Abstract group 2806008480.b: $C_3^{11}.C_2.M_{11}$

• Label: 2806008480.b
• Order: \( 2^{5} \cdot 3^{13} \cdot 5 \cdot 11 \)
• Exponent: \( 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{13} \cdot 5 \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $33$
• Trans deg.: $33$
• Rank: $2$

Group information

Description:	$C_3^{11}.C_2.M_{11}$
Order:	\(2806008480\)\(\medspace = 2^{5} \cdot 3^{13} \cdot 5 \cdot 11 \)
Exponent:	\(3960\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \)
Automorphism group:	Group of order \(2806008480\)\(\medspace = 2^{5} \cdot 3^{13} \cdot 5 \cdot 11 \)
Composition factors:	$C_2$, $C_3$ x 11, $M_{11}$
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	8	9	10	11	12	15	18	22	24	33
Elements	1	551367	3063986	8660520	10392624	239955210	155889360	75057840	280600848	85030560	342090540	270208224	363741840	255091680	545612760	170061120	2806008480
Conjugacy classes	1	3	59	2	1	99	4	30	1	2	29	13	15	2	10	2	273
Divisions	1	3	59	2	1	99	2	28	1	1	23	9	14	1	5	1	250
Autjugacy classes	1	3	59	2	1	99	4	30	1	2	29	13	15	2	10	2	273

Minimal presentations

Permutation degree:	$33$
Transitive degree:	$33$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	22	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $33$ $\langle(1,15,26,7,31,4,10,22,21,29,16)(2,13,25,9,32,6,12,23,19,28,17,3,14,27,8,33,5,11,24,20,30,18) \!\cdots\! \rangle$
Transitive group:	33T120				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:	$(C_3^{11}.M_{11})$ . $C_2$	$(C_3^{11}.C_2)$ . $M_{11}$	$C_3^{11}$ . $(C_2\times M_{11})$	$C_3^5$ . $(C_3^6.(C_2\times M_{11}))$	all 5

Elements of the group are displayed as permutations of degree 33.

Homology

Abelianization:	$C_{2} $
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 7 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_1$
Commutator:	a subgroup isomorphic to $C_3^{11}.M_{11}$
Frattini:	a subgroup isomorphic to $C_3^6$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^9.C_3^4$

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 5832 (order at least 481140) are shown, as well as all normal subgroups of any index.

Order 2806008480: $C_3^{11}.C_2.M_{11}$

Order 1403004240: $C_3^{11}.M_{11}$

Order 255091680: $C_3^{11}.C_2.A_6.C_2$

Order 233834040: $C_3^{10}.\PSL(2,11).S_3$

Order 127545840: $C_3^{11}.A_6.C_2$ x 2, $C_3^{11}.C_2.A_6$

Order 116917020: $C_3^{11}.\PSL(2,11)$

Order 85030560: $C_3^{10}.A_6.C_2^2$

Order 77944680: $C_3^{10}.C_2.\PSL(2,11)$

Order 63772920: $C_3^{11}.A_6$

Order 51018336: $C_3^8.C_3^5.C_4.C_2^3$

Order 42515280: $C_3^{10}.A_6.C_2$ x 2, $C_3^{11}.C_2.S_5$, $C_3^{10}.C_2.A_6$

Order 38972340: $C_3^{10}.\PSL(2,11)$

Order 25509168: $C_3^8.C_3^5.\SD_{16}$ x 2, $C_3^{11}.(C_2\times F_9)$, $C_3^{10}.F_9:S_3$, $C_3^8.C_3^5.C_2^3.C_2$, $C_3^8.C_3^4.D_6.C_2^2$, $C_3^8.C_3^4.C_4.C_6.C_2$

Order 21257640: $C_3^{11}.S_5$ x 2, $C_3^{11}.C_2.A_5$ x 2, $C_3^{10}.A_6$

Order 19486170: $(C_3^{10}.C_{33}):C_{10}$

Order 17006112: $C_3^{10}.Q_8.S_3^2$, $C_3^8.C_3^4.C_4.C_2^3$

Order 14171760: $C_3^{10}.C_2.S_5$, $C_3^9.S_3.S_5$

Order 12754584: $C_3^8.C_3^4.C_{12}.C_2$ x 4, $C_3^{11}.C_3:S_3.C_2^2$, $C_3^{10}.C_3^3.C_8$, $C_3^9.C_3^4.C_8$, $C_3^9.C_3^4.C_2^3$, $C_3^9.C_3^3.D_6.C_2$, $C_3^8.C_3^5.Q_8$, $C_3^8.C_3^5.D_4$

