Abstract group 79833600.d: $C_2^2\times A_{11}$

• Label: 79833600.d
• Order: \( 2^{9} \cdot 3^{4} \cdot 5^{2} \cdot 7 \cdot 11 \)
• Exponent: \( 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: 4
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{5} \cdot 5^{2} \cdot 7 \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \)
• Perm deg.: $15$
• Trans deg.: $44$
• Rank: $2$

Group information

Description:	$C_2^2\times A_{11}$
Order:	\(79833600\)\(\medspace = 2^{9} \cdot 3^{4} \cdot 5^{2} \cdot 7 \cdot 11 \)
Exponent:	\(27720\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Automorphism group:	$S_3\times S_{11}$, of order \(239500800\)\(\medspace = 2^{9} \cdot 3^{5} \cdot 5^{2} \cdot 7 \cdot 11 \)
Composition factors:	$C_2$ x 2, $A_{11}$
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	7	8	9	10	11	12	14	15	18	20	21	22	30	42
Elements	1	73263	142010	1829520	809424	8603430	237600	9979200	2217600	4424112	3628800	11642400	3564000	887040	6652800	3991680	1900800	10886400	2661120	5702400	79833600
Conjugacy classes	1	11	3	12	2	29	1	4	1	10	2	12	7	2	3	4	2	6	6	6	124
Divisions	1	11	3	12	2	29	1	4	1	10	1	12	7	2	3	4	1	3	6	3	116
Autjugacy classes	1	5	3	6	2	13	1	2	1	4	1	6	3	2	1	2	1	1	2	1	58

Dimension	1	10	44	45	110	120	126	132	165	210	231	252	330	385	462	550	594	660	693	825	924	990	1100	1155	1188	1232	1320	1540	2310
Irr. complex chars.	4	4	4	4	4	4	8	4	4	4	4	0	4	4	4	4	12	4	4	4	4	8	4	4	0	4	4	4	4	124
Irr. rational chars.	4	4	4	4	4	4	0	4	4	4	4	4	4	4	4	4	4	4	4	4	4	8	4	4	4	4	4	4	4	116

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$44$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $15$ $\langle(1,3,4)(2,5,9,7,6)(12,14)(13,15), (1,2,4,7,5,8,9,11,3,6,10)(12,13)(14,15)\rangle$
Transitive group:	44T1260				more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$A_{11}$ . $C_2^2$	$C_2^2$ . $A_{11}$	$(C_2\times A_{11})$ . $C_2$ (3)	$C_2$ . $(C_2\times A_{11})$ (3)	more information

Elements of the group are displayed as permutations of degree 15.

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 10 normal subgroups (4 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 $C_2^2$
Commutator:	a subgroup isomorphic to $A_{11}$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed

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 53760 (order at least 1485) are shown, as well as all normal subgroups of any index.

Order 79833600: $C_2^2\times A_{11}$

Order 39916800: $C_2\times A_{11}$ x 3

Order 19958400: $A_{11}$

Order 7257600: $C_2^2\times A_{10}$

Order 3628800: $C_2\times A_{10}$ x 3

Order 1814400: $A_{10}$

Order 1451520: $C_2^2\times S_9$

Order 725760: $C_2\times S_9$ x 6, $C_2^2\times A_9$

Order 483840: $(C_2\times C_6).S_8$

Order 362880: $S_9$ x 4, $C_2\times A_9$ x 3

Order 241920: $C_6:S_8$ x 6, $C_2\times C_6\times A_8$, $(C_2^2\times A_4).S_7$

Order 181440: $A_9$

Order 172800: $C_2^2\times A_5:S_6$

Order 161280: $C_2^2\times S_8$

Order 120960: $C_2\times A_4:S_7$ x 6, $C_3:S_8$ x 4, $C_6\times A_8$ x 3, $C_2^2\times A_4\times A_7$

Order 86400: $C_2\times A_5:S_6$ x 6, $C_2^2\times A_5\times A_6$

Order 80640: $C_2\times S_8$ x 6, $C_2^4:S_7$, $C_2^2\times A_8$

Order 60480: $A_4:S_7$ x 4, $C_2\times A_4\times A_7$ x 3, $C_2\times C_6:S_7$, $C_3\times A_8$

Order 57600: $A_5^2:C_4\times C_2^2$

Order 43200: $A_5:S_6$ x 4, $C_2\times A_5\times A_6$ x 3

Order 40320: $C_2^3:S_7$ x 12, $S_8$ x 4, $C_2\times A_8$ x 3, $C_2^4\times A_7$, $C_2^3.S_7$, $C_2^3\times S_7$

