Abstract group 193616640.a: $C_7^5:(C_2\wr S_5\times C_3)$

• Label: 193616640.a
• Order: \( 2^{8} \cdot 3^{2} \cdot 5 \cdot 7^{5} \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{2} \cdot 5 \cdot 7^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $35$
• Trans deg.: $35$
• Rank: $2$

Group information

Description:	$C_7^5:(C_2\wr S_5\times C_3)$
Order:	\(193616640\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5 \cdot 7^{5} \)
Exponent:	\(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Automorphism group:	$C_7^5:(C_2\wr S_5\times C_3)$, of order \(193616640\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5 \cdot 7^{5} \)
Composition factors:	$C_2$ x 6, $C_3$, $C_7$ x 5, $A_5$
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	7	8	10	12	14	15	21	24	28	30	35	42	56	84
Elements	1	127967	421694	2642080	921984	30951634	16806	4609920	6453888	42833840	5099010	12907776	3645600	16134720	14164920	12907776	5531904	23872800	3457440	6914880	193616640
Conjugacy classes	1	11	5	10	1	43	19	2	1	26	85	2	15	4	24	2	1	22	1	3	278
Divisions	1	11	3	10	1	25	19	2	1	14	85	1	14	2	24	1	1	19	1	2	237
Autjugacy classes	1	11	5	10	1	43	19	2	1	26	85	2	15	4	24	2	1	22	1	3	278

Minimal presentations

Permutation degree:	$35$
Transitive degree:	$35$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	30	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $35$ $\langle(1,3,5)(2,7,6)(8,28,21,32,9,25,16,35,12,27,15,33,13,22,17,31,10,26,18,34,14,23,19,30) \!\cdots\! \rangle$
Transitive group:	35T223				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_7^5$ $\,\rtimes\,$ $(C_2\wr S_5\times C_3)$				more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_7^5.C_2^5.S_5)$ . $C_3$	$(C_7^5.C_6.C_2^4)$ . $S_5$	$(C_7^5:C_2\wr A_5)$ . $C_6$	$(C_7^5:C_3)$ . $(C_2\wr S_5)$	all 16
Aut. group:	$\Aut(C_7^5:(C_2^4:A_5))$	$\Aut(C_7^5:C_2\wr A_5)$	$\Aut(C_7^5:(C_2^4:S_5))$	$\Aut(C_7^5:(C_2^4:S_5))$	all 9

Elements of the group are displayed as permutations of degree 35.

Homology

Abelianization:	$C_{2} \times C_{6} \simeq C_{2}^{2} \times C_{3}$
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 19 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 $C_7^5:(C_2^4:A_5)$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^5$

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 315 (order at least 614656) are shown, as well as all normal subgroups of any index.

Order 193616640: $C_7^5:(C_2\wr S_5\times C_3)$

Order 96808320: $C_7^5:(C_3\times C_2^4:S_5)$, $C_7^5:(C_3\times C_2^4:S_5)$, $C_7^5:(C_2^5:\GL(2,4))$

Order 64538880: $C_7^5.C_2^5.S_5$

Order 48404160: $C_7^5:(C_2^4:\GL(2,4))$

Order 38723328: $C_7^4.C_2^3.(A_4\times F_7).C_2^2$

Order 32269440: $C_7^5:(C_2^4:S_5)$, $C_7^5:(C_2^4:S_5)$, $C_7^5:C_2\wr A_5$, $C_7^5.C_2^4.(C_6\times F_5)$

Order 19361664: $C_7^4.C_2^3.(A_4\times F_7).C_2$ x 3, $C_7^4.C_2.C_2^4.C_{21}.C_6.C_2$ x 2, $C_7^5.A_4.C_6.C_2^4$, $C_7^4.Q_8.(A_4\times F_7).C_2$, $C_7^4.C_{14}.C_2^4.C_6^2$

Order 16134720: $C_7^5:(C_2^4:A_5)$, $C_7^5:(C_3\times C_2^4:F_5)$, $C_7^5:(C_3\times C_2^4:F_5)$, $D_7\wr D_5:C_3$

