Abstract group 3513840.b: $(C_{11}^2\times C_{110}).D_{132}$

• Label: 3513840.b
• Order: \( 2^{4} \cdot 3 \cdot 5 \cdot 11^{4} \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 5 \cdot 11 \)
• $\card{Z(G)}$: 110
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3 \cdot 5^{3} \cdot 11^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 5^{3} \)
• Perm deg.: not computed
• Trans deg.: $2640$
• Rank: $2$

Group information

Description:	$(C_{11}^2\times C_{110}).D_{132}$
Order:	\(3513840\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 11^{4} \)
Exponent:	\(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
Automorphism group:	Group of order \(255552000\)\(\medspace = 2^{9} \cdot 3 \cdot 5^{3} \cdot 11^{3} \)
Composition factors:	$C_2$ x 4, $C_3$, $C_5$, $C_{11}$ x 4
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	8	10	11	12	15	20	22	24	30	33	40	44	55	60	66	88	110	120	132	165	220	264	330	440	660	1320
Elements	1	1453	242	1694	4	242	484	5812	14640	484	968	6776	188880	968	968	29040	1936	203280	58560	1936	29040	58080	755520	3872	58080	116160	813120	116160	116160	232320	232320	464640	3513840
Conjugacy classes	1	2	1	2	4	1	2	8	725	2	4	8	845	4	4	120	8	240	2900	8	120	240	3380	16	240	480	960	480	480	960	960	1920	15125
Divisions	1	2	1	2	1	1	1	2	79	1	1	2	91	1	1	7	1	14	79	1	7	7	91	1	7	7	14	7	7	7	7	7	458

Minimal presentations

Permutation degree:	not computed
Transitive degree:	$2640$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of linear representations for this group have not been computed

Constructions

Presentation:	${\langle a, b, c, d \mid a^{110}=b^{264}=c^{11}=d^{11}=[c,d]=1, b^{a}=b^{131}d^{7}, c^{a}= \!\cdots\! \rangle}$
Matrix group:	$\left\langle \left(\begin{array}{rr} 120 & 1 \\ 120 & 0 \end{array}\right), \left(\begin{array}{rr} 27 & 0 \\ 0 & 27 \end{array}\right), \left(\begin{array}{rr} 56 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 34 & 110 \\ 22 & 23 \end{array}\right), \left(\begin{array}{rr} 103 & 72 \\ 98 & 18 \end{array}\right), \left(\begin{array}{rr} 31 & 86 \\ 62 & 90 \end{array}\right), \left(\begin{array}{rr} 111 & 0 \\ 0 & 111 \end{array}\right), \left(\begin{array}{rr} 120 & 0 \\ 0 & 120 \end{array}\right), \left(\begin{array}{rr} 45 & 99 \\ 22 & 12 \end{array}\right), \left(\begin{array}{rr} 103 & 23 \\ 49 & 18 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/121\Z)$
Direct product:	not computed
Semidirect product:	$C_{11}^4$ $\,\rtimes\,$ $(C_{120}:C_2)$	$(C_{11}^4:C_{15})$ $\,\rtimes\,$ $\SD_{16}$	$(C_{11}^3\times C_{55})$ $\,\rtimes\,$ $(C_{24}:C_2)$	$(C_{11}^4:C_3)$ $\,\rtimes\,$ $(C_5\times \SD_{16})$	more information
Trans. wreath product:	not computed
Possibly split product:	$(C_{11}^4:C_{40})$ . $S_3$	$(C_{11}^4:C_{30})$ . $D_4$	$(C_{11}^4:C_{20})$ . $D_6$	$(C_{11}^4:D_{12})$ . $C_{10}$	all 82

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{121}\Z)$.

Homology

Abelianization:	$C_{2} \times C_{110} \simeq C_{2}^{2} \times C_{5} \times C_{11}$
Schur multiplier:	not computed
Commutator length:	not computed

Subgroups

There are 92 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_{110}$
Commutator:	a subgroup isomorphic to $C_{11}^2:C_{132}$
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:	$P_{ 11 } \simeq$ $C_{11}^4$

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 2662 (order at least 1320) are shown, as well as all normal subgroups of any index.

