Abstract group 16134720.d: $C_7^5:(C_2^4:A_5)$

• Label: 16134720.d
• Order: \( 2^{6} \cdot 3 \cdot 5 \cdot 7^{5} \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 1 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{2} \cdot 5 \cdot 7^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \)
• Perm deg.: $35$
• Trans deg.: $35$
• Rank: $2$

Group information

Description:	$C_7^5:(C_2^4:A_5)$
Order:	\(16134720\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \cdot 7^{5} \)
Exponent:	\(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
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 4, $C_7$ x 5, $A_5$
Derived length:	$0$

This group is nonabelian and perfect (hence nonsolvable).

Group statistics

Order	1	2	3	4	5	6	7	14	21	28	35	42
Elements	1	15435	3920	432180	921984	576240	16806	1245090	1340640	2593080	5531904	3457440	16134720
Conjugacy classes	1	3	1	2	2	3	76	91	90	9	12	9	299
Divisions	1	3	1	2	1	2	19	21	13	2	1	2	68
Autjugacy classes	1	3	1	2	1	2	19	21	13	2	1	2	68

Dimension	1	3	4	5	6	10	15	16	20	30	40	48	60	64	80	90	96	120	160	180	240	320	360	384	480	576	720	960	1440	1920	2880	3840
Irr. complex chars.	1	2	1	4	0	11	1	6	1	15	15	12	0	6	75	0	0	6	30	0	42	48	0	0	20	0	0	3	0	0	0	0	299
Irr. rational chars.	1	0	1	2	1	3	1	0	1	1	0	0	1	0	0	1	1	3	0	2	6	0	2	1	10	1	6	8	6	5	3	1	68

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	10	10	30
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $35$ $\langle(1,9,20,31,22)(2,14,15,33,25)(3,8,21,34,27)(4,10,17,30,28)(5,11,19,32,24) \!\cdots\! \rangle$
Transitive group:	35T164				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_7^5.C_2^4)$ . $A_5$				more information

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

Homology

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

Subgroups

There are 4 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_7^5:(C_2^4:A_5)$
Commutator:	$G' \simeq$ $C_7^5:(C_2^4:A_5)$	$G/G' \simeq$ $C_1$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_7^5:(C_2^4:A_5)$
Fitting:	$\operatorname{Fit} \simeq$ $C_7^5$	$G/\operatorname{Fit} \simeq$ $C_2^4:A_5$
Radical:	$R \simeq$ $C_7^5.C_2^4$	$G/R \simeq$ $A_5$
Socle:	$\operatorname{soc} \simeq$ $C_7^5$	$G/\operatorname{soc} \simeq$ $C_2^4:A_5$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\wr C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
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 20580 (order at least 784) are shown, as well as all normal subgroups of any index.

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

Order 3226944: $C_7^4.C_{14}.C_2^4.C_6$

Order 2689120: $C_7^5:(C_2^4:D_5)$

Order 1613472: $C_7^5.A_4.D_4$, $C_7^4.C_2.C_2^4.C_{21}$

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

Order 1075648: $C_7^5.C_2\wr C_2^2$

Order 1008420: $C_7\wr A_5$

Order 806736: $C_7^3.C_{14}^2.D_6$, $C_7^3.C_{14}^2.C_6.C_2$, $C_7^5.A_4.C_2^2$

Order 537824: $C_7^5.C_2^4.C_2$, $C_7^5.C_2^3.C_2^2$, $C_7\times C_7^4.C_2.C_2^4$

Order 460992: $D_7\wr A_4$

Order 403368: $C_7:D_7\times C_7^3:A_4$, $C_7^3.C_{14}^2.C_6$, $C_7\times C_7^4.A_4.C_2$, $C_7\times C_7^4.Q_8.C_3$, $C_7^3.(C_7\times A_4).C_{14}$, $C_7^5.D_6.C_2$

