Abstract group 4148928.a: $C_7^3.(C_6\times \GL(2,7))$

• Label: 4148928.a
• Order: \( 2^{6} \cdot 3^{3} \cdot 7^{4} \)
• Exponent: \( 2^{4} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3^{2} \)
• $\card{Z(G)}$: 6
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{3} \cdot 7^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $75$
• Trans deg.: $336$
• Rank: $2$

Group information

Description:	$C_7^3.(C_6\times \GL(2,7))$
Order:	\(4148928\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 7^{4} \)
Exponent:	\(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \)
Automorphism group:	$C_2^2\times C_7^3.C_6.\SO(3,7)\times S_3$, of order \(16595712\)\(\medspace = 2^{8} \cdot 3^{3} \cdot 7^{4} \)
Composition factors:	$C_2$ x 3, $C_3$ x 2, $C_7$ x 3, $\PSL(2,7)$
Derived length:	$1$

This group is nonabelian and nonsolvable. Whether it is almost simple has not been computed.

Group statistics

Order	1	2	3	4	6	7	8	12	14	16	21	24	28	42	48	56	84	112	168	336
Elements	1	3823	26756	2352	1009892	16806	32928	177576	85014	65856	280572	411600	26460	910908	823200	24696	52920	49392	49392	98784	4148928
Conjugacy classes	1	5	17	2	103	5	4	16	12	8	25	32	3	69	64	2	6	4	4	8	390
Divisions	1	5	9	2	53	5	2	8	12	2	13	8	3	35	8	1	3	1	1	1	173
Autjugacy classes	1	4	7	2	28	5	3	6	10	4	11	9	3	22	12	2	3	2	2	2	138

Minimal presentations

Permutation degree:	$75$
Transitive degree:	$336$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	48	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $75$ $\langle(1,2,5,15,30,23,12)(3,17,21,9,37,16,10)(4,11,26,49,51,44,45)(6,29,13,32,27,20,8) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 6 & 0 & 0 & 0 \\ 0 & 6 & 0 & 0 \\ 0 & 0 & 6 & 0 \\ 0 & 0 & 0 & 6 \end{array}\right), \left(\begin{array}{rrrr} 3 & 0 & 0 & 0 \\ 5 & 6 & 1 & 0 \\ 1 & 0 & 2 & 0 \\ 5 & 2 & 4 & 4 \end{array}\right), \left(\begin{array}{rrrr} 3 & 4 & 1 & 1 \\ 6 & 6 & 3 & 0 \\ 4 & 1 & 3 & 0 \\ 0 & 6 & 5 & 5 \end{array}\right), \left(\begin{array}{rrrr} 0 & 5 & 3 & 2 \\ 1 & 1 & 1 & 3 \\ 3 & 5 & 0 & 2 \\ 6 & 3 & 6 & 1 \end{array}\right), \left(\begin{array}{rrrr} 2 & 5 & 4 & 6 \\ 4 & 1 & 1 & 1 \\ 2 & 3 & 3 & 1 \\ 1 & 1 & 4 & 3 \end{array}\right), \left(\begin{array}{rrrr} 1 & 3 & 3 & 2 \\ 0 & 3 & 6 & 5 \\ 2 & 6 & 5 & 0 \\ 2 & 5 & 4 & 4 \end{array}\right), \left(\begin{array}{rrrr} 1 & 3 & 6 & 6 \\ 2 & 2 & 6 & 6 \\ 6 & 2 & 3 & 4 \\ 4 & 6 & 5 & 4 \end{array}\right), \left(\begin{array}{rrrr} 5 & 6 & 5 & 4 \\ 4 & 2 & 6 & 5 \\ 5 & 2 & 3 & 1 \\ 5 & 5 & 3 & 0 \end{array}\right), \left(\begin{array}{rrrr} 6 & 3 & 6 & 5 \\ 6 & 0 & 5 & 6 \\ 4 & 1 & 3 & 4 \\ 3 & 4 & 1 & 4 \end{array}\right), \left(\begin{array}{rrrr} 6 & 3 & 6 & 0 \\ 2 & 5 & 5 & 6 \\ 6 & 6 & 4 & 4 \\ 6 & 6 & 5 & 3 \end{array}\right), \left(\begin{array}{rrrr} 4 & 4 & 1 & 6 \\ 0 & 1 & 6 & 3 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{7})$
Direct product:	$C_3$ $\, \times\, $ $(C_7^3.(C_2\times \GL(2,7)))$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_7^3.\GL(2,7))$ . $C_6$ (3)	$(C_7^3.\GL(2,7))$ . $C_6$	$(C_7^3.\GL(2,7))$ . $C_6$ (3)	$(C_7^3.\GL(2,7))$ . $C_6$	all 38

Elements of the group are displayed as matrices in $\GL_{4}(\F_{7})$.