Order 3513840: $(C_{11}^2\times C_{110}).D_{132}$

Order 1756920: $(C_{11}^3\times C_{110}).D_6$, $C_{11}^4:C_{120}$, $C_5\times C_{11}^4:D_{12}$

Order 1171280: $C_5\times C_{11}^4:\SD_{16}$

Order 878460: $(C_{11}^3\times C_{110}).S_3$, $C_{11}^4:C_{60}$, $C_5\times C_{11}^4:D_6$

Order 702768: $(C_{11}^2\times C_{22}).D_{132}$

Order 585640: $C_5\times C_{11}^3:D_{44}$, $C_5\times C_{11}^4:Q_8$, $C_{11}^4:C_{40}$

Order 439230: $C_{11}^4:C_{30}$, $C_5\times C_{11}^3:D_{33}$

Order 351384: $(C_{11}^2\times C_{22}).D_{66}$, $C_{11}^3:C_{264}$, $C_{11}^4:D_{12}$

Order 319440: $C_{11}^2.C_{24}:C_{110}$, $C_{11}^3.C_{30}.D_4$

Order 292820: $C_{11}^4:C_{20}$ x 2, $C_5\times C_{11}^4:C_2^2$

Order 234256: $C_{11}^4:\SD_{16}$

Order 219615: $C_{11}^4:C_{15}$

Order 175692: $(C_{11}^2\times C_{22}).D_{33}$, $C_{11}^4:C_{12}$, $C_{11}^4:D_6$

Order 159720: $C_{11}^2:C_{1320}$ x 7, $C_5\times C_{11}^3:D_{12}$, $C_{11}^2:C_{165}:Q_8$, $(C_{11}\times C_{110}).D_{66}$, $C_5\times C_{11}^2:D_{132}$

Order 146410: $C_{11}^3\times C_{110}$, $C_{11}^4:C_{10}$

Order 117128: $C_{11}^3:D_{44}$, $C_{11}^4:Q_8$, $C_{11}^3:C_{88}$

Order 106480: $C_{11}^2:\SD_{16}\times C_{55}$, $C_5\times C_{11}^3:\SD_{16}$

Order 87846: $C_{11}^4:C_6$, $C_{11}^3:D_{33}$

Order 79860: $C_{11}^2:C_{660}$ x 7, $C_{10}\times C_{11}\wr S_3$, $C_{11}\wr C_3:C_{20}$, $C_{11}\wr C_3:C_{20}$, $C_5\times C_{11}^2:D_{66}$

Order 73205: $C_{11}^3\times C_{55}$

Order 63888: $(C_{11}\times C_{22}).D_{132}$, $(C_{11}^2\times C_{22}).D_{12}$

Order 58564: $C_{11}^4:C_4$ x 2, $C_{11}^4:C_2^2$

Order 53240: $C_{11}^2:C_{440}$ x 7, $C_{110}.D_{11}^2$ x 2, $C_{110}.D_{11}^2$, $C_{110}.D_{11}^2$, $C_{110}.D_{11}^2$, $C_{110}.D_{11}^2$

Order 43923: $C_{11}^4:C_3$

Order 39930: $C_{11}^2:C_{330}$ x 7, $C_5\times C_{11}\wr S_3$, $C_5\times C_{11}^2:D_{33}$

Order 31944: $C_{11}^2:C_{264}$ x 7, $(C_{11}\times C_{22}).D_{66}$, $C_{11}^2:D_{132}$, $C_{11}^3:D_{12}$, $C_{11}\wr C_3:Q_8$

Order 29282: $C_{11}^3\times C_{22}$, $C_{11}^3:C_{22}$

Order 29040: $C_{1320}:C_{22}$, $(C_{11}\times C_{110}).D_{12}$

Order 26620: $C_{11}\times C_{55}\times C_{11}:C_4$ x 8, $C_{55}\times C_{11}^2:C_4$ x 6, $C_{11}\times C_{110}\times D_{11}$ x 6, $C_{11}^2:C_{220}$ x 5, $C_{10}\times C_{11}^3:C_2$ x 4, $C_5\times C_{11}^3:C_4$ x 3, $C_{11}:D_{11}\times C_{110}$ x 2, $C_{110}\times C_{11}\times D_{11}$, $C_{110}\times C_{11}:D_{11}$, $C_{11}^2:C_{220}$