Order 12907776: $C_7^4.C_{14}.C_2^3.C_6.C_2^3$, $C_7^4.C_2\wr A_4.C_2\times D_7$

Order 12101040: $C_7^5:(C_6\times S_5)$

Order 10756480: $D_7\wr F_5$

Order 9680832: $C_7^5.A_4.C_6.C_2^3$ x 11, $C_7^4.C_{14}.C_2^4.C_3.C_6$ x 3, $C_7^4.C_2.C_2^4.C_{21}.C_6$ x 2, $C_7^3.C_{14}^2.C_6^2.C_2^2$ x 2, $C_7^5.C_2^3.C_6^2.C_2$, $C_7^4.Q_8.A_4:F_7$, $C_7^4.C_2^3.A_4:F_7$, $C_7^3.C_{14}^2.C_3.C_6.C_2^3$

Order 8067360: $C_7^5:(C_3\times C_2^4:D_5)$, $C_7^5:(C_3\times C_2^4:D_5)$, $D_7\wr C_5:C_3$

Order 6453888: $C_7^4.C_{14}.C_2^3.C_6.C_2^2$ x 8, $C_7^5.C_2^3.C_6.C_2^3$ x 4, $C_7^4.C_2^3.(D_7\times S_4)$ x 2, $C_7^5.C_2^2.C_6.C_2^4$, $C_7^5.C_2.C_6.C_2^5$, $C_7^5.A_4.D_4.C_2^2$, $C_7^5.(C_6\times D_4).C_2^3$, $C_7^4.Q_8.S_4\times D_7$, $C_7^4.C_{14}.C_2^4.C_6.C_2$, $C_7^4.C_2^3.S_4\times D_7$, $C_7^4.C_2\wr A_4\times D_7$, $C_7^4.C_2.C_2^4.D_{42}$, $C_7\times C_7^4.C_2\wr A_4.C_2$

Order 6050520: $C_7^5:(C_3\times S_5)$, $C_7^5:(C_3\times S_5)$, $C_7^5:(C_6\times A_5)$

Order 5531904: $C_2\times C_7^4.C_2^3.(C_6\times S_4)$

Order 5378240: $C_7^5:(C_2^4:F_5)$, $C_7^5:(C_2^4:F_5)$, $D_7\wr D_5$

Order 4840416: $C_7^3.C_{14}^2.C_6^2.C_2$ x 20, $C_7^5.A_4.C_6.C_2^2$ x 8, $C_7^5.C_2^3.C_6^2$ x 4, $C_7^5.A_4.C_{12}.C_2$ x 4, $C_7^3.C_{14}^2.(C_{12}\times S_3)$ x 4, $C_7^5.C_2^2.C_6^2.C_2$ x 2, $C_7^5.C_6^2.C_2^3$, $C_7^5.(C_{12}\times A_4).C_2$, $C_7^4.Q_8.(S_3\times F_7)$, $C_7^4.C_{14}.C_2^4.C_3^2$, $C_7^3.C_{14}^2.C_3^2.C_2^3$

Order 4302592: $D_7\times C_7^4.D_4^2.C_2$

Order 4033680: $C_7^5:(C_2\times S_5)$, $C_7^5:(C_2^4:C_{15})$

Order 3226944: $C_7^5.C_2^3.C_6.C_2^2$ x 10, $C_7^5.(C_3\times D_4^2)$ x 10, $C_7^5.(D_4\times S_4)$ x 8, $C_7^5.(C_3\times C_2\wr C_4)$ x 8, $C_7^5.C_2.C_6.C_2^4$ x 7, $C_7^5.(C_3\times D_4:D_4)$ x 6, $C_7^5.C_2^2.C_6.C_2^3$ x 4, $C_7^5.(C_6\times D_4).C_2^2$ x 4, $C_7^4.C_{14}.C_2^4.C_6$ x 4, $C_7^5.C_6.C_2^5$ x 3, $C_7^3.C_{14}^2.C_6.C_2^3$ x 3, $C_7^5.A_4.C_2^3.C_2$ x 2, $C_7^5.(D_4\times A_4).C_2$ x 2, $C_7^4.C_2^3.C_7:S_4$ x 2, $C_7\times C_7^4.C_2^3.S_4$ x 2, $C_7^5.C_{12}.C_2^4$, $C_7^5.C_6.C_4.C_2^3$, $C_7^5.C_4.C_6.C_2^3$, $C_7^5.A_4.C_4.C_2^2$, $C_7^4.C_2^3.A_4\times D_7$, $C_7\times C_7^4.C_2\wr A_4$

