Abstract group 193261200.a: $C_{11}^4.\GL(2,11)$

• Label: 193261200.a
• Order: \( 2^{4} \cdot 3 \cdot 5^{2} \cdot 11^{5} \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \cdot 11^{2} \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 5 \cdot 11 \)
• $\card{Z(G)}$: 110
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3 \cdot 5^{2} \cdot 11^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 5 \)
• Perm deg.: not computed
• Trans deg.: $13200$
• Rank: $2$

Group information

Description:	$C_{11}^4.\GL(2,11)$
Order:	\(193261200\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \cdot 11^{5} \)
Exponent:	\(14520\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11^{2} \)
Automorphism group:	Group of order \(140553600\)\(\medspace = 2^{7} \cdot 3 \cdot 5^{2} \cdot 11^{4} \)
Composition factors:	$C_2$ x 2, $C_5$, $C_{11}$ x 4, $\PSL(2,11)$
Derived length:	$1$

This group is nonabelian and nonsolvable.

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	121	132	165	220	242	264	330	440	605	660	1210	1320
Elements	1	15973	13310	13310	159724	13310	26620	543052	14640	26620	53240	53240	1931280	53240	53240	1597200	106480	1597200	19224960	106480	1597200	3194400	65224320	212960	1756920	3194400	6388800	6388800	1756920	6388800	6388800	12777600	7027680	12777600	7027680	25555200	193261200
Conjugacy classes	1	2	1	1	14	1	2	38	131	2	4	4	251	4	4	120	8	120	1724	8	120	240	4604	16	121	240	480	480	121	480	480	960	484	960	484	1920	14630
Divisions	1	2	1	1	4	1	1	10	15	1	1	1	27	1	1	7	1	7	46	1	7	7	118	1	14	7	7	7	14	7	7	7	14	7	14	7	375

Minimal presentations

Permutation degree:	not computed
Transitive degree:	$13200$
Rank:	$2$
Inequivalent generating pairs:	not computed

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

Constructions

Permutation group:	Degree $172$ $\langle(1,6)(2,14)(3,20)(4,27)(5,34)(7,28)(8,50)(9,40)(10,41)(11,36)(12,63)(13,66) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 120 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 56 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 81 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 87 & 35 \\ 86 & 36 \end{array}\right), \left(\begin{array}{rr} 36 & 120 \\ 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/121\Z)$
Direct product:	$C_5$ $\, \times\, $ $C_{11}$ $\, \times\, $ $((C_{11}^2\times C_{22}).\PGL(2,11))$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_{11}^4$ . $\GL(2,11)$	$(C_{11}^4.\SL(2,11))$ . $C_{10}$	$(C_{11}^3.\GL(2,11))$ . $C_{11}$	$C_{11}$ . $(C_{11}^3.\GL(2,11))$	all 22
Aut. group:	$\Aut(C_{121}^2)$	$\Aut(C_{121}\times C_{242})$

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

Homology

Abelianization:	$C_{110} \simeq C_{2} \times C_{5} \times C_{11}$
Schur multiplier:	$C_1$
Commutator length:	not computed

Subgroups

There are 24 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 $C_{11}^3.\SL(2,11)$
Frattini:	a subgroup isomorphic to $C_{11}^2\times C_{22}$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}^4.C_{11}$

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 13310 (order at least 14520) are shown, as well as all normal subgroups of any index.

Order 193261200: $C_{11}^4.\GL(2,11)$

Order 96630600: $C_{11}^3.(C_{55}\times \SL(2,11))$

Order 38652240: $(C_{11}^3\times C_{22}).\PGL(2,11)$

Order 19326120: $C_{11}^4.\SL(2,11)$

Order 17569200: $C_{11}^3.\GL(2,11)$

Order 16105100: $C_{11}^3.(C_{110}\times F_{11})$

Order 8784600: $(C_{11}^2\times C_{55}).\SL(2,11)$, $C_5\times C_{11}^4.\SL(2,5)$

Order 8052550: $C_{11}^3.(C_{55}\times F_{11})$ x 2, $C_{11}^3.C_{55}:C_{110}$

Order 4026275: $C_{11}^3.C_{55}:C_{55}$

Order 3513840: $(C_{11}^2\times C_{22}).\PGL(2,11)$, $(C_{11}^2\times C_{110}).D_{132}$, $C_5\times C_{11}^4:\GL(2,3)$

Order 3221020: $C_{11}^3.(C_{22}\times F_{11})$ x 5, $C_{11}^3.C_{11}.C_{110}.C_2$

Order 2928200: $C_{11}^4:C_{10}\wr C_2$

Order 1756920: $C_{11}^3.\SL(2,11)$, $(C_{11}^3\times C_{110}).D_6$, $C_{11}^4:C_{120}$, $C_5\times C_{11}^4:D_{12}$, $C_5\times C_{11}^4:\SL(2,3)$, $C_{11}^4.\SL(2,5)$