Order 268912: $C_7^5.C_2^4$, $C_7^5.C_2^3.C_2$, $C_7\times C_7^4.D_4:C_2$, $C_7^5.C_2^3.C_2$, $C_7^4.C_{14}.C_2^3$, $C_7\times C_7^4.C_2^3.C_2$, $C_7^5.C_2^3.C_2$

Order 230496: $C_7:D_7^3:A_4$

Order 201684: $C_7^3.C_{14}^2.C_3$ x 2, $C_7^5.C_6.C_2$, $C_7^5.C_6.C_2$, $C_7^4.C_{42}.C_2$

Order 168070: $C_7^5:D_5$

Order 153664: $D_7\wr C_2^2$

Order 144060: $C_7^4:A_5$

Order 134456: $C_7^5.C_2^3$ x 2, $C_7^4.(C_7\times D_4)$ x 2, $C_7\times C_7^4:C_2^3$, $C_7\times C_7^4:D_4$, $C_7^4.(C_7\times D_4)$, $C_7^5.C_2^2.C_2$, $C_7\times C_7^4:(C_2\times C_4)$, $C_7^4.D_{14}.C_2$, $C_7^5.D_4$, $C_7^5.C_2^3$, $C_7\times C_7^4:Q_8$

Order 115248: $D_7\times C_7^3:S_4$, $D_7\times D_7\wr C_3$

Order 100842: $C_7^5:S_3$, $C_7^5:C_6$, $C_7^5:C_6$

Order 84035: $C_7\wr C_5$

Order 76832: $C_7:D_7^3:C_2^2$, $D_7^2\wr C_2$, $C_7:D_7^3:C_4$

Order 67228: $C_7\times C_7^4:C_2^2$ x 2, $C_7^5:C_2^2$ x 2, $C_7^5:C_4$, $C_7^5:C_4$, $C_7^5:C_2^2$, $C_7^5:C_2^2$

Order 57624: $C_7^3:A_4\times D_7$ x 3, $C_7^4:(C_2\times A_4)$ x 2, $D_7^3:C_{21}$ x 2, $C_7^4.D_6.C_2$, $C_7^4:\SL(2,3)$, $C_7\wr S_4$, $C_7^4:S_4$

Order 50421: $C_7^2\times C_7^3:C_3$

Order 38416: $C_7^2\wr C_2:D_4$ x 2, $C_7^4.C_2^4$, $C_7^4.C_2^3.C_2$, $C_7^2\wr C_2.D_4$, $D_7^2:D_7^2$, $C_7^4:D_4:C_2$, $C_7^2\wr C_2.D_4$

Order 33614: $C_7^2\times C_7^3:C_2$, $C_7\times C_7^4:C_2$, $C_7^5:C_2$

Order 32928: $C_7^3.A_4.D_4$

Order 28812: $C_7\wr A_4$ x 8, $C_7^4:(C_3:C_4)$, $C_7^4:(C_2\times C_6)$, $C_7^4:(C_2\times C_6)$, $C_7^4:D_6$, $C_7^4:D_6$, $C_7^3:C_6\times D_7$

Order 24010: $C_7^4:D_5$

Order 19208: $C_7^3:D_{28}$ x 5, $C_7^4:C_2^3$ x 3, $C_7^4:D_4$ x 3, $C_7:D_7^3$ x 3, $C_7^4:C_2^3$ x 2, $C_7^4:(C_2\times C_4)$ x 2, $C_7\wr D_4$ x 2, $C_7^4:(C_2\times C_4)$, $C_7^4:D_4$, $C_7^4:C_2^3$, $C_7^4:D_4$, $C_7\times C_7^3:C_2^3$, $C_7^4:Q_8$, $C_7^4:(C_2\times C_4)$

Order 16807: $C_7^5$

Order 16464: $C_7^3.A_4.C_4$, $C_2\times C_7^3:S_4$, $D_7^3:C_6$

Order 14406: $C_7^3:D_{21}$ x 5, $C_7^3:C_{42}$ x 5, $C_7^3:C_{42}$ x 3, $C_7^4:C_6$ x 2, $C_7\times C_7:C_3\times C_7:D_7$, $C_7^4:S_3$, $C_7^4:C_6$, $C_7:(C_7:C_3)\times C_7:D_7$