Homology

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

Subgroups

There are 64 normal subgroups (28 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 $C_6$
Commutator:	a subgroup isomorphic to $C_7^3.\SL(2,7)$
Frattini:	a subgroup isomorphic to $C_2$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times \SD_{32}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^3:C_7$

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 588 (order at least 7056) are shown, as well as all normal subgroups of any index.

Order 4148928: $C_7^3.(C_6\times \GL(2,7))$

Order 2074464: $C_7^3:C_6^2.\PSL(2,7)$, $(C_7^2\times C_{21}).\GL(2,7)$, $(C_7^2\times C_{21}).\GL(2,7)$

Order 1382976: $C_7^3.(C_2\times \GL(2,7))$ x 3, $C_7^3.(C_2\times \GL(2,7))$

Order 1037232: $C_7^3.C_3^2.\SL(2,7)$

Order 691488: $C_7^3.(C_6\times \SL(2,7))$ x 3, $C_7^3.\GL(2,7)$ x 3, $C_7^3.\GL(2,7)$ x 3, $C_7^3.(C_6\times \SL(2,7))$, $C_7^3.\GL(2,7)$, $C_7^3.\GL(2,7)$

Order 518616: $(C_7\times C_{42}).F_7^2$

Order 460992: $C_7^2:D_{14}.\PGL(2,7)$

Order 345744: $C_7^3.(C_3\times \SL(2,7))$ x 3, $(C_7^2\times C_{21}):\SL(2,7)$

Order 296352: $C_7^3:C_6^2.S_4$

Order 259308: $C_3\times C_7^2.F_7^2$ x 4, $C_7^3:D_{14}:C_3^3$, $C_7^3.C_{21}:C_6^2$, $C_7^3:D_{14}:C_3^3$

Order 230496: $C_7^3.(C_2\times \SL(2,7))$, $(C_7^2\times C_{14}).\PGL(2,7)$, $(C_7^2\times C_{14}).\PGL(2,7)$

Order 197568: $C_7^2.C_{112}:C_6^2$

Order 172872: $C_2\times C_7^2.F_7^2$ x 9, $\He_7:(D_7\times C_6^2)$, $C_7^3:C_{14}:C_6^2$, $C_7^3:(D_7\times C_6^2)$, $C_7^3:C_{14}:C_6^2$

Order 148176: $C_7^3:(C_6^2.A_4)$, $(C_7^2\times C_{14}).(C_3^2\times S_4)$, $C_{14}:F_7^2.C_6$, $(C_7^2\times C_{42}).(C_3\times S_4)$

Order 129654: $C_7^3:(C_3^2\times F_7)$ x 2, $\He_7:(C_3^2\times F_7)$ x 2, $C_7^3:(C_3^2\times F_7)$ x 2, $\He_7.C_7.C_3^3.C_2$

Order 115248: $C_7^3.\SL(2,7)$

Order 98784: $(C_7^2\times C_{14}).(C_6\times S_4)$ x 3, $C_7^3:(C_6\times C_{48})$, $C_6.D_7^3:C_6$, $C_7^3:(C_3^2\times \SD_{32})$, $C_7^3:(C_3^2\times \SD_{32})$, $C_6.D_7\wr S_3$, $C_6.D_7^3:C_6$, $C_7^3:(C_3^2\times \SD_{32})$, $C_7^3:(C_3^2\times \SD_{32})$