Order 34560: $C_2^2\times A_4:S_6$

Order 31680: $C_2^2\times M_{11}$ x 2

Order 30240: $C_6:S_7$ x 6, $C_2\times C_6\times A_7$, $A_4\times A_7$

Order 28800: $C_2\times A_5^2:C_4$ x 6, $A_5^2:C_2^3$ x 2, $C_2\times D_{10}.S_6$

Order 21600: $A_5\times A_6$

Order 20160: $C_2^2:S_7$ x 16, $C_2^2\times S_7$ x 10, $C_2^3\times A_7$ x 7, $C_2\times A_7:C_4$ x 6, $A_8$

Order 17280: $C_2\times A_4:S_6$ x 6, $(C_6^2\times A_5):D_4$, $C_2\times D_6\times S_6$, $C_2^2\times A_4\times A_6$

Order 16128: $C_6\times C_2^4:\GL(3,2)$

Order 15840: $C_2\times M_{11}$ x 6

Order 15120: $C_3:S_7$ x 4, $C_6\times A_7$ x 3

Order 14400: $A_5^2:C_2^2$ x 12, $D_{10}.S_6$ x 6, $A_5^2:C_4$ x 4, $C_2^2\times A_5^2$ x 2, $C_2\times D_{10}\times A_6$

Order 13824: $C_6:S_4^2:C_2^2$

Order 11520: $C_2^2\times S_4\times S_5$ x 2, $C_2^4:S_6$

Order 10080: $C_2\times S_7$ x 16, $C_2^2\times A_7$ x 12, $A_7:C_4$ x 4

Order 8640: $D_6\times S_6$ x 12, $(C_6\times \GL(2,4)):D_4$ x 12, $A_4:S_6$ x 4, $C_2\times A_4\times A_6$ x 3, $A_5:D_6^2$ x 2, $C_6^2:C_4\times A_5$, $C_2\times D_6\times A_6$, $C_2\times C_6:S_6$, $C_2\times C_6\times S_6$

Order 8064: $C_6\times C_2^3:\GL(3,2)$ x 3, $C_2^2\times A_4\times \GL(3,2)$

Order 7920: $M_{11}$ x 2

Order 7680: $C_2^6:S_5$

Order 7560: $C_3\times A_7$

Order 7200: $\PSOPlus(4,5)$ x 8, $C_2\times A_5^2$ x 6, $D_{10}\times A_6$ x 6, $A_6:F_5$ x 4, $C_2\times C_{10}\times A_6$

Order 6912: $(C_6\times A_4^2):D_4$ x 12, $C_2^6:C_3^3:C_4$, $C_2\times C_6:S_4^2$, $C_6\times A_4^2:C_2^3$

Order 6048: $C_2^2\times {}^2G(2,3)$

Order 5760: $C_2\times S_4\times S_5$ x 24, $C_2^3:S_6$ x 12, $C_2^2\times A_4:S_5$ x 3, $C_2^2\times A_4\times S_5$ x 2, $C_2^2\times S_4\times A_5$ x 2, $C_2^4\times A_6$, $C_2^3.S_6$, $C_2^3\times S_6$

Order 5376: $C_2^5:\GL(3,2)$

Order 5184: $C_2^2\times S_3\wr S_3$

Order 5040: $C_2\times A_7$ x 7, $S_7$ x 4

Order 4800: $C_2\times D_{10}.S_5$ x 2

Order 4608: $(D_6\times C_2^4):S_4$, $S_4^2:C_2^3$

Order 4320: $C_2\times A_5:S_3^2$ x 18, $A_5:\SOPlus(4,2)$ x 16, $S_3\times S_6$ x 16, $(C_6\times \GL(2,4)):C_4$ x 6, $C_6:S_6$ x 6, $D_6\times A_6$ x 6, $C_6\times S_6$ x 6, $C_6^2:S_5$ x 2, $C_6:D_6\times A_5$, $C_2\times C_6\times A_6$, $A_4\times A_6$

Order 4032: $C_2\times A_4\times \GL(3,2)$ x 3, $C_2\times C_6:\PGL(2,7)$, $C_3\times C_2^3:\GL(3,2)$