Order 11547360: $C_3^5.(S_3\times M_{11})$

Order 10628820: $C_3^{11}.A_5$ x 2

Order 9743085: $C_3^{10}.C_{11}.C_{15}$

Order 8503056: $C_3^{10}.C_3:S_3.D_4$ x 3, $C_3^{10}.Q_8.C_3.S_3$, $C_3^{10}.Q_8.C_3.C_6$, $C_3^{10}.C_6.A_4.C_2$, $C_3^8.C_6^2.S_3^2$, $C_3^8.C_3^4.\SD_{16}$, $C_3^8.C_3^4.C_4.C_2^2$, $C_3^8.C_3^4.C_2^3.C_2$, $C_3^8.C_3^3.D_6.C_2^2$, $C_3^8.C_3^2.D_6:D_6$, $C_3^6.C_3^5.D_6.C_2^2$

Order 7085880: $C_3^{10}.S_5$ x 4, $C_3^9.S_3.A_5$ x 4, $C_3^9.S_3.A_5$ x 2, $C_3^{10}.C_2.A_5$, $C_3^8.C_5.C_3^3.C_4.C_2$

Order 6495390: $(C_3^{10}.C_{11}):C_{10}$

Order 6377292: $C_3^9.C_3^4.C_4$ x 2, $C_3^9.C_3^4.C_2^2$ x 2, $C_3^{11}.C_3.D_6$, $C_3^{10}.C_3^3.C_4$, $C_3^9.C_3^3.D_6$, $C_3^8.C_3^5.C_4$, $C_3^8.C_3^5.C_2^2$

Order 5773680: $C_3^6:M_{11}$

Order 5668704: $C_3^{11}.C_4.C_2^3$, $C_3^{10}.Q_8.D_6$, $C_3^8.F_9:D_6$

Order 4723920: $C_3^9.C_2.S_5$

Order 4251528: $C_3^{10}.C_3:S_3.C_2^2$ x 5, $C_3^8.C_3^4.Q_8$ x 4, $C_3^9.S_3^3$ x 3, $C_3^8.C_3^4.D_4$ x 3, $C_3^8.C_3^3.C_{12}.C_2$ x 2, $C_3^8.C_3^2.C_6^2.C_2$ x 2, $C_3^{11}.A_4.C_2$, $C_3^{10}.Q_8.C_3^2$, $C_3^{10}.F_9$, $C_3^9.C_3^2.D_6.C_2$, $C_3^8.C_3^4.C_2^2.C_2$, $C_3^8.C_3^3.C_6.C_2^2$, $C_3^7.C_6^2.\He_3.C_2$, $C_3^7.C_6^2.C_3^3.C_2$, $C_3^7.C_3^5.C_2^3$, $C_3^6.C_3^5.D_6.C_2$, $C_3^6.C_3^5.C_{12}.C_2$, $C_3^6.C_3^5.C_3.D_4$, $C_3^6.C_3^4.F_9$

Order 3897234: $C_3^{11}.C_{22}$

Order 3849120: $C_3^5:(C_2\times M_{11})$

Order 3542940: $C_3^{10}.A_5$ x 4, $C_3^8.C_5.C_3^3.C_4$ x 2, $C_3^{10}.A_5$ x 2, $C_3^{10}.A_5$, $C_3^8.C_{15}.C_3.D_6$

Order 3247695: $(C_3^{10}.C_{11}):C_5$

Order 3188646: $C_3^9.C_3^4.C_2$ x 2, $C_3^{11}.C_3.S_3$, $C_3^{11}.C_3.C_6$, $C_3^7.C_3^3.C_3^3.C_2$

Order 3149280: $C_3^7.C_2.A_6.C_2$

Order 2834352: $C_3^{10}.Q_8.S_3$ x 3, $C_3^{11}.\SD_{16}$ x 2, $C_3^8.C_6^2.D_6$ x 2, $C_3^{11}.C_2^3.C_2$, $C_3^{10}.Q_8.C_6$, $C_3^{10}.D_6.C_2^2$, $C_3^{10}.C_4.C_6.C_2$, $C_3^9.C_4.C_6^2$, $C_3^9.(C_2\times F_9)$, $C_3^8.C_3^2.D_6.C_2^2$, $C_3^8.A_4:S_3^2$, $C_3^8.A_4.S_3^2$, $C_3^8.(S_3\times S_3\wr C_2)$, $C_3^7.C_6^2.S_3^2$, $C_3^8.F_9:S_3$, $C_3^8.F_9:C_6$, $C_3^8.S_3^3:C_2$, $C_3^8.C_3^3:\SD_{16}$, $C_3^8.C_3^3:\SD_{16}$, $C_3^8.(S_3\times \PSU(3,2))$, $C_3^8.(S_3\times F_9)$

Order 2361960: $C_3^8:D_5.S_3^2$ x 4, $C_3^9.S_5$ x 2, $C_3^9.C_2.A_5$, $C_3^{10}.C_{10}.C_4$, $C_3^9.C_2.A_5$