Order 86436: $C_7^2.F_7^2$ x 36, $\He_7:(C_6\times F_7)$ x 9, $C_7^3:(C_6\times F_7)$ x 9, $C_7^3:(C_6\times F_7)$ x 9, $C_7^3:(C_6\times F_7)$ x 4, $C_7^3:(C_6\times F_7)$ x 4, $C_7^3:(C_6\times F_7)$ x 4, $C_7^3:(C_3^2\times D_{14})$ x 4, $\He_7.C_7.C_6^2$, $C_7^3:(C_6\times F_7)$, $C_7^3:(C_6\times F_7)$, $C_7^3:(C_3^2\times D_{14})$, $\He_7:(C_6\times F_7)$, $C_7^3:(C_6\times F_7)$, $\He_7:(C_3^2\times D_{14})$, $C_6\times \He_7:C_{42}$, $\He_7:(C_6\times F_7)$, $C_6\times C_7^3:C_{42}$, $C_7.\He_7.C_6^2$, $C_7^3:(C_6\times F_7)$

Order 74088: $\He_7:C_6^3$, $C_{42}:F_7^2$, $C_7^3:(C_3^2\times \SL(2,3))$, $C_7^3:(S_3\times C_6^2)$, $C_7^2:C_{42}.C_6^2$, $C_7^3:C_6^2:C_6$, $C_7^3:C_6^2:C_6$, $C_{42}:F_7:F_7$, $C_7^3:C_6^2:C_6$

Order 65856: $C_7^3:(C_6\times \SD_{32})$ x 3, $C_7^3:(C_6\times \SD_{32})$

Order 64827: $(C_7.\He_7):C_3^3$

Order 57624: $\He_7.F_7.C_2^2$ x 3, $C_7^3:(C_6\times D_{14})$ x 3, $\He_7:(C_6\times D_{14})$ x 3, $\He_7.F_7.C_2^2$ x 3, $C_6\times C_7^3.D_{14}$

Order 49392: $C_7^3:(C_6\times \SL(2,3))$ x 6, $C_7^3:(C_6.S_4)$ x 3, $C_7^3:(C_6.S_4)$ x 3, $C_7^3:(C_6^2:C_2^2)$ x 3, $C_7^3:(C_6\times \SL(2,3))$ x 2, $C_{21}.C_{14}^2:D_6$, $C_7^3:(C_3^2\times Q_{16})$, $C_7^3:(C_3^2\times D_8)$, $(D_{14}\times C_7^2):C_6^2$, $C_7^3:(C_3^2\times D_8)$, $C_{42}.D_{14}:F_7$, $C_7^3:(C_6\times C_{24})$, $C_3\times C_7^3:C_{48}$, $C_7^3:(C_3^2\times D_8)$, $(C_7^2\times C_{42}).S_4$, $C_7^3:(C_3^2\times Q_{16})$, $C_7^3:(C_3^2\times Q_{16})$, $C_3\times C_7^3:C_{48}$, $(C_7^2\times C_{42}).S_4$

Order 43218: $C_7^3:(C_3\times F_7)$ x 18, $C_7^3:(C_3\times F_7)$ x 18, $\He_7:(C_3\times F_7)$ x 18, $C_7^3:(C_{21}:C_6)$ x 9, $C_3\times C_7^3:F_7$ x 2, $C_3\times C_7^3:F_7$ x 2, $C_3\times C_7^3:F_7$ x 2, $(C_3\times \He_7):F_7$ x 2, $C_3\times C_7^3:F_7$ x 2, $C_3\times \He_7:F_7$ x 2, $C_3\times C_7^3:C_{42}$ x 2, $\He_7:(C_3^2\times D_7)$ x 2, $C_3\times \He_7:F_7$ x 2, $C_3\times C_7^3:F_7$ x 2, $C_7^3:(C_3^2\times D_7)$ x 2, $C_3\times \He_7:C_{42}$ x 2, $C_3\times \He_7:C_{42}$, $C_7^3:(C_{21}:C_6)$, $C_3\times C_7^3:C_{42}$, $C_7^3:(C_{21}:C_6)$

Order 37044: $C_{21}.F_7^2$ x 4, $C_7\wr C_3:C_6^2$ x 2, $C_{21}:F_7^2$ x 2, $C_3\times \He_7:C_6^2$ x 2, $C_3\times C_7^3:C_6^2$, $C_3\times \He_7:C_6^2$, $C_3\times C_7^3:C_6^2$, $C_3\times C_7^3:C_6^2$, $C_7\wr C_3:C_6^2$, $C_7\wr C_3:C_6^2$, $C_7^3:C_3^2:C_{12}$, $C_7^3:C_3^2:C_{12}$