Order 12005: $C_7^4:C_5$

Order 10976: $C_7^3.C_2^4.C_2$

Order 9604: $C_7\wr C_2^2$ x 13, $C_7^4:C_2^2$ x 7, $C_7^4:C_2^2$ x 6, $C_7\wr C_4$ x 5, $C_7^4:C_2^2$ x 3, $C_7^4:C_2^2$ x 2, $C_7^4:C_4$, $C_7^4:C_2^2$, $C_7\times C_7^3:C_2^2$, $C_7^4:C_4$

Order 8232: $C_7\times C_7^2:\SL(2,3)$ x 2, $D_7\wr C_3$ x 2, $C_7^3:(C_3:D_4)$, $C_7^3:(C_3:D_4)$, $C_2\times C_7^3:A_4$, $C_7^3:S_4$

Order 7203: $C_7^3:C_{21}$ x 13, $C_7^3\times C_7:C_3$

Order 5488: $C_7\times C_7^2:(C_2\times D_4)$ x 2, $C_7^3:(C_2^2:C_4)$, $C_7^3:(C_2^2:C_4)$, $C_7^3:(C_2^2:C_4)$, $C_7^3:(C_2\times D_4)$, $C_7^3:(C_2\times D_4)$, $C_2\times D_7^2\times D_7$, $D_7^3:C_2$

Order 4802: $C_7^2\wr C_2$ x 20, $C_7\times C_7^3:C_2$ x 13, $C_7^3\times D_7$ x 6, $C_7^4:C_2$

Order 4704: $C_7^2.A_4.D_4$

Order 4116: $D_7\times C_7:F_7$ x 2, $C_7^3:A_4$ x 2, $C_7^3:(C_2\times C_6)$, $C_7^3:(C_3:C_4)$, $C_7^3:D_6$, $C_7^3:(C_2\times C_6)$, $C_7\wr C_3:C_4$, $C_7^3:D_6$, $C_7^3:D_6$, $C_2\times C_7^3:C_6$, $D_7\times C_7:F_7$

Order 2744: $D_7^2:C_{14}$ x 10, $C_7^3:D_4$ x 7, $D_7^3$ x 6, $C_7\times C_2\times D_7^2$ x 4, $C_7\times C_7:D_7.C_2^2$ x 2, $C_7\times C_2\times C_7:D_7.C_2$ x 2, $C_7^3:(C_2\times C_4)$, $C_7^3:(C_2\times C_4)$, $C_7^3:C_2^3$, $C_7^3:D_4$, $C_{14}:D_7^2$, $C_7^3:D_4$, $C_7^2:C_4\times D_7$, $C_7^2:D_{28}$

Order 2401: $C_7^4$ x 19

Order 2352: $C_7^2:(C_2\times S_4)$, $C_7^2:(C_2^2\times A_4)$, $C_7^2:(A_4:C_4)$

Order 2058: $C_7^2:C_{42}$ x 10, $C_7\wr S_3$ x 5, $C_7^2:C_{42}$ x 5, $C_7^3:C_6$ x 5, $C_7^3:C_6$ x 3, $C_7^3:C_6$ x 2, $C_7^2:C_{42}$ x 2, $C_7^3:S_3$, $C_7:D_7\times C_7:C_3$, $C_7^2:D_{21}$, $C_7^2:C_{42}$, $C_7^2:C_{42}$

Order 1568: $D_{14}\wr C_2$

Order 1372: $C_7\times D_7^2$ x 40, $C_7:D_7^2$ x 15, $C_7^2:C_{28}$ x 12, $C_7:D_7^2$ x 10, $C_7^3:C_4$ x 5, $D_{14}\times C_7^2$ x 4, $C_7^2:D_{14}$ x 2, $C_7^2:D_{14}$, $C_7^2:C_{28}$