Order 3840: $C_2\wr S_5$ x 6, $C_2^2\times D_4\times S_5$, $C_2^6:A_5$

Order 3600: $D_5\times A_6$ x 4, $C_{10}\times A_6$ x 3, $A_5^2$ x 2

Order 3456: $(C_3\times A_4^2):D_4$ x 16, $C_6:S_4^2$ x 9, $C_6\times \POPlus(4,3)$ x 9, $(C_6\times A_4^2):C_4$ x 6, $A_4^2:C_2^2\times C_6$, $A_4^2:C_2^2\times C_6$, $C_2^2\times C_6^2:D_{12}$, $C_2\times A_4^2:D_6$

Order 3024: $\SL(2,8):C_6$ x 3

Order 2880: $S_4\times S_5$ x 32, $C_2\times A_4:S_5$ x 18, $C_2^2:S_6$ x 16, $C_2\times S_4\times A_5$ x 12, $C_2\times A_4\times S_5$ x 12, $C_2^2\times S_6$ x 10, $C_2^3\times A_6$ x 7, $C_2^2.S_6$ x 6, $C_2^4:\GL(2,4)$ x 3, $\GL(2,4):C_2^4$ x 3, $(D_5\times C_6^2).D_4$, $A_6.C_2^3$

Order 2688: $C_2^4:\GL(3,2)$ x 3, $C_2^4\times \GL(3,2)$

Order 2640: $C_2^2\times \PSL(2,11)$ x 2

Order 2592: $S_3^3:D_6$ x 12, $C_2^2\times C_3^3:S_4$, $C_2^2\times C_3^3:S_4$, $C_2^2\times S_3\wr C_3$

Order 2520: $A_7$

Order 2400: $D_{10}.S_5$ x 12, $C_2\times D_{10}\times A_5$ x 2

Order 2304: $(C_2^3\times D_6):S_4$ x 12, $S_4^2:C_2^2$ x 12, $C_2^2\times S_4^2$ x 2, $C_6\times C_2^4:S_4$, $(D_6\times C_2^4):A_4$, $(C_2^4\times C_6).S_4$, $C_2^4:D_6^2$, $C_6\times C_2^4:S_4$, $A_4^2:C_4\times C_2^2$, $C_2^4:D_6^2$

Order 2160: $A_5:S_3^2$ x 20, $\GL(2,4):D_6$ x 12, $\GL(2,4):D_6$ x 6, $C_3^2:C_4\times A_5$ x 4, $C_3:S_6$ x 4, $S_3\times A_6$ x 4, $C_3\times S_6$ x 4, $C_6\times A_6$ x 3, $C_6^2\times A_5$

Order 2016: $C_6:\PGL(2,7)$ x 6, $C_2\times C_6\times \GL(3,2)$ x 2, $F_8:C_6^2$, $C_{14}:(C_6\times S_4)$, $C_2^2\times \SL(2,8)$, $A_4\times \GL(3,2)$

Order 1920: $C_2\times D_4\times S_5$ x 24, $C_2^4:S_5$ x 4, $C_2\wr A_5$ x 3, $C_2^4:S_5$ x 3, $C_2^4\times S_5$ x 2, $C_2^2\times F_5\times S_4$ x 2, $C_2^2\times D_4\times A_5$, $C_2^2\times C_4:S_5$, $C_4\times C_2^2\times S_5$

Order 1800: $C_5\times A_6$

Order 1728: $C_2\times C_6^2:D_{12}$ x 12, $C_3\times \POPlus(4,3)$ x 10, $C_3:S_4^2$ x 10, $C_6\times \PSOPlus(4,3)$ x 6, $A_4\wr C_2\times C_6$ x 6, $A_4^2:D_6$ x 6, $(C_3\times A_4^2):C_4$ x 4, $A_4:D_6^2$ x 2, $C_2^6:C_3^3$, $A_4\times C_6^2:C_4$, $D_6^2:D_6$, $C_6^2:\GL(2,3)$

Order 1600: $C_{10}^2:\OD_{16}$

Order 1536: $C_2^6:S_4$, $(C_2^3\times D_{12}):D_4$

Order 1512: ${}^2G(2,3)$

Order 4: $C_2^2$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 79833600: $C_2^2\times A_{11}$ ($C_1$)

Order 39916800: $C_2\times A_{11}$ ($C_2$) x 3

Order 19958400: $A_{11}$ ($C_2^2$)

Order 4: $C_2^2$ ($A_{11}$)

Order 2: $C_2$ ($C_2\times A_{11}$) x 3

Order 1: $C_1$ ($C_2^2\times A_{11}$)

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

Supergroups

This group is a maximal subgroup of 5 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 $124 \times 124$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $116 \times 116$ rational character table.