Order 32928: $C_7^3:(C_2\times C_{48})$ x 3, $C_7^3:(C_6\times Q_{16})$ x 3, $C_7^3:(C_6\times D_8)$ x 3, $C_7^3:(C_3\times \SD_{32})$ x 3, $C_7^3:(C_3\times \SD_{32})$ x 3, $C_7^3:(C_3\times \SD_{32})$ x 3, $C_7^3:(C_3\times \SD_{32})$ x 3, $C_3\times C_7^3:\SD_{32}$, $C_7^3:(C_6\times D_8)$, $C_7^3:(C_6\times Q_{16})$, $C_7^2:D_{14}.S_4$, $C_7^2:(D_7\times C_{48})$, $C_3\times C_7^3:\SD_{32}$, $C_{21}\times C_7^2:\SD_{32}$, $C_3\times C_7^3:\SD_{32}$

Order 28812: $C_7^3.(C_3\times D_{14})$ x 12, $C_7^3:(C_2\times F_7)$ x 12, $C_7^3:(C_2\times F_7)$ x 12, $C_7^3:(C_2\times F_7)$ x 12, $(C_7^2\times C_{21}):D_{14}$ x 4, $C_2\times \He_7:F_7$ x 3, $C_2\times \He_7:F_7$ x 3, $C_2\times C_7^3:F_7$ x 3, $\He_7:(C_3\times D_{14})$ x 3, $C_2\times C_7^3:F_7$ x 3, $C_2\times \He_7:C_{42}$ x 3, $C_7^3:(C_3\times D_{14})$ x 3, $C_2\times C_7^3:C_{42}$ x 3, $C_7^3:(C_3\times D_{14})$ x 3, $C_2\times C_7^3:F_7$ x 3, $C_2\times C_7^3:F_7$ x 3, $C_2\times C_7^3:F_7$ x 3, $C_2\times \He_7:C_{42}$, $C_2\times C_7^3:C_{42}$, $C_6\times C_7^3:D_7$

Order 28224: $C_7^2:C_{16}:C_6^2$

Order 24696: $C_{14}.F_7^2$ x 9, $C_{14}:F_7^2$ x 6, $C_7^3:(C_3\times \SL(2,3))$ x 6, $C_7^3:(C_6\times D_6)$ x 3, $C_7^3:C_6\wr C_2$ x 3, $C_7^3:C_6\wr C_2$ x 3, $C_7^3:(C_6:C_{12})$ x 3, $C_7^3:C_6\wr C_2$ x 3, $C_7^3:C_6\wr C_2$ x 3, $C_2\times \He_7:C_6^2$ x 2, $C_3\times C_7^3:\SL(2,3)$ x 2, $C_7^3:(C_6\times C_{12})$, $(C_{21}\times D_{14}):F_7$, $C_7^3:(D_4\times C_3^2)$, $(C_{21}\times D_{14}):F_7$, $C_7^3:(Q_8\times C_3^2)$, $C_3\times C_7^3:C_{24}$, $C_3\times C_7^3:C_{24}$, $C_7^3:(Q_8\times C_3^2)$, $C_2\times \He_7:C_6^2$, $C_7^3:C_6\wr C_2$, $(C_7^2\times C_{42}).D_6$, $C_2\times \He_7:C_6^2$, $C_2\times C_7^3:C_6^2$, $D_7\times C_{42}:F_7$, $C_2\times C_7^3:C_6^2$, $C_7^3:C_6\wr C_2$, $C_7^3:C_6\wr C_2$, $C_7^3:D_6\times C_6$, $(C_7\times C_{42}).D_{42}$

Order 21952: $C_7^3.C_8.C_2^3$

Order 21609: $C_7^3:C_{21}:C_3$ x 9, $C_7^3:(C_{21}:C_3)$, $C_3\times \He_7:C_{21}$, $C_3\times C_7^3:C_{21}$, $C_7^3:(C_{21}:C_3)$