Order 2125764: $C_3^9.C_3^3.C_4$ x 11, $C_3^{10}.S_3^2$ x 6, $C_3^8.C_3^4.C_2^2$ x 5, $C_3^{10}.C_3.D_6$ x 3, $C_3^8.C_3^4.C_4$ x 3, $C_3^8.C_3^3.D_6$ x 3, $C_3^7.C_3^5.C_2^2$ x 3, $C_3^9.C_3^3.C_2^2$ x 2, $C_3^9.C_3^2.D_6$ x 2, $C_3^8.C_3^3.C_6.C_2$ x 2, $C_3^7.C_3^4.C_6.C_2$ x 2, $C_3^{11}.D_6$, $C_3^{11}.C_6.C_2$, $C_3^{10}.(C_6\times S_3)$, $C_3^8.C_6^2.C_3^2$, $C_3^7.C_3^4.D_6$, $C_3^7.C_3^3.C_6^2$, $C_3^6.C_3^5.D_6$, $C_3^5.C_3^5.(C_6\times S_3)$

Order 1948617: $C_3^{10}.C_{33}$

Order 1924560: $C_3^5:M_{11}$

Order 1889568: $C_3^{10}.C_4.C_2^3$ x 5, $C_3^8.Q_8.S_3^2$, $C_3^7.C_3^3.C_4.C_2^3$, $C_3^8.F_9:C_2^2$

Order 1771470: $C_3^{11}.C_{10}$, $C_3^8.C_{15}.C_3.C_6$, $C_3^8.C_3^2:D_{15}$

Order 1594323: $C_3^9.C_3^4$

Order 1574640: $C_3^7.A_6.C_2$ x 4, $C_3^7.C_2.A_6$

Order 1417176: $C_3^7.C_6^2.C_3.C_6$ x 12, $C_3^{10}.A_4.C_2$ x 9, $C_3^8.S_3^3$ x 8, $C_3^9.C_3.D_6.C_2$ x 3, $C_3^8.C_3^3.Q_8$ x 3, $C_3^8.C_3^3.D_4$ x 3, $C_3^4.\He_3^2:(C_4\times S_3)$ x 3, $C_3^{11}.C_8$ x 2, $C_3^{10}.Q_8.C_3$ x 2, $C_3^{10}.C_6.C_2^2$ x 2, $C_3^8.C_3^3.C_2^3$ x 2, $C_3^8.C_3:S_3.C_6.C_2$ x 2, $C_3^6.C_3^5.C_2^3$ x 2, $C_3^6.C_3^5.C_2^2.C_2$ x 2, $C_3^8.S_3^3$ x 2, $C_3^8.C_3^3:Q_8$ x 2, $C_3^{11}.D_4$, $C_3^{11}.C_2^3$, $C_3^{10}.D_6.C_2$, $C_3^{10}.C_{12}.C_2$, $C_3^9.C_6^2.C_2$, $C_3^9.C_3^2.C_4.C_2$, $C_3^9.C_2.C_6^2$, $C_3^8.C_6.C_6^2$, $C_3^7.C_6^2.C_3.S_3$, $C_3^7.C_3^4.C_4.C_2$, $C_3^7.C_3^4.C_2^3$, $C_3^6.C_2^2.C_9.C_3^3.C_2$, $C_3^6.C_2^2.C_3^4.C_6$, $C_3^8.S_3^2:C_6$, $C_3^8.C_3^3:Q_8$, $C_3^9.F_9$, $C_3^8.(C_3\times F_9)$, $C_3^8.C_3^2:D_{12}$, $C_3^8.S_3^2:S_3$

Order 1299078: $C_3^{10}.C_{22}$

Order 1180980: $C_3^8.C_{15}:D_6$ x 9, $C_3^5.C_3\wr F_5$ x 8, $C_3^9.A_5$, $C_3^9.A_5$, $C_3^5.C_3\wr F_5$, $C_3^2.C_3^8:F_5$

Order 1062882: $C_3^{10}.C_3.S_3$ x 12, $C_3^8.C_3^4.C_2$ x 12, $C_3^9.C_3^3.C_2$ x 11, $C_3^{10}.C_3.C_6$ x 9, $C_3^{11}.C_6$ x 3, $C_3^7.C_3^5.C_2$ x 2, $C_3^8.\He_3.C_6$, $C_3^8.C_3^3.C_6$

