C_2$), $\He_{11}:(C_5\times D_{12})$ ($C_2$)

Order 79860: $\He_{11}:C_{60}$ ($C_2^2$)

Order 63888: $\He_{11}:C_{24}:C_2$ ($C_5$)

Order 53240: $\He_{11}:C_{40}$ ($S_3$)

Order 39930: $\He_{11}:C_{30}$ ($D_4$)

Order 31944: $\He_{11}:C_{24}$ ($C_{10}$), $\He_{11}:C_3:Q_8$ ($C_{10}$), $\He_{11}:D_{12}$ ($C_{10}$)

Order 26620: $\He_{11}:C_{20}$ ($D_6$)

Order 19965: $\He_{11}:C_{15}$ ($\SD_{16}$)

Order 15972: $\He_{11}:C_{12}$ ($C_2\times C_{10}$)

Order 13310: $C_{11}^2:F_{11}$ ($D_{12}$)

Order 10648: $\He_{11}:C_8$ ($C_5\times S_3$)

Order 7986: $\He_{11}:C_6$ ($C_5\times D_4$)

Order 6655: $\He_{11}:C_5$ ($C_{24}:C_2$)

Order 5324: $\He_{11}:C_4$ ($S_3\times C_{10}$)

Order 3993: $\He_{11}:C_3$ ($C_5\times \SD_{16}$)

Order 2662: $C_{11}^2:D_{11}$ ($C_5\times D_{12}$)

Order 1331: $\He_{11}$ ($C_{120}:C_2$)

Order 11: $C_{11}$ ($F_{121}:C_2$)

Order 1: $C_1$ ($\He_{11}:C_{120}:C_2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 319440: $\He_{11}:C_{120}:C_2$ ($C_1$)

Order 159720: $\He_{11}:C_{120}$ ($C_2$), $\He_{11}:C_{15}:Q_8$ ($C_2$), $\He_{11}:(C_5\times D_{12})$ ($C_2$)

Order 79860: $\He_{11}:C_{60}$ ($C_2^2$)

Order 63888: $\He_{11}:C_{24}:C_2$ ($C_5$)

Order 53240: $\He_{11}:C_{40}$ ($S_3$)

Order 39930: $\He_{11}:C_{30}$ ($D_4$)

Order 31944: $\He_{11}:C_{24}$ ($C_{10}$), $\He_{11}:C_3:Q_8$ ($C_{10}$), $\He_{11}:D_{12}$ ($C_{10}$)

Order 26620: $\He_{11}:C_{20}$ ($D_6$)

Order 19965: $\He_{11}:C_{15}$ ($\SD_{16}$)

Order 15972: $\He_{11}:C_{12}$ ($C_2\times C_{10}$)

Order 13310: $C_{11}^2:F_{11}$ ($D_{12}$)

Order 10648: $\He_{11}:C_8$ ($C_5\times S_3$)

Order 7986: $\He_{11}:C_6$ ($C_5\times D_4$)

Order 6655: $\He_{11}:C_5$ ($C_{24}:C_2$)

Order 5324: $\He_{11}:C_4$ ($S_3\times C_{10}$)

Order 3993: $\He_{11}:C_3$ ($C_5\times \SD_{16}$)

Order 2662: $C_{11}^2:D_{11}$ ($C_5\times D_{12}$)

Order 1331: $\He_{11}$ ($C_{120}:C_2$)

Order 11: $C_{11}$ ($F_{121}:C_2$)

Order 1: $C_1$ ($\He_{11}:C_{120}:C_2$)

Series

Derived series

$\He_{11}:C_{120}:C_2$

$\rhd$

$\He_{11}:C_{12}$

$\rhd$

$\He_{11}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Chief series

$\He_{11}:C_{120}:C_2$

$\rhd$

$\He_{11}:(C_5\times D_{12})$

$\rhd$

$\He_{11}:C_{60}$

$\rhd$

$\He_{11}:C_{12}$

$\rhd$

$\He_{11}:C_6$

$\rhd$

$\He_{11}:C_3$

$\rhd$

$\He_{11}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Lower central series

$\He_{11}:C_{120}:C_2$

$\rhd$

$\He_{11}:C_{12}$

$\rhd$

$\He_{11}:C_6$

$\rhd$

$\He_{11}:C_3$

Upper central series

$C_1$

Supergroups

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

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

Character theory

Complex character table

See the $86 \times 86$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $29 \times 29$ rational character table.