Order 21168: $C_{14}^2:(C_3^2\times D_6)$

Order 19208: $(C_7^2\times C_{14}):D_{14}$

Order 18522: $C_7^2:(C_3^2\times F_7)$ x 4, $C_7^2:(C_3^2\times F_7)$ x 2, $C_7^2:(C_3^2\times F_7)$ x 2, $C_7^2:(C_3^2\times F_7)$, $C_6\times \He_7:C_3^2$, $C_7^2:(C_3^2\times F_7)$, $C_7^3:(S_3\times C_3^2)$, $C_7^3:(S_3\times C_3^2)$, $C_6\times C_7^3:C_3^2$

Order 16464: $C_7^3:(C_2\times \SL(2,3))$ x 5, $C_7^3:(C_3\times D_8)$ x 3, $C_7^3:C_{48}$ x 3, $C_7^3:(C_3\times Q_{16})$ x 3, $C_7^3:C_{48}$ x 3, $C_7^3:(C_3\times Q_{16})$ x 3, $C_7^3:(C_6\times D_4)$ x 3, $C_7^3:(C_3\times D_8)$ x 3, $C_7^3:(C_3\times D_8)$ x 3, $C_7^3:(C_6\times Q_8)$ x 3, $C_7^3:(C_2\times C_{24})$ x 3, $C_7^3:(C_3\times Q_{16})$ x 3, $C_3\times C_7^3:Q_{16}$, $C_7^3:C_{48}$, $(D_{14}\times C_7^2):D_6$, $C_7^2:Q_{16}\times C_{21}$, $C_7^2:D_8\times C_{21}$, $(C_7^2\times C_{14}).S_4$, $C_3\times C_7^2:D_{56}$, $C_3\times C_7^3:Q_{16}$, $C_6.D_7^3$, $C_6.D_7^3$, $C_7^2:(D_7\times C_{24})$, $(C_7^2\times C_{14}).S_4$, $C_3\times C_7^3:D_8$, $C_7^2:C_{336}$

Order 14406: $\He_7.F_7$ x 6, $C_7^3:C_{42}$ x 6, $C_7^3:F_7$ x 6, $C_7^3:F_7$ x 6, $\He_7:C_{42}$ x 6, $C_7^3:F_7$ x 6, $\He_7:(C_3\times D_7)$ x 6, $C_7^3.(C_3\times D_7)$ x 6, $\He_7:F_7$ x 6, $\He_7:F_7$ x 6, $C_7^3:F_7$ x 6, $C_7^3:F_7$ x 6, $C_7^3:(C_7:C_6)$ x 3, $C_7^3:C_{42}$ x 3, $\He_7:C_{42}$ x 3, $C_7^3:(C_7:C_6)$ x 3, $C_3\times C_7^3:D_7$ x 2, $C_7^3:C_{42}$ x 2, $\He_7:C_{42}$ x 2, $C_7^3:C_{42}$

Order 14112: $C_7^2:(C_3^2\times \SD_{32})$ x 4, $C_{84}:D_{14}:C_6$, $C_{21}:F_7:Q_{16}$, $C_7^2:(C_6\times C_{48})$

Order 12348: $\He_7:C_6^2$ x 36, $\He_7:C_6^2$ x 18, $C_7:F_7^2$ x 18, $\He_7:C_6^2$ x 9, $\He_7:C_6^2$ x 8, $C_7^3:(C_6\times S_3)$ x 6, $C_7^3:C_6^2$ x 6, $C_7^3:C_6^2$ x 6, $C_7^3:C_6^2$ x 6, $\He_7:C_6^2$ x 4, $\He_7:C_6^2$ x 4, $C_7^3:C_6^2$ x 4, $C_7\wr C_3:C_{12}$ x 3, $C_7\wr C_3:C_{12}$ x 3, $C_7^3:(C_6\times S_3)$ x 3, $C_7^3:(C_6\times S_3)$ x 3, $\He_7:C_6^2$ x 3, $C_7^3:C_6^2$ x 2, $C_7^3:C_6^2$ x 2, $C_7^3:C_6^2$ x 2, $\He_7:C_6^2$ x 2, $C_3\times C_7^3:D_6$ x 2, $\He_7:C_6^2$ x 2, $\He_7:C_6^2$ x 2, $\He_7:C_6^2$ x 2, $C_7^3:C_6^2$, $C_7^3:C_6^2$, $C_7^3:C_6^2$, $C_6\times C_7\wr S_3$, $C_7\wr C_3:C_{12}$, $C_7\wr C_3:C_{12}$, $C_3\times C_7^2:D_{42}$, $\He_7:C_6^2$, $C_7^3:C_6^2$, $C_7^3:C_6^2$, $C_7^3:C_6^2$, $\He_7:C_6^2$, $C_3\times C_7^3:C_{12}$, $C_7^3:C_6^2$, $C_3\times C_7^3:C_{12}$

