Abstract group 49392.bb: $\He_7:(C_6^2:C_2^2)$

• Label: 49392.bb
• Order: \( 2^{4} \cdot 3^{2} \cdot 7^{3} \)
• Exponent: \( 2^{2} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{2} \cdot 7^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $100$
• Trans deg.: $196$
• Rank: $3$

Group information

Description:	$\He_7:(C_6^2:C_2^2)$
Order:	\(49392\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7^{3} \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	$\He_7.C_6^2.C_2^3$, of order \(98784\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7^{3} \)
Composition factors:	$C_2$ x 4, $C_3$ x 2, $C_7$ x 3
Derived length:	$4$

This group is nonabelian and solvable.

Group statistics

Order	1	2	3	4	6	7	12	14	21	28	42
Elements	1	883	980	588	22148	342	8232	5634	1764	3528	5292	49392
Conjugacy classes	1	7	5	2	35	3	4	9	3	4	9	82
Divisions	1	7	3	2	19	3	2	9	2	2	6	56
Autjugacy classes	1	5	5	1	25	3	2	7	3	2	7	61

Dimension	1	2	4	12	24	36	42	84
Irr. complex chars.	24	30	0	12	0	4	8	4	82
Irr. rational chars.	8	14	12	4	4	4	4	6	56

Minimal presentations

Permutation degree:	$100$
Transitive degree:	$196$
Rank:	$3$
Inequivalent generating triples:	$342451200$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	42	42	42
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{6}=b^{6}=c^{14}=d^{7}=e^{14}=[b,e]=[d,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $100$ $\langle(2,8,26)(3,16,67)(4,23,14)(5,31,90)(6,18,38)(7,12,41)(9,36,27)(10,49,92) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 1 & 6 & 0 & 0 \\ 3 & 0 & 6 & 0 \\ 3 & 2 & 4 & 4 \end{array}\right), \left(\begin{array}{rrrr} 6 & 0 & 0 & 0 \\ 0 & 6 & 0 & 0 \\ 0 & 0 & 6 & 0 \\ 0 & 0 & 0 & 6 \end{array}\right), \left(\begin{array}{rrrr} 3 & 1 & 1 & 3 \\ 3 & 6 & 5 & 1 \\ 3 & 5 & 3 & 6 \\ 6 & 3 & 4 & 6 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 2 & 1 \\ 1 & 2 & 0 & 6 \\ 1 & 2 & 4 & 6 \\ 2 & 0 & 0 & 5 \end{array}\right), \left(\begin{array}{rrrr} 6 & 3 & 5 & 4 \\ 2 & 1 & 6 & 5 \\ 5 & 3 & 2 & 6 \\ 1 & 5 & 4 & 5 \end{array}\right), \left(\begin{array}{rrrr} 5 & 0 & 6 & 1 \\ 3 & 6 & 1 & 5 \\ 2 & 6 & 5 & 2 \\ 1 & 3 & 0 & 5 \end{array}\right), \left(\begin{array}{rrrr} 3 & 0 & 0 & 0 \\ 6 & 6 & 0 & 0 \\ 4 & 0 & 6 & 0 \\ 3 & 1 & 2 & 5 \end{array}\right), \left(\begin{array}{rrrr} 4 & 5 & 3 & 2 \\ 1 & 5 & 1 & 3 \\ 3 & 5 & 4 & 2 \\ 6 & 3 & 6 & 5 \end{array}\right), \left(\begin{array}{rrrr} 6 & 1 & 3 & 3 \\ 3 & 2 & 4 & 3 \\ 1 & 5 & 3 & 6 \\ 5 & 1 & 4 & 6 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{7})$
Direct product:	$C_2$ $\, \times\, $ $(\He_7:C_6\wr C_2)$
Semidirect product:	$\He_7$ $\,\rtimes\,$ $(C_6^2:C_2^2)$	$(\He_7:C_3)$ $\,\rtimes\,$ $(C_6\times D_4)$	$(\He_7:C_3)$ $\,\rtimes\,$ $(C_6:D_4)$	$(C_7^2:D_{14})$ $\,\rtimes\,$ $(C_6\times S_3)$ (2)	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2\times \He_7:C_6)$ . $D_6$	$(C_7^2:D_7)$ . $(C_6\times D_6)$	$(\He_7:C_6)$ . $(C_2\times D_6)$	$(C_7^2:D_{14})$ . $(C_6\times S_3)$	all 10
Aut. group:	$\Aut(C_{21}.D_7^2)$	$\Aut(C_3\times \He_7:D_6)$	$\Aut(\He_7:C_6^2)$	$\Aut(\He_7:C_6\wr C_2)$	all 5