Order 1344: $(C_2^2\times D_{14}):A_4$

Order 1176: $C_{14}^2:C_6$ x 2, $C_7^2:\SL(2,3)$ x 2, $(C_7\times C_{14}).D_6$, $C_{14}^2:C_6$, $C_7^2:S_4$, $D_7^2:S_3$

Order 1029: $C_7^2:C_{21}$ x 21, $C_7\wr C_3$ x 13, $C_7^2\times C_{21}$

Order 960: $C_2^4:A_5$

Order 784: $D_7^2:C_2^2$ x 3, $D_7^2:C_4$ x 2, $D_{14}^2$, $D_{14}:D_{14}$, $D_{14}:D_{14}$, $C_{14}^2:C_4$

Order 64: $C_2\wr C_2^2$

Order 5: $C_5$

Order 3: $C_3$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 20580 (order at least 784) are shown, as well as all normal subgroups of any index.

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

Order 3226944: $C_7^4.C_{14}.C_2^4.C_6$

Order 2689120: $C_7^5:(C_2^4:D_5)$

Order 1613472: $C_7^5.A_4.D_4$, $C_7^4.C_2.C_2^4.C_{21}$

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

Order 1075648: $C_7^5.C_2\wr C_2^2$

Order 1008420: $C_7\wr A_5$

Order 806736: $C_7^3.C_{14}^2.D_6$, $C_7^3.C_{14}^2.C_6.C_2$, $C_7^5.A_4.C_2^2$

Order 537824: $C_7^5.C_2^4.C_2$, $C_7^5.C_2^3.C_2^2$, $C_7\times C_7^4.C_2.C_2^4$

Order 460992: $D_7\wr A_4$

Order 403368: $C_7:D_7\times C_7^3:A_4$, $C_7^3.C_{14}^2.C_6$, $C_7\times C_7^4.A_4.C_2$, $C_7\times C_7^4.Q_8.C_3$, $C_7^3.(C_7\times A_4).C_{14}$, $C_7^5.D_6.C_2$

Order 268912: $C_7^5.C_2^4$, $C_7^5.C_2^3.C_2$, $C_7\times C_7^4.D_4:C_2$, $C_7^5.C_2^3.C_2$, $C_7^4.C_{14}.C_2^3$, $C_7\times C_7^4.C_2^3.C_2$, $C_7^5.C_2^3.C_2$

Order 230496: $C_7:D_7^3:A_4$

Order 201684: $C_7^3.C_{14}^2.C_3$ x 2, $C_7^5.C_6.C_2$, $C_7^5.C_6.C_2$, $C_7^4.C_{42}.C_2$

Order 168070: $C_7^5:D_5$

Order 153664: $D_7\wr C_2^2$

Order 144060: $C_7^4:A_5$

Order 134456: $C_7^5.C_2^3$ x 2, $C_7\times C_7^4:C_2^3$, $C_7^4.(C_7\times D_4)$, $C_7\times C_7^4:D_4$, $C_7^4.(C_7\times D_4)$, $C_7^5.C_2^2.C_2$, $C_7\times C_7^4:(C_2\times C_4)$, $C_7^4.D_{14}.C_2$, $C_7^5.D_4$, $C_7^5.C_2^3$, $C_7\times C_7^4:Q_8$

Order 115248: $D_7\times C_7^3:S_4$, $D_7\times D_7\wr C_3$

Order 100842: $C_7^5:S_3$, $C_7^5:C_6$, $C_7^5:C_6$

Order 84035: $C_7\wr C_5$

Order 76832: $C_7:D_7^3:C_2^2$, $D_7^2\wr C_2$, $C_7:D_7^3:C_4$

Order 67228: $C_7\times C_7^4:C_2^2$ x 2, $C_7^5:C_2^2$ x 2, $C_7^5:C_4$, $C_7^5:C_4$, $C_7^5:C_2^2$, $C_7^5:C_2^2$