Order 12096: $C_6\times \GL(2,7)$

Order 10976: $C_7^3:\SD_{32}$, $C_7^2:(D_7\times D_8)$, $C_7^3:\SD_{32}$, $C_7^2:(D_7\times Q_{16})$, $C_7^2:(D_7\times C_{16})$, $C_7^3:\SD_{32}$, $C_7^3:\SD_{32}$

Order 10584: $C_6\times F_7^2$ x 2, $C_{14}^2:(S_3\times C_3^2)$ x 2, $C_{42}:F_7:C_6$ x 2, $C_7^2:(S_3\times C_6^2)$, $C_{21}:F_7:C_{12}$, $C_7^2:C_6^3$

Order 9604: $C_7^3.D_{14}$ x 4, $C_7^3:D_{14}$, $\He_7:D_{14}$, $C_7^3:D_{14}$

Order 9408: $C_7^2:(C_6\times \SD_{32})$ x 3, $(C_7\times C_{84}).D_8$

Order 9261: $C_7^3:C_3^3$, $\He_7:C_3^3$

Order 8232: $C_7^2:(C_6\times D_{14})$ x 6, $C_7^3:\SL(2,3)$ x 5, $C_7^3:(C_2\times C_{12})$ x 3, $C_7^3:(C_3\times Q_8)$ x 3, $C_7^3:C_{24}$ x 3, $(C_7\times D_{14}):F_7$ x 3, $C_7^3:C_{24}$ x 3, $C_7^3:(C_3\times Q_8)$ x 3, $(C_7\times D_{14}):F_7$ x 3, $C_7^3:(C_3\times D_4)$ x 3, $C_7^2:(C_2^2\times F_7)$ x 3, $(C_7\times D_{14}):F_7$ x 3, $C_7:(D_{14}\times F_7)$ x 3, $C_7^2:(C_2^2\times F_7)$ x 3, $D_7\times C_{14}:F_7$ x 2, $C_2\times C_7^3:D_6$, $C_{42}.D_7^2$, $C_{42}.D_7^2$, $C_{42}.D_7^2$, $(C_7\times C_{14}).D_{42}$, $C_7\wr C_3:D_4$, $C_{42}.D_7^2$, $C_{42}.D_7^2$, $(C_7^2\times C_{14}).D_6$, $(D_{14}\times C_7^2):S_3$, $C_7^3:(C_3:D_4)$, $C_{42}:D_7^2$, $C_7^3:C_{24}$, $C_{42}.D_7^2$, $C_{42}.D_7^2$, $C_7^2:C_{168}$

Order 7203: $C_7^3:C_7:C_3$ x 3, $C_7^3:C_7:C_3$ x 3, $\He_7:C_{21}$ x 3, $C_7^3:C_{21}$ x 3, $C_7^3:C_{21}$

Order 7056: $C_{14}^2:(C_6\times S_3)$ x 3, $C_7^2:(C_3^2\times D_8)$ x 3, $C_7^2:(C_3^2\times Q_{16})$ x 3, $C_3\times C_7^2:C_{48}$ x 2, $C_3\times D_{28}:F_7$, $C_7^2:(C_6\times C_{24})$, $C_{21}:(Q_8\times F_7)$, $C_3\times C_{14}^2:D_6$, $C_{14}^2:C_6^2$

Order 6174: $C_3\times C_7^3:C_6$ x 2, $C_3\times C_7^3:C_6$

Order 4116: $C_2\times C_7^3:C_6$ x 3, $C_2\times C_7^3:C_6$

Order 3087: $C_7^3:C_3^2$

Order 2401: $C_7^3:C_7$

Order 2058: $C_7^3:C_6$ x 6, $C_7^3:C_6$ x 3, $C_7^3:C_6$ x 2, $C_7^2\times C_{42}$