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

Homology

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

Subgroups

There are 98484 subgroups in 628 conjugacy classes, 58 normal (34 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $\He_7.C_6.C_6.C_2$
Commutator:	$G' \simeq$ $\He_7:C_6$	$G/G' \simeq$ $C_2^2\times C_6$
Frattini:	$\Phi \simeq$ $C_7$	$G/\Phi \simeq$ $C_7^2.(C_6\times C_3:D_4)$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times \He_7$	$G/\operatorname{Fit} \simeq$ $C_6\wr C_2$
Radical:	$R \simeq$ $\He_7:(C_6^2:C_2^2)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{14}$	$G/\operatorname{soc} \simeq$ $C_7^2:(C_3\times C_3:D_4)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $\He_7$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 49392: $\He_7:(C_6^2:C_2^2)$

Order 24696: $\He_7.C_6.C_6.C_2$ x 3, $C_2\times \He_7:(C_6\times S_3)$, $\He_7.C_6^2.C_2$, $C_2\times \He_7:(C_3\times C_3:C_4)$, $C_2\times \He_7:C_6^2$

Order 16464: $C_2\times \He_7:(C_3:D_4)$, $C_2\times \He_7:(C_3\times D_4)$

Order 12348: $\He_7:(C_6\times S_3)$ x 3, $\He_7:C_6^2$ x 3, $\He_7:(C_3\times C_3:C_4)$ x 2, $C_2\times \He_7:(C_3\times C_6)$, $C_2\times \He_7:(C_3\times S_3)$, $C_2\times \He_7:(C_3\times C_6)$

Order 8232: $\He_7:(C_3:D_4)$ x 4, $\He_7:(C_3\times D_4)$ x 4, $C_2\times \He_7:(C_2\times C_6)$ x 2, $\He_7:(C_2\times C_3:C_4)$, $C_2\times \He_7:(C_2\times C_6)$, $C_2\times \He_7:D_6$, $C_2\times \He_7:(C_2\times C_6)$, $C_2\times \He_7:C_{12}$

Order 6174: $\He_7:(C_3\times C_6)$ x 2, $\He_7:(C_3\times C_6)$ x 2, $\He_7:(C_3\times S_3)$ x 2, $C_2\times \He_7:C_3^2$

Order 5488: $C_2\times \He_7:D_4$

Order 4116: $\He_7:(C_2\times C_6)$ x 6, $\He_7:(C_2\times C_6)$ x 4, $\He_7:D_6$ x 3, $\He_7:(C_2\times C_6)$ x 3, $C_2\times \He_7:C_6$ x 2, $\He_7:C_{12}$ x 2, $\He_7:(C_3:C_4)$ x 2, $C_2\times \He_7:C_6$, $\He_7:(C_2\times C_6)$, $C_2\times \He_7:C_6$, $C_2\times \He_7:C_6$, $C_2\times \He_7:C_6$, $C_2\times \He_7:S_3$, $C_2\times \He_7:C_6$

Order 3528: $C_2\times C_7^2:C_6^2$

Order 3087: $\He_7:C_3^2$

Order 2744: $\He_7:D_4$ x 4, $C_2\times \He_7:C_2^2$ x 2, $\He_7:(C_2\times C_4)$

Order 2058: $\He_7:C_6$ x 4, $\He_7:C_6$ x 2, $\He_7:S_3$ x 2, $\He_7:C_6$ x 2, $\He_7:C_6$ x 2, $\He_7:C_6$ x 2, $\He_7:C_6$ x 2, $\He_7:C_6$ x 2, $C_2\times \He_7:C_3$, $C_2\times \He_7:C_3$, $\He_7:C_6$

Order 1764: $F_7^2$ x 4, $C_7^2:C_6^2$ x 2, $C_7^2:C_6^2$

Order 1372: $C_7^2:D_{14}$ x 6, $C_7^2:D_{14}$ x 2, $\He_7:C_4$ x 2, $C_7^2:D_{14}$