Order 3025260: $C_7^5:\GL(2,4)$

Order 2765952: $C_7^4.C_2^3.(C_6\times S_4)$ x 4, $C_2\times C_7^4.C_2^3.A_4.C_6$ x 3

Order 2689120: $C_7^5:(C_2^4:D_5)$, $C_7^5:(C_2^4:D_5)$, $D_7\wr C_5$

Order 2420208: $C_7^3.C_{14}^2.(C_6\times S_3)$ x 36, $C_7^5.C_6^2.C_2^2$ x 11, $C_7^5.C_2^2.C_6^2$ x 8, $C_7^5.C_6.C_6.C_2^2$ x 2, $C_7^5.A_4.C_6.C_2$ x 2, $C_7^5.(C_{12}\times A_4)$ x 2, $C_7^3.C_{14}^2.C_3.C_{12}$ x 2, $C_7^5.C_3.C_6.C_2^3$, $C_7^5.(C_6\times D_{12})$, $C_7^4.Q_8.C_{21}.C_6$, $C_7^4.C_{14}.A_4.C_6$, $C_7^4.C_2.A_4:F_7$

Order 2151296: $C_7^4.C_{14}.C_2\wr C_2^2$ x 4, $C_7^4.C_7:D_4^2.C_2$ x 2, $C_7^4.C_2\wr C_2^2\times D_7$ x 2, $D_7\times C_7^4.C_2\wr C_4$, $C_7^5.C_2^4.C_2^3$, $C_7^4.D_4:D_4\times D_7$, $C_7^4.C_4^2:C_4\times D_7$, $C_7^4.C_2\wr C_4\times D_7$, $C_7^4.(C_7\times D_4^2).C_2$, $C_7\times C_7^4.D_4^2.C_2$

Order 2016840: $C_7^5:S_5$, $C_7\wr S_5$, $C_7^5:(C_2\times A_5)$, $C_7^5:(C_6\times F_5)$

Order 1843968: $C_2\times C_7^4.C_2\wr A_4.C_2$, $C_2\times C_7^4.(C_3\times D_4^2).C_2$

Order 1728720: $C_7^4:(C_6\times S_5)$

Order 1613472: $C_7^5.C_6.C_2^4$ x 46, $C_7^5.(C_6\times D_4).C_2$ x 29, $C_7^5.C_2.C_6.C_2^3$ x 27, $C_7^3.C_{14}^2.C_6.C_2^2$ x 18, $C_7^5.A_4.D_4$ x 12, $C_7^5.C_{12}.C_2^3$ x 8, $C_7^5.(D_4\times A_4)$ x 8, $C_7^5.(C_{12}\times D_4)$ x 8, $C_7^5.C_2.C_6.C_4.C_2$ x 5, $C_7^5.C_{12}.C_4.C_2$ x 4, $C_7^5.C_6.C_4.C_2^2$ x 4, $C_7^5.A_4.C_2^3$ x 4, $C_7^3.C_{14}^2.C_{12}.C_2$ x 4, $C_7^5.C_6.C_2^3.C_2$ x 2, $C_7^5.C_6.C_2.C_2^3$ x 2, $C_7^5.A_4.C_4.C_2$ x 2, $C_7^3.(C_7\times A_4).C_7.C_2^3$ x 2, $C_7^5.C_3.C_4^2.C_2$, $C_7^5.(C_6\times D_8)$, $C_7^4.Q_8.S_3\times D_7$, $C_7^4.C_{14}.C_2^4.C_3$, $C_7^3.(D_7\times S_4)\times D_7$, $C_7\times C_7^4.C_2^3.A_4$