Order 21296: $C_{11}^3:\SD_{16}$, $C_{11}^3:\SD_{16}$

Order 19965: $C_{11}^2:C_{165}$ x 7

Order 15972: $C_{11}^2:C_{132}$ x 7, $C_{11}\wr C_3:C_4$, $C_{11}^2:D_{66}$, $C_{11}^3:D_6$, $C_{11}\wr C_3:C_4$

Order 14641: $C_{11}^4$

Order 14520: $C_{11}\times C_{1320}$, $(C_{11}\times C_{110}).D_6$, $C_{11}^2:C_{120}$, $C_{132}.C_{110}$, $C_{55}\times D_{132}$, $C_5\times C_{11}^2:D_{12}$

Order 13310: $C_{11}^2\times C_{110}$ x 77, $C_{11}\times C_{55}\times D_{11}$ x 11, $C_5\times C_{11}^3:C_2$ x 6, $C_{11}^2:C_{110}$ x 5, $C_{11}^2\times C_{110}$ x 2, $C_{55}\times C_{11}\times D_{11}$, $C_{55}\times C_{11}:D_{11}$

Order 10648: $C_{11}^2:C_{88}$ x 7, $C_{11}^3:Q_8$ x 2, $C_{11}^2:D_{44}$, $C_{11}^3:Q_8$, $C_{11}^2:D_{44}$, $C_{11}^3:D_4$

Order 9680: $C_{88}:C_{110}$, $C_5\times C_{11}^2:\SD_{16}$

Order 7986: $C_{11}^2:C_{66}$ x 7, $C_{11}^2:D_{33}$, $C_{11}\wr S_3$

Order 7260: $C_{11}\times C_{660}$, $C_{33}:C_{220}$, $C_{55}\times D_{66}$, $C_{10}\times C_{11}^2:S_3$, $(C_{11}\times C_{110}).S_3$, $C_{11}^2:C_{60}$

Order 6655: $C_{11}^2\times C_{55}$ x 79

Order 5808: $C_{264}:C_{22}$, $(C_{11}\times C_{22}).D_{12}$

Order 5324: $C_{11}^2:C_{44}$ x 11, $C_{11}^2\times C_{11}:C_4$ x 8, $C_{11}\times C_{22}\times D_{11}$ x 6, $C_2\times C_{11}^3:C_2$ x 5, $C_{11}^3:C_4$ x 3, $C_{22}\times C_{11}\times D_{11}$, $C_{22}\times C_{11}:D_{11}$, $C_{11}\times C_{11}\times C_{11}:C_4$, $C_{11}^2:D_{22}$

Order 4840: $C_{44}.C_{110}$ x 2, $C_{55}\times D_{22}:C_2$, $C_{11}\times C_{440}$, $C_5\times C_{11}^2:Q_8$, $C_5\times C_{11}^2:D_4$, $C_{55}\times D_{44}$, $C_{110}.D_{22}$, $C_{110}.D_{22}$, $C_{11}^2:C_{40}$

Order 3993: $C_{11}\wr C_3$ x 7

Order 3630: $C_{11}\times C_{330}$, $C_{55}\times D_{33}$, $C_5\times C_{11}^2:S_3$, $C_{11}^2:C_{30}$

Order 2904: $C_{11}\times C_{264}$, $C_{132}.C_{22}$, $C_{11}\times D_{132}$, $C_{11}^2:D_{12}$, $(C_{11}\times C_{22}).D_6$, $C_{11}^2:C_{24}$

Order 2662: $C_{11}^2\times C_{22}$ x 79, $C_{11}^2:C_{22}$ x 12, $D_{11}\times C_{11}^2$ x 12

Order 2640: $C_{55}\times C_8:S_3$, $C_5\times C_{33}:\SD_{16}$

Order 2420: $C_{11}:C_{220}$ x 84, $D_{11}\times C_{110}$ x 71, $C_5\times C_{11}\times C_{11}:C_4$ x 4, $C_{10}\times C_{11}\times D_{11}$ x 3, $C_{11}\times C_{220}$ x 2, $C_{22}\times C_{110}$, $C_{10}\times C_{11}:D_{11}$, $C_5\times C_{11}^2:C_4$, $C_{11}^2:C_{20}$

Order 1936: $C_{88}:C_{22}$, $C_{11}^2:\SD_{16}$

Order 1815: $C_{11}\times C_{165}$, $C_{11}^2:C_{15}$

Order 1452: $C_{11}\times D_{66}$, $C_{11}^2:D_6$, $C_{11}\times C_{132}$, $C_{33}:C_{44}$, $C_{11}^2:C_{12}$, $(C_{11}\times C_{22}).S_3$