Order 1176: $D_{14}\times F_7$ x 2, $D_{14}:F_7$ x 2, $D_{14}:F_7$

Order 1029: $C_7^2:C_{21}$, $\He_7:C_3$, $\He_7:C_3$

Order 1008: $C_2\times D_{42}:C_6$

Order 882: $C_7:C_3\times F_7$ x 4, $C_7:(C_3\times F_7)$ x 2, $C_2\times C_7^2:C_3^2$

Order 686: $C_7^2:C_{14}$ x 4, $C_7^2:D_7$ x 2, $C_2\times \He_7$

Order 588: $D_7:F_7$ x 8, $D_7\times F_7$ x 8, $C_{14}:F_7$ x 4, $D_7:F_7$ x 4, $C_{14}:F_7$ x 2, $C_{14}:F_7$ x 2, $C_{14}:F_7$ x 2, $C_7:C_3\times D_{14}$ x 2, $C_{14}\times F_7$ x 2, $C_{14}:F_7$

Order 504: $D_{42}:C_6$ x 4, $C_{14}:C_6^2$ x 2, $C_{42}:C_{12}$, $D_{42}:C_6$

Order 441: $C_7^2:C_3^2$

Order 392: $D_7\times D_{14}$ x 2

Order 343: $\He_7$

Order 336: $C_6:D_{28}$, $D_{28}:C_6$

Order 294: $C_7:F_7$ x 8, $C_7^2:C_6$ x 4, $C_7\times F_7$ x 4, $C_7:F_7$ x 4, $C_7:F_7$ x 4, $C_7:F_7$ x 4, $C_7^2:C_6$ x 2, $C_7:C_{42}$ x 2, $C_7:F_7$ x 2, $C_7^2:C_6$

Order 252: $C_6\times F_7$ x 10, $C_{42}:C_6$ x 4, $C_{42}:C_6$ x 2, $C_{21}:C_{12}$ x 2

Order 196: $D_7^2$ x 8, $C_7\times D_{14}$ x 4, $C_7:D_{14}$ x 2

Order 168: $C_2^2\times F_7$ x 7, $C_{28}:C_6$ x 4, $C_3:D_{28}$ x 4, $C_6\times D_{14}$ x 2, $C_2\times D_{42}$, $C_6:C_{28}$, $C_{14}:C_{12}$

Order 147: $C_7^2:C_3$ x 2, $C_7:C_{21}$ x 2, $C_7^2:C_3$

Order 144: $C_6^2:C_2^2$

Order 126: $C_3\times F_7$ x 6, $C_{21}:C_6$ x 6, $C_{21}:C_6$ x 2

Order 112: $C_2\times D_{28}$

Order 98: $C_7\times D_7$ x 8, $C_7:D_7$ x 4, $C_7\times C_{14}$ x 2

Order 84: $C_2\times F_7$ x 38, $C_3\times D_{14}$ x 10, $C_{14}:C_6$ x 6, $D_{42}$ x 4, $C_3:C_{28}$ x 2, $C_7:C_{12}$ x 2, $C_2\times C_{42}$ x 2

Order 72: $C_6\wr C_2$ x 4, $C_2\times C_6^2$, $C_6\times D_6$, $C_6:C_{12}$

Order 63: $C_{21}:C_3$ x 2

Order 56: $D_{28}$ x 4, $C_2\times D_{14}$ x 4, $C_2\times C_{28}$

Order 49: $C_7^2$ x 2

Order 48: $C_6\times D_4$, $C_6:D_4$

Order 42: $F_7$ x 24, $C_7:C_6$ x 18, $C_{42}$ x 6, $C_3\times D_7$ x 6, $D_{21}$ x 2

Order 36: $C_6^2$ x 5, $C_6\times S_3$ x 4, $C_3:C_{12}$ x 2

Order 28: $D_{14}$ x 20, $C_2\times C_{14}$ x 3, $C_{28}$ x 2

Order 24: $C_3:D_4$ x 4, $C_2^2\times C_6$ x 4, $C_3\times D_4$ x 4, $C_2\times C_{12}$, $C_6:C_4$, $C_2\times D_6$

Order 21: $C_7:C_3$ x 6, $C_{21}$ x 2

Order 18: $C_3\times C_6$ x 5, $C_3\times S_3$ x 2

Order 16: $C_2\times D_4$

Order 14: $D_7$ x 12, $C_{14}$ x 9

Order 12: $C_2\times C_6$ x 21, $D_6$ x 4, $C_{12}$ x 2, $C_3:C_4$ x 2

Order 9: $C_3^2$

Order 8: $D_4$ x 4, $C_2^3$ x 2, $C_2\times C_4$

Order 7: $C_7$ x 3

Order 6: $C_6$ x 19, $S_3$ x 2

Order 4: $C_2^2$ x 9, $C_4$ x 2

Order 3: $C_3$ x 3

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 49392: $\He_7:(C_6^2:C_2^2)$