Order 1382976: $D_7\wr A_4:C_3$ x 2, $C_7:D_7^3:(C_3\times S_4)$ x 2, $C_7^4:(C_3\times Q_8:S_4)$ x 2, $C_7^3.(A_4\times F_7).C_2^3$, $C_2\times C_7^4.C_2^3.A_4.C_3$, $C_2\times C_7^3.(A_4\times F_7).C_2^2$

Order 1344560: $C_7^5:(C_2^4:C_5)$

Order 1210104: $C_7^5.C_3^2.C_2^3$ x 16, $C_7^3.C_{14}^2.C_3.C_6$ x 12, $C_7^5.C_6^2.C_2$ x 11, $C_7^5.(C_6\times A_4)$ x 7, $C_7^5.C_6.C_6.C_2$ x 4, $C_7^5.(C_{12}\times S_3)$ x 4, $C_7^5.A_4.C_6$ x 3, $C_7^4.D_{42}:C_6$ x 2, $C_7^4.D_6:F_7$ x 2, $C_7^5.C_3.C_{12}.C_2$, $C_7^5.C_2.C_6^2$, $C_7^5.(C_6\times C_{12})$, $C_7^4.D_{84}:C_3$, $C_7^4.C_{14}.A_4.C_3$

Order 1075648: $C_7^5.D_4^2$ x 9, $C_7^5.C_2\wr C_2^2$ x 8, $C_7^5.C_2^3.C_2^3$ x 6, $C_7^5.C_2\wr C_4$ x 6, $C_7^5.D_4:D_4$ x 5, $C_7^5.C_4^2:C_4$ x 3, $C_7^4.C_2^3.C_2^2\times D_7$ x 3, $C_7^5.C_2^5.C_2$ x 2, $C_7\times C_7^4.C_2\wr C_4$ x 2, $C_7\times C_7^4.C_2\wr C_2^2$ x 2, $D_7\times C_7^4.C_4\wr C_2$, $D_7\times C_7^4.C_4.C_2^3$, $C_7^5.C_2.C_2^5$, $C_7^4.C_2^4.C_2\times D_7$, $C_7^4.C_2.C_2^4\times D_7$, $C_7^4.(C_7\times D_4^2)$, $C_7^4.(C_2^3.C_4)\times D_7$, $C_7\times C_7^4.D_4:D_4$, $C_7\times C_7^4.C_4^2:C_4$

Order 1008420: $C_7\wr A_5$, $C_7\wr C_5:C_{12}$, $C_7\wr C_5:C_{12}$, $C_7^5:(C_3\times D_{10})$

Order 921984: $D_7\wr D_4:C_3$ x 8, $D_7\wr S_4$ x 4, $C_2\times C_7^4.C_2^3.C_6.C_2^2$ x 2, $C_2\times C_7^4.C_2\wr A_4$ x 2, $C_7^4.C_2^3.C_6.C_2^3$, $C_7^4.C_2.C_6.C_2^5$, $C_2\times C_7^4.Q_8.S_4$, $C_2\times C_7^4.C_2^3.S_4$, $C_2\times C_7^4.(C_6\times D_4).C_2^2$, $C_2\times C_7^4.(C_3\times D_4:D_4)$, $C_2\times C_7^4.(C_3\times C_2\wr C_4)$

Order 864360: $C_7^4:(C_3\times S_5)$, $C_7^4:(C_3\times S_5)$, $C_7^4:(C_6\times A_5)$