Order 1331: $C_{11}^3$ x 79

Order 1320: $C_{1320}$ x 7, $C_5\times D_{132}$, $C_{165}:Q_8$, $D_{12}\times C_{55}$, $C_{165}:Q_8$

Order 1210: $C_{11}\times C_{110}$ x 2

Order 968: $C_{11}^2:C_8$

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

Order 605: $C_{11}\times C_{55}$ x 2

Order 484: $C_{11}^2:C_4$

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

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

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

Order 110: $C_{110}$ x 2

Order 55: $C_{55}$ x 2

Order 22: $C_{22}$ x 2

Order 11: $C_{11}$ x 2

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 3513840: $(C_{11}^2\times C_{110}).D_{132}$ ($C_1$)

Order 1756920: $(C_{11}^3\times C_{110}).D_6$ ($C_2$), $C_{11}^4:C_{120}$ ($C_2$), $C_5\times C_{11}^4:D_{12}$ ($C_2$)

Order 878460: $C_{11}^4:C_{60}$ ($C_2^2$)

Order 702768: $(C_{11}^2\times C_{22}).D_{132}$ ($C_5$)

Order 585640: $C_{11}^4:C_{40}$ ($S_3$)

Order 439230: $C_{11}^4:C_{30}$ ($D_4$)

Order 351384: $(C_{11}^2\times C_{22}).D_{66}$ ($C_{10}$), $C_{11}^3:C_{264}$ ($C_{10}$), $C_{11}^4:D_{12}$ ($C_{10}$)

Order 319440: $C_{11}^3.C_{30}.D_4$ ($C_{11}$)

Order 292820: $C_{11}^4:C_{20}$ ($D_6$)

Order 219615: $C_{11}^4:C_{15}$ ($\SD_{16}$)

Order 175692: $C_{11}^4:C_{12}$ ($C_2\times C_{10}$)

Order 159720: $(C_{11}\times C_{110}).D_{66}$ ($C_{22}$), $C_{11}^2:C_{1320}$ ($C_{22}$), $C_{11}^2:C_{1320}$ ($D_{11}$), $C_5\times C_{11}^2:D_{132}$ ($C_{22}$)

Order 146410: $C_{11}^3\times C_{110}$ ($D_{12}$)

Order 117128: $C_{11}^3:C_{88}$ ($C_5\times S_3$)

Order 87846: $C_{11}^4:C_6$ ($C_5\times D_4$)

Order 79860: $C_{11}^2:C_{660}$ ($C_2\times C_{22}$), $C_{11}^2:C_{660}$ ($D_{22}$)

Order 73205: $C_{11}^3\times C_{55}$ ($C_{24}:C_2$)

Order 63888: $(C_{11}\times C_{22}).D_{132}$ ($C_{55}$)

Order 58564: $C_{11}^4:C_4$ ($S_3\times C_{10}$)

Order 53240: $C_{11}^2:C_{440}$ ($D_{33}$), $C_{11}^2:C_{440}$ ($S_3\times C_{11}$)

Order 43923: $C_{11}^4:C_3$ ($C_5\times \SD_{16}$)

Order 39930: $C_{11}^2:C_{330}$ ($D_4\times C_{11}$), $C_{11}^2:C_{330}$ ($D_{44}$)

Order 31944: $(C_{11}\times C_{22}).D_{66}$ ($C_{110}$), $C_{11}^2:D_{132}$ ($C_{110}$), $C_{11}^2:C_{264}$ ($C_{110}$), $C_{11}^2:C_{264}$ ($C_5\times D_{11}$)

Order 29282: $C_{11}^3\times C_{22}$ ($C_5\times D_{12}$)

Order 26620: $C_{11}^2:C_{220}$ ($D_{66}$), $C_{11}^2:C_{220}$ ($S_3\times C_{22}$)

Order 19965: $C_{11}^2:C_{165}$ ($C_{88}:C_2$), $C_{11}^2:C_{165}$ ($C_{11}\times \SD_{16}$)

Order 15972: $C_{11}^2:C_{132}$ ($C_2\times C_{110}$), $C_{11}^2:C_{132}$ ($C_5\times D_{22}$)