Order 1610510: $C_{11}^4.F_{11}$ x 10, $C_{11}^3.C_{11}:C_{110}$ x 5, $C_{10}:(C_{11}^4.C_{11})$, $(C_{11}^4.C_{11}):C_{10}$, $(C_{11}\times C_{11}^3.C_{11}):C_{10}$

Order 1464100: $C_{11}^4.C_{10}^2$, $(C_{11}^3\times C_{110}).D_5$, $C_{11}^4:(C_5\times D_{10})$, $C_{121}.C_{110}^2$, $C_{10}\times C_{11}^3.F_{11}$

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 805255: $C_{11}^3.C_{11}:C_{55}$ x 5, $C_5:(C_{11}^4.C_{11})$

Order 798600: $C_5\times C_{11}^3:\SL(2,5)$

Order 732050: $D_{121}:C_{55}^2$ x 2, $C_5\times C_{11}^3.F_{11}$ x 2, $C_2\times C_{11}^2.C_{55}^2$, $C_{242}.C_{55}^2$, $C_{11}^4:(C_5\times C_{10})$, $C_{11}^4:(C_5\times D_5)$, $C_{11}^3.C_{110}:C_5$

Order 702768: $(C_{11}^2\times C_{22}).D_{132}$, $C_{11}^4:\GL(2,3)$

Order 644204: $C_{11}^4.D_{22}$

Order 585640: $(C_{11}^2\times C_{22}).D_{110}$, $C_5\times C_{11}^3:D_{44}$, $C_5\times C_{11}^4:Q_8$, $C_{11}^4:C_{40}$

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

Order 366025: $C_{11}^2.C_{55}^2$, $C_{121}:C_{55}^2$, $C_{11}^3.C_{55}:C_5$

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

Order 322102: $C_{11}^4.D_{11}$ x 2, $C_{11}^4.C_{22}$

Order 319440: $C_{11}^2.C_{24}:C_{110}$, $C_{11}^3.C_{30}.D_4$, $(C_{11}^2\times C_{110}).S_4$

Order 292820: $C_2\times C_{11}^3.F_{11}$ x 5, $C_{11}\times C_{242}:C_{110}$ x 5, $C_{11}^4:(C_2\times C_{10})$ x 3, $C_{605}.C_{22}^2$ x 3, $(C_{11}^2\times C_{22}).D_{55}$, $C_{11}^4:D_{10}$, $C_5\times C_{11}^4:C_2^2$, $C_{11}^4:C_{20}$, $C_{11}^3.(C_5\times D_{22})$

Order 266200: $C_{11}^3:C_{10}\wr C_2$, $C_{11}^3.C_5.C_{20}.C_2$

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 161051: $C_{11}^4.C_{11}$

Order 159720: $C_{11}^2:C_{1320}$ x 7, $C_5\times C_{11}^3:D_{12}$, $C_{11}^2:C_{165}:Q_8$, $C_{11}^3:\SL(2,5)$, $(C_{11}\times C_{110}).D_{66}$, $C_5\times C_{11}^2:D_{132}$, $C_5\times C_{11}^3:\SL(2,3)$

Order 146410: $C_{11}^3.C_{110}$ x 13, $C_{11}^3.F_{11}$ x 10, $C_{11}^3.F_{11}$ x 10, $C_5\times D_{121}:C_{11}^2$ x 6, $C_{11}^3.C_{11}:C_{10}$ x 5, $C_{11}\times C_{121}:C_{110}$ x 5, $C_{11}^3:C_{110}$ x 5, $C_{11}^3:C_{110}$ x 3, $C_{1210}:C_{11}^2$ x 3, $C_5\times C_{11}^3.D_{11}$ x 2, $C_{11}^3:D_{55}$, $C_{11}^3\times C_{110}$, $C_{11}^4:C_{10}$

Order 133100: $C_{1210}:C_{110}$ x 11, $C_{11}^3:C_{10}^2$ x 7, $C_{1210}:C_{110}$, $C_{110}\times C_{11}^2:D_5$, $(C_{11}^2\times C_{110}).D_5$, $C_{11}:C_{110}^2$, $(C_{11}\times C_{110}).D_{55}$, $C_5\times C_{11}^2:D_{110}$

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: $C_{11}^3.C_{55}$ x 13, $C_{11}^3.C_{11}:C_5$ x 5, $C_{11}\times C_{121}:C_{55}$ x 5, $C_{11}^3:C_{55}$ x 3, $C_{605}:C_{11}^2$ x 3, $C_{11}^3\times C_{55}$

Order 66550: $C_{605}:C_{110}$ x 22, $C_{11}\times C_{55}:F_{11}$ x 12, $C_{1210}:C_{55}$ x 11, $C_5\times C_{11}^2:C_{110}$ x 7, $C_{605}:C_{110}$ x 2, $D_{11}:C_{55}^2$ x 2, $C_{1210}.C_{55}$, $C_{22}:C_{55}^2$, $C_{11}^2:D_5\times C_{55}$, $C_5\times C_{11}^2:D_{55}$