Order 57624: $C_7^3:A_4\times D_7$ x 3, $C_7^4:(C_2\times A_4)$ x 2, $D_7^3:C_{21}$ x 2, $C_7^4.D_6.C_2$, $C_7^4:\SL(2,3)$, $C_7\wr S_4$, $C_7^4:S_4$

Order 50421: $C_7^2\times C_7^3:C_3$

Order 38416: $C_7^2\wr C_2:D_4$ x 2, $C_7^4.C_2^4$, $C_7^4.C_2^3.C_2$, $C_7^2\wr C_2.D_4$, $D_7^2:D_7^2$, $C_7^4:D_4:C_2$, $C_7^2\wr C_2.D_4$

Order 33614: $C_7^2\times C_7^3:C_2$, $C_7\times C_7^4:C_2$, $C_7^5:C_2$

Order 32928: $C_7^3.A_4.D_4$

Order 28812: $C_7\wr A_4$ x 8, $C_7^4:(C_3:C_4)$, $C_7^4:(C_2\times C_6)$, $C_7^4:(C_2\times C_6)$, $C_7^4:D_6$, $C_7^4:D_6$, $C_7^3:C_6\times D_7$

Order 24010: $C_7^4:D_5$

Order 19208: $C_7^3:D_{28}$ x 5, $C_7^4:C_2^3$ x 3, $C_7:D_7^3$ x 3, $C_7^4:C_2^3$ x 2, $C_7^4:D_4$ x 2, $C_7^4:(C_2\times C_4)$ x 2, $C_7\wr D_4$ x 2, $C_7^4:(C_2\times C_4)$, $C_7^4:D_4$, $C_7^4:C_2^3$, $C_7^4:D_4$, $C_7\times C_7^3:C_2^3$, $C_7^4:Q_8$, $C_7^4:(C_2\times C_4)$

Order 16807: $C_7^5$

Order 16464: $C_7^3.A_4.C_4$, $C_2\times C_7^3:S_4$, $D_7^3:C_6$

Order 14406: $C_7^3:D_{21}$ x 5, $C_7^3:C_{42}$ x 5, $C_7^3:C_{42}$ x 3, $C_7^4:C_6$, $C_7\times C_7:C_3\times C_7:D_7$, $C_7^4:S_3$, $C_7^4:C_6$, $C_7:(C_7:C_3)\times C_7:D_7$

Order 12005: $C_7^4:C_5$

Order 10976: $C_7^3.C_2^4.C_2$

Order 9604: $C_7\wr C_2^2$ x 13, $C_7^4:C_2^2$ x 7, $C_7^4:C_2^2$ x 6, $C_7\wr C_4$ x 5, $C_7^4:C_2^2$ x 3, $C_7^4:C_2^2$ x 2, $C_7^4:C_4$, $C_7^4:C_2^2$, $C_7\times C_7^3:C_2^2$, $C_7^4:C_4$

Order 8232: $D_7\wr C_3$ x 2, $C_7^3:(C_3:D_4)$, $C_7\times C_7^2:\SL(2,3)$, $C_7^3:(C_3:D_4)$, $C_2\times C_7^3:A_4$, $C_7^3:S_4$

Order 7203: $C_7^3:C_{21}$ x 13, $C_7^3\times C_7:C_3$

Order 5488: $C_7\times C_7^2:(C_2\times D_4)$ x 2, $C_7^3:(C_2^2:C_4)$, $C_7^3:(C_2^2:C_4)$, $C_7^3:(C_2^2:C_4)$, $C_7^3:(C_2\times D_4)$, $C_7^3:(C_2\times D_4)$, $C_2\times D_7^2\times D_7$, $D_7^3:C_2$

Order 4802: $C_7^2\wr C_2$ x 20, $C_7\times C_7^3:C_2$ x 12, $C_7^3\times D_7$ x 6, $C_7^4:C_2$