Order 14641: $C_{11}^4$ ($C_{120}:C_2$)

Order 14520: $C_{11}^2:C_{120}$ ($C_{11}\times D_{11}$)

Order 13310: $C_{11}^2\times C_{110}$ ($D_{132}$), $C_{11}^2\times C_{110}$ ($C_{11}\times D_{12}$)

Order 10648: $C_{11}^2:C_{88}$ ($C_5\times D_{33}$), $C_{11}^2:C_{88}$ ($S_3\times C_{55}$)

Order 7986: $C_{11}^2:C_{66}$ ($D_4\times C_{55}$), $C_{11}^2:C_{66}$ ($C_5\times D_{44}$)

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

Order 6655: $C_{11}^2\times C_{55}$ ($C_{264}:C_2$), $C_{11}^2\times C_{55}$ ($C_{24}:C_{22}$)

Order 5324: $C_{11}^2:C_{44}$ ($C_5\times D_{66}$), $C_{11}^2:C_{44}$ ($S_3\times C_{110}$)

Order 4840: $C_{11}^2:C_{40}$ ($C_{11}\times D_{33}$)

Order 3993: $C_{11}\wr C_3$ ($C_{440}:C_2$), $C_{11}\wr C_3$ ($\SD_{16}\times C_{55}$)

Order 3630: $C_{11}^2:C_{30}$ ($C_{11}\times D_{44}$)

Order 2904: $C_{11}^2:C_{24}$ ($D_{11}\times C_{55}$)

Order 2662: $C_{11}^2\times C_{22}$ ($C_5\times D_{132}$), $C_{11}^2\times C_{22}$ ($D_{12}\times C_{55}$)

Order 2420: $C_{11}^2:C_{20}$ ($C_{11}\times D_{66}$)

Order 1815: $C_{11}^2:C_{15}$ ($C_{88}:C_{22}$)

Order 1452: $C_{11}^2:C_{12}$ ($D_{11}\times C_{110}$)

Order 1331: $C_{11}^3$ ($C_{24}:C_{110}$), $C_{11}^3$ ($C_{1320}:C_2$)

Order 1210: $C_{11}\times C_{110}$ ($C_{11}^2:D_{12}$), $C_{11}\times C_{110}$ ($C_{11}\times D_{132}$)

Order 968: $C_{11}^2:C_8$ ($C_{55}\times D_{33}$)

Order 726: $C_{11}^2:C_6$ ($C_{55}\times D_{44}$)

Order 605: $C_{11}\times C_{55}$ ($C_{264}:C_{22}$), $C_{11}\times C_{55}$ ($(C_{11}\times C_{22}).D_{12}$)

Order 484: $C_{11}^2:C_4$ ($C_{55}\times D_{66}$)

Order 363: $C_{11}^2:C_3$ ($C_{88}:C_{110}$)

Order 242: $C_{11}\times C_{22}$ ($C_{11}^2:(C_5\times D_{12})$), $C_{11}\times C_{22}$ ($C_{55}\times D_{132}$)

Order 121: $C_{11}^2$ ($C_{1320}:C_{22}$), $C_{11}^2$ ($(C_{11}\times C_{110}).D_{12}$)

Order 110: $C_{110}$ ($C_{11}^3:D_{12}$) x 2

Order 55: $C_{55}$ ($(C_{11}\times C_{22}).D_{132}$), $C_{55}$ ($(C_{11}^2\times C_{22}).D_{12}$)

Order 22: $C_{22}$ ($C_{11}^3.C_{60}.C_2$), $C_{22}$ ($C_{11}^2.C_{132}.C_{10}$)

Order 11: $C_{11}$ ($C_{11}^2.C_{24}:C_{110}$), $C_{11}$ ($C_{11}^3.C_{30}.D_4$)

Order 10: $C_{10}$ ($C_{11}^3.C_{132}.C_2$)

Order 5: $C_5$ ($(C_{11}^2\times C_{22}).D_{132}$)

Order 2: $C_2$ ($C_{11}^3.C_{132}.C_{10}$)

Order 1: $C_1$ ($(C_{11}^2\times C_{110}).D_{132}$)

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 1 larger groups in the database.

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

Character theory

Complex character table

The $15125 \times 15125$ character table is not available for this group.

Rational character table

The $458 \times 458$ rational character table is not available for this group.