Order 1049760: $C_3^6:(A_6.C_2^2)$ x 3, $C_3^6.C_2.A_6.C_2$ x 2

Order 962280: $C_3^5:(S_3\times \PSL(2,11))$

Order 944784: $C_3^9.D_6.C_2^2$ x 12, $C_3^9.C_6.D_4$ x 11, $C_3^9.C_4.C_6.C_2$ x 8, $C_3^7.C_6^2.D_6$ x 7, $C_3^8.D_6:D_6$ x 6, $C_3^7.A_4.C_3.D_6$ x 4, $C_3^{10}.C_4.C_2^2$ x 3, $C_3^8.(C_2\times F_9)$ x 3, $C_3^6.C_6^2.S_3^2$ x 2, $C_3^3.C_3^4.C_3^3.\SD_{16}$ x 2, $C_3^8.Q_8.C_3.S_3$, $C_3^8.Q_8.C_3.C_6$, $C_3^8.C_6.A_4.C_2$, $C_3^8.C_4.C_6^2$, $C_3^8.C_3:S_3.D_4$, $C_3^7.F_9:S_3$, $C_3^7.C_3^2.C_4.C_6.C_2$, $C_3^6.A_4.C_3^2.D_6$, $C_3^3.C_3^5.C_4:S_3^2$, $C_3^3.C_3^5.C_3.D_6.C_2^2$, $C_3^8.F_9:C_2$, $C_3^8.F_9:C_2$, $C_3^8.F_9:C_2$, $C_3^8.F_9:C_2$, $C_3^8.(C_2\times \PSU(3,2))$, $C_3^8.S_3^2:C_2^2$, $C_3^8.(C_2\times F_9)$

Order 885735: $C_3^8.C_5.C_3^3$

Order 787320: $C_3^8.(S_3\times F_5)$ x 5, $C_3^7.A_6$ x 4

Order 708588: $C_3^9.(C_6\times S_3)$ x 32, $C_3^{10}.D_6$ x 31, $C_3^{10}.C_6.C_2$ x 16, $C_3^8.C_6^2.C_3$ x 16, $C_3^7.C_3^3.D_6$ x 16, $C_3^8.C_3^3.C_2^2$ x 15, $C_3^9.S_3^2$ x 15, $C_3^8.C_3.C_6^2$ x 14, $C_3^6.C_3^4.D_6$ x 11, $C_3^8.C_3^2.D_6$ x 7, $C_3^7.C_3^4.C_2^2$ x 7, $C_3^9.S_3^2$ x 6, $C_3^9.C_3^2.C_4$ x 4, $C_3^8.C_3^3.C_4$ x 4, $C_3^5.C_3^5.C_6.C_2$ x 3, $C_3^8.C_3^2:C_{12}$ x 3, $C_3^8.C_3^3:C_4$ x 3, $C_3^8.C_3:S_3^2$ x 3, $C_3^9.(C_6\times S_3)$ x 3, $C_3^{11}.C_4$ x 2, $C_3^9.C_6^2$ x 2, $C_3^9.C_3.D_6$ x 2, $C_3^7.C_6^2.C_3^2$ x 2, $C_3^9.S_3^2$ x 2, $C_3^8.(C_3\times S_3^2)$ x 2, $C_3^6.\He_3.C_6^2$, $C_3^6.C_3^5.C_2^2$, $C_3^6.C_3^4.C_6.C_2$, $C_3^5.C_3^5.D_6$, $C_3^5.C_3^4.C_6^2$

Order 649539: $C_3^{10}.C_{11}$

Order 629856: $C_3^9.C_4.C_2^3$ x 11, $C_3^7.(S_3\times \GL(2,3))$ x 4, $C_3^8.Q_8.D_6$, $C_3^5.C_3^4.C_4.C_2^3$

Order 590490: $C_3^5.C_3\wr D_5$ x 9, $C_3^8.C_3:D_{15}$ x 9, $C_3^{10}.C_{10}$ x 9

Order 531441: $C_3^9.C_3^3$ x 14, $C_3^{11}.C_3$

Order 524880: $C_3^6.A_6.C_2$ x 6, $C_3^5:A_6.C_6$ x 6, $C_3^5.(S_3\times A_6)$ x 6, $C_3^5.A_6.S_3$ x 4, $C_3^6.C_2.A_6$ x 3, $C_3\wr A_5.D_6$

Order 481140: $C_3^6:\PSL(2,11)$

Order 354294: $C_3^{11}.C_2$

Order 177147: $C_3^{11}$

Order 729: $C_3^6$

Order 243: $C_3^5$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 2806008480: $C_3^{11}.C_2.M_{11}$ ($C_1$)

Order 1403004240: $C_3^{11}.M_{11}$ ($C_2$)

Order 354294: $C_3^{11}.C_2$ ($M_{11}$)

Order 177147: $C_3^{11}$ ($C_2\times M_{11}$)

Order 729: $C_3^6$ ($C_3^2:(C_3^3:C_2).M_{11}$)

Order 243: $C_3^5$ ($C_3^6.(C_2\times M_{11})$)

Order 1: $C_1$ ($C_3^{11}.C_2.M_{11}$)

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 1 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

The $273 \times 273$ character table is not available for this group.

Rational character table

The $250 \times 250$ rational character table is not available for this group.