Order 4704: $C_7^2.A_4.D_4$

Order 4116: $C_7^3:A_4$ x 2, $C_7^3:(C_2\times C_6)$, $C_7^3:(C_3:C_4)$, $C_7^3:D_6$, $C_7^3:(C_2\times C_6)$, $C_7\wr C_3:C_4$, $C_7^3:D_6$, $C_7^3:D_6$, $D_7\times C_7:F_7$, $C_2\times C_7^3:C_6$, $D_7\times C_7:F_7$

Order 2744: $C_7^3:D_4$ x 7, $D_7^3$ x 6, $D_7^2:C_{14}$ x 6, $C_7\times C_2\times D_7^2$ x 4, $C_7\times C_2\times C_7:D_7.C_2$ x 2, $C_7^3:(C_2\times C_4)$, $C_7^3:(C_2\times C_4)$, $C_7\times C_7:D_7.C_2^2$, $C_7^3:C_2^3$, $C_7^3:D_4$, $C_{14}:D_7^2$, $C_7^3:D_4$, $C_7^2:C_4\times D_7$, $C_7^2:D_{28}$

Order 2401: $C_7^4$ x 19

Order 2352: $C_7^2:(C_2\times S_4)$, $C_7^2:(C_2^2\times A_4)$, $C_7^2:(A_4:C_4)$

Order 2058: $C_7\wr S_3$ x 5, $C_7^2:C_{42}$ x 5, $C_7^3:C_6$ x 5, $C_7^2:C_{42}$ x 3, $C_7^3:C_6$ x 3, $C_7^2:C_{42}$ x 2, $C_7^3:S_3$, $C_7^3:C_6$, $C_7:D_7\times C_7:C_3$, $C_7^2:D_{21}$, $C_7^2:C_{42}$, $C_7^2:C_{42}$

Order 1568: $D_{14}\wr C_2$

Order 1372: $C_7\times D_7^2$ x 33, $C_7:D_7^2$ x 13, $C_7:D_7^2$ x 10, $C_7^2:C_{28}$ x 8, $C_7^3:C_4$ x 5, $D_{14}\times C_7^2$ x 4, $C_7^2:D_{14}$ x 2, $C_7^2:D_{14}$, $C_7^2:C_{28}$

Order 1344: $(C_2^2\times D_{14}):A_4$

Order 1176: $C_{14}^2:C_6$ x 2, $(C_7\times C_{14}).D_6$, $C_{14}^2:C_6$, $C_7^2:S_4$, $D_7^2:S_3$, $C_7^2:\SL(2,3)$

Order 1029: $C_7^2:C_{21}$ x 13, $C_7\wr C_3$ x 13, $C_7^2\times C_{21}$

Order 960: $C_2^4:A_5$

Order 784: $D_7^2:C_2^2$ x 3, $D_7^2:C_4$ x 2, $D_{14}^2$, $D_{14}:D_{14}$, $D_{14}:D_{14}$, $C_{14}^2:C_4$

Order 64: $C_2\wr C_2^2$

Order 5: $C_5$

Order 3: $C_3$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 16134720: $C_7^5:(C_2^4:A_5)$ ($C_1$)

Order 268912: $C_7^5.C_2^4$ ($A_5$)

Order 16807: $C_7^5$ ($C_2^4:A_5$)

Order 1: $C_1$ ($C_7^5:(C_2^4:A_5)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 16134720: $C_7^5:(C_2^4:A_5)$ ($C_1$)

Order 268912: $C_7^5.C_2^4$ ($A_5$)

Order 16807: $C_7^5$ ($C_2^4:A_5$)

Order 1: $C_1$ ($C_7^5:(C_2^4:A_5)$)

Series

Derived series	$C_7^5:(C_2^4:A_5)$
Chief series	$C_7^5:(C_2^4:A_5)$	$\rhd$	$C_7^5.C_2^4$	$\rhd$	$C_7^5$	$\rhd$	$C_1$
Lower central series	$C_7^5:(C_2^4:A_5)$
Upper central series	$C_1$

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

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

Character theory

Complex character table

See the $299 \times 299$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $68 \times 68$ rational character table.