Order 63888: $(C_{11}\times C_{22}).D_{132}$, $C_{11}^3:\GL(2,3)$, $(C_{11}^2\times C_{22}).D_{12}$

Order 58564: $C_{121}:C_{22}^2$ x 3, $C_{11}^3.D_{22}$, $C_{11}^4:C_2^2$, $C_{11}^4:C_4$

Order 53240: $C_{11}^2:C_{440}$ x 7, $(C_{11}\times C_{22}).D_{110}$, $(C_{11}^2\times C_{22}).D_{10}$, $C_{110}.D_{11}^2$, $C_{110}.D_{11}^2$, $C_{110}.D_{11}^2$, $C_{110}.D_{11}^2$, $C_{110}.D_{11}^2$

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 33275: $C_{605}:C_{55}$ x 11, $C_{11}^3:C_5^2$ x 7, $C_{605}:C_{55}$, $C_{11}:C_{55}^2$

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

Order 29282: $C_{11}^3.C_{22}$ x 13, $D_{121}:C_{11}^2$ x 6, $C_{242}:C_{11}^2$ x 3, $C_{11}^3.D_{11}$ x 2, $C_{11}^3\times C_{22}$, $C_{11}^3:C_{22}$

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

Order 26620: $C_{242}:C_{110}$ x 55, $C_{1210}:C_{22}$ x 33, $C_{11}\times C_{22}:F_{11}$ x 31, $C_{11}:D_{11}\times C_{110}$ x 7, $C_{11}^2:C_{220}$ x 7, $C_{242}:C_{110}$ x 5, $C_{11}\times C_{22}\times F_{11}$ x 5, $C_{1210}.C_{22}$ x 3, $C_{55}.C_{22}^2$ x 3, $C_{11}^2:C_{220}$ x 2, $(C_{11}\times C_{22}).D_{55}$, $C_{11}^2:D_{110}$, $C_{11}^3:D_{10}$, $(C_{11}^2\times C_{22}).D_5$

Order 24200: $C_{110}\wr C_2$, $(C_{11}\times C_{110}).D_{10}$

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: $C_{11}^3.C_{11}$ x 13, $C_{121}:C_{11}^2$ x 3, $C_{11}^4$

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

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

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

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

Order 1331: $C_{11}^3$

Order 110: $C_{110}$

Order 55: $C_{55}$

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 193261200: $C_{11}^4.\GL(2,11)$ ($C_1$)

Order 96630600: $C_{11}^3.(C_{55}\times \SL(2,11))$ ($C_2$)

Order 38652240: $(C_{11}^3\times C_{22}).\PGL(2,11)$ ($C_5$)

Order 19326120: $C_{11}^4.\SL(2,11)$ ($C_{10}$)

Order 17569200: $C_{11}^3.\GL(2,11)$ ($C_{11}$)

Order 8784600: $(C_{11}^2\times C_{55}).\SL(2,11)$ ($C_{22}$)

Order 3513840: $(C_{11}^2\times C_{22}).\PGL(2,11)$ ($C_{55}$)

Order 1756920: $C_{11}^3.\SL(2,11)$ ($C_{110}$)

Order 146410: $C_{11}^3\times C_{110}$ ($\PGL(2,11)$)

Order 73205: $C_{11}^3\times C_{55}$ ($\SL(2,11):C_2$)

Order 29282: $C_{11}^3\times C_{22}$ ($C_5.\PSL(2,11).C_2$)

Order 14641: $C_{11}^4$ ($\GL(2,11)$)

Order 13310: $C_{11}^2\times C_{110}$ ($C_{11}\times \PSL(2,11).C_2$)

Order 6655: $C_{11}^2\times C_{55}$ ($C_{11}\times C_2.\PSL(2,11).C_2$)

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

Order 1331: $C_{11}^3$ ($C_{11}\times C_{10}.\PSL(2,11).C_2$)

Order 110: $C_{110}$ ($C_{11}^3.\PSL(2,11).C_2$)

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

Order 22: $C_{22}$ ($(C_{11}^2\times C_{55}).\PSL(2,11).C_2$)

Order 11: $C_{11}$ ($C_{11}^3.\GL(2,11)$)

Order 10: $C_{10}$ ($C_{11}^4.\PSL(2,11).C_2$)

Order 5: $C_5$ ($(C_{11}^3\times C_{22}).\PGL(2,11)$)

Order 2: $C_2$ ($(C_{11}^3\times C_{55}).\PSL(2,11).C_2$)

Order 1: $C_1$ ($C_{11}^4.\GL(2,11)$)

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

Character theory

Complex character table

The $14630 \times 14630$ character table is not available for this group.

Rational character table

The $375 \times 375$ rational character table is not available for this group.