Order 1372: $C_7^2:D_{14}$

Order 1029: $C_7^3:C_3$ x 3, $C_7^2\times C_{21}$

Order 686: $C_7^3:C_2$ x 2, $C_7^2\times C_{14}$

Order 343: $C_7^3$

Order 64: $C_2\times \SD_{32}$

Order 27: $C_3^3$

Order 6: $C_6$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 4148928: $C_7^3.(C_6\times \GL(2,7))$ ($C_1$)

Order 2074464: $C_7^3:C_6^2.\PSL(2,7)$ ($C_2$), $(C_7^2\times C_{21}).\GL(2,7)$ ($C_2$), $(C_7^2\times C_{21}).\GL(2,7)$ ($C_2$)

Order 1382976: $C_7^3.(C_2\times \GL(2,7))$ ($C_3$) x 3, $C_7^3.(C_2\times \GL(2,7))$ ($C_3$)

Order 1037232: $C_7^3.C_3^2.\SL(2,7)$ ($C_2^2$)

Order 691488: $C_7^3.(C_6\times \SL(2,7))$ ($C_6$) x 3, $C_7^3.\GL(2,7)$ ($C_6$) x 3, $C_7^3.\GL(2,7)$ ($C_6$) x 3, $C_7^3.(C_6\times \SL(2,7))$ ($C_6$), $C_7^3.\GL(2,7)$ ($C_6$), $C_7^3.\GL(2,7)$ ($C_6$)

Order 460992: $C_7^2:D_{14}.\PGL(2,7)$ ($C_3^2$)

Order 345744: $C_7^3.(C_3\times \SL(2,7))$ ($C_2\times C_6$) x 3, $(C_7^2\times C_{21}):\SL(2,7)$ ($C_2\times C_6$)

Order 230496: $C_7^3.(C_2\times \SL(2,7))$ ($C_3\times C_6$), $(C_7^2\times C_{14}).\PGL(2,7)$ ($C_3\times C_6$), $(C_7^2\times C_{14}).\PGL(2,7)$ ($C_3\times C_6$)

Order 115248: $C_7^3.\SL(2,7)$ ($C_6^2$)

Order 12348: $C_7^3:C_6^2$ ($\PGL(2,7)$)

Order 6174: $C_3\times C_7^3:C_6$ ($\SL(2,7):C_2$) x 2, $C_3\times C_7^3:C_6$ ($C_2\times \PGL(2,7)$)

Order 4116: $C_2\times C_7^3:C_6$ ($C_3\times \PGL(2,7)$) x 3, $C_2\times C_7^3:C_6$ ($C_3\times \PGL(2,7)$)

Order 3087: $C_7^3:C_3^2$ ($\SL(2,7):C_2^2$)

Order 2058: $C_7^3:C_6$ ($\GL(2,7)$) x 6, $C_7^3:C_6$ ($C_6\times \PGL(2,7)$) x 3, $C_7^3:C_6$ ($\GL(2,7)$) x 2, $C_7^2\times C_{42}$ ($C_6\times \PGL(2,7)$)

Order 1372: $C_7^2:D_{14}$ ($C_3^2\times \PGL(2,7)$)

Order 1029: $C_7^3:C_3$ ($C_2\times \GL(2,7)$) x 3, $C_7^2\times C_{21}$ ($C_2\times \GL(2,7)$)

Order 686: $C_7^3:C_2$ ($C_3\times \GL(2,7)$) x 2, $C_7^2\times C_{14}$ ($(C_3\times C_6).\SO(3,7)$)

Order 343: $C_7^3$ ($C_6\times \GL(2,7)$)

Order 6: $C_6$ ($C_7^3.C_6.\SO(3,7)$)

Order 3: $C_3$ ($C_7^3.(C_2\times \GL(2,7))$)

Order 2: $C_2$ ($C_7^3.C_3^2.C_2.\SO(3,7)$)

Order 1: $C_1$ ($C_7^3.(C_6\times \GL(2,7))$)

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 $390 \times 390$ character table is not available for this group.

Rational character table

The $173 \times 173$ rational character table is not available for this group.