Order 806736: $C_7^5.C_6.C_2^3$ x 172, $C_7^3.C_{14}^2.D_6$ x 22, $C_7^5.A_4.C_2^2$ x 17, $C_7^5.C_6.C_2^2.C_2$ x 15, $C_7^5.C_6.C_4.C_2$ x 10, $C_7^3.(C_7\times A_4).C_{14}.C_2$ x 10, $C_7^5.A_4.C_4$ x 4, $C_7^4.(C_2^3.F_7)$ x 4, $C_7^3.C_{14}^2.C_6.C_2$ x 4, $C_7^5.C_{12}.C_2^2$ x 3, $C_7^5.C_4.C_6.C_2$ x 3, $C_7^5.C_3.C_4^2$ x 3, $C_7^5.(C_3\times D_8)$ x 3, $C_7^5.D_{12}.C_2$ x 2, $C_7^4.D_{21}:D_4$ x 2, $C_7^4.D_4:F_7$ x 2, $D_7\times C_7^3.C_7:S_4$, $C_7^4.Q_8:F_7$, $C_7^4.Q_8.D_{21}$, $C_7^4.Q_8.C_3\times D_7$, $C_7^4.D_{56}:C_3$, $C_7^4.C_{42}.C_2^3$, $C_7^4.C_8:F_7$, $C_7^4.C_2.D_7:A_4$, $C_7^4.A_4.C_2\times D_7$, $C_7^4.(D_4.F_7)$, $C_7^3.A_4.C_{14}^2$, $C_7^3.A_4.C_{14}\times D_7$, $C_7\times C_7^4.Q_8.S_3$, $C_7\times C_7^3.(D_7\times S_4)$

Order 691488: $D_7^3.(S_3\times F_7)$ x 18, $C_7^3.C_2^2.C_7.C_6^2.C_2$ x 4, $C_7^3.(A_4\times F_7).C_2^2$ x 2, $C_2\times C_7^3.C_2^2.(S_3\times F_7)$ x 2, $C_2\times C_7^3.A_4:F_7.C_2$ x 2, $C_2\times C_7^3.(A_4\times F_7).C_2$ x 2, $C_7^4.C_6^2.C_2^3$, $C_7^4.C_2^2.C_6^2.C_2$, $C_2\times C_7^4.Q_8.C_3.C_6$, $C_2\times C_7^4.C_2^2.C_6^2$, $C_2\times C_7^3.(C_7\times A_4).C_6.C_2$, $C_7:D_7^3:(C_3\times A_4)$

Order 672280: $C_7^5:(C_2\times F_5)$

Order 614656: $C_2\times C_7^4.D_4^2.C_2$

Order 537824: $C_7^5.C_2^5$

Order 268912: $C_7^5.C_2^4$

Order 100842: $C_7^5:C_6$

Order 50421: $C_7^5:C_3$

Order 33614: $C_7^5:C_2$

Order 16807: $C_7^5$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 193616640: $C_7^5:(C_2\wr S_5\times C_3)$ ($C_1$)

Order 96808320: $C_7^5:(C_3\times C_2^4:S_5)$ ($C_2$), $C_7^5:(C_3\times C_2^4:S_5)$ ($C_2$), $C_7^5:(C_2^5:\GL(2,4))$ ($C_2$)

Order 64538880: $C_7^5.C_2^5.S_5$ ($C_3$)

Order 48404160: $C_7^5:(C_2^4:\GL(2,4))$ ($C_2^2$)

Order 32269440: $C_7^5:(C_2^4:S_5)$ ($C_6$), $C_7^5:(C_2^4:S_5)$ ($C_6$), $C_7^5:C_2\wr A_5$ ($C_6$)

Order 16134720: $C_7^5:(C_2^4:A_5)$ ($C_2\times C_6$)

Order 1613472: $C_7^5.C_6.C_2^4$ ($S_5$)

Order 806736: $C_7^5.C_6.C_2^3$ ($C_2\times S_5$)

Order 537824: $C_7^5.C_2^5$ ($C_3\times S_5$)

Order 268912: $C_7^5.C_2^4$ ($C_6\times S_5$)

Order 100842: $C_7^5:C_6$ ($C_2^4:S_5$)

Order 50421: $C_7^5:C_3$ ($C_2\wr S_5$)

Order 33614: $C_7^5:C_2$ ($C_3\times C_2^4:S_5$)

Order 16807: $C_7^5$ ($C_2\wr S_5\times C_3$)

Order 1: $C_1$ ($C_7^5:(C_2\wr S_5\times C_3)$)

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 2 larger groups in the database.

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

Character theory

Complex character table

The $278 \times 278$ character table is not available for this group.

Rational character table

The $237 \times 237$ rational character table is not available for this group.