Order 24696: $C_2\times \He_7:(C_6\times S_3)$, $\He_7.C_6.C_6.C_2$, $C_2\times \He_7:(C_3\times C_3:C_4)$, $C_2\times \He_7:C_6^2$

Order 16464: $C_2\times \He_7:(C_3:D_4)$, $C_2\times \He_7:(C_3\times D_4)$

Order 12348: $\He_7:(C_6\times S_3)$ x 2, $\He_7:C_6^2$ x 2, $\He_7:(C_3\times C_3:C_4)$, $C_2\times \He_7:(C_3\times C_6)$, $C_2\times \He_7:(C_3\times S_3)$, $C_2\times \He_7:(C_3\times C_6)$

Order 8232: $C_2\times \He_7:(C_2\times C_6)$ x 2, $\He_7:(C_2\times C_3:C_4)$, $C_2\times \He_7:(C_2\times C_6)$, $\He_7:(C_3:D_4)$, $\He_7:(C_3\times D_4)$, $C_2\times \He_7:D_6$, $C_2\times \He_7:(C_2\times C_6)$, $C_2\times \He_7:C_{12}$

Order 6174: $\He_7:(C_3\times C_6)$ x 2, $\He_7:(C_3\times C_6)$, $C_2\times \He_7:C_3^2$, $\He_7:(C_3\times S_3)$

Order 5488: $C_2\times \He_7:D_4$

Order 4116: $\He_7:(C_2\times C_6)$ x 4, $C_2\times \He_7:C_6$ x 2, $\He_7:(C_2\times C_6)$ x 2, $\He_7:D_6$ x 2, $\He_7:(C_2\times C_6)$ x 2, $C_2\times \He_7:C_6$, $\He_7:(C_2\times C_6)$, $C_2\times \He_7:C_6$, $C_2\times \He_7:C_6$, $\He_7:C_{12}$, $C_2\times \He_7:C_6$, $\He_7:(C_3:C_4)$, $C_2\times \He_7:S_3$, $C_2\times \He_7:C_6$

Order 3528: $C_2\times C_7^2:C_6^2$

Order 3087: $\He_7:C_3^2$

Order 2744: $C_2\times \He_7:C_2^2$ x 2, $\He_7:(C_2\times C_4)$, $\He_7:D_4$

Order 2058: $\He_7:C_6$ x 2, $\He_7:C_6$ x 2, $\He_7:C_6$ x 2, $\He_7:C_6$ x 2, $\He_7:C_6$, $\He_7:S_3$, $C_2\times \He_7:C_3$, $C_2\times \He_7:C_3$, $\He_7:C_6$, $\He_7:C_6$, $\He_7:C_6$

Order 1764: $C_7^2:C_6^2$ x 2, $F_7^2$ x 2, $C_7^2:C_6^2$

Order 1372: $C_7^2:D_{14}$ x 4, $C_7^2:D_{14}$ x 2, $C_7^2:D_{14}$, $\He_7:C_4$

Order 1176: $D_{14}\times F_7$ x 2, $D_{14}:F_7$ x 2, $D_{14}:F_7$

Order 1029: $C_7^2:C_{21}$, $\He_7:C_3$, $\He_7:C_3$

Order 1008: $C_2\times D_{42}:C_6$

Order 882: $C_7:C_3\times F_7$ x 3, $C_2\times C_7^2:C_3^2$, $C_7:(C_3\times F_7)$

Order 686: $C_7^2:D_7$ x 2, $C_7^2:C_{14}$ x 2, $C_2\times \He_7$

Order 588: $C_{14}:F_7$ x 4, $D_7:F_7$ x 4, $D_7\times F_7$ x 4, $C_{14}:F_7$ x 2, $C_{14}:F_7$ x 2, $C_{14}:F_7$ x 2, $C_7:C_3\times D_{14}$ x 2, $C_{14}\times F_7$ x 2, $D_7:F_7$ x 2, $C_{14}:F_7$

Order 504: $C_{14}:C_6^2$ x 2, $C_{42}:C_{12}$, $D_{42}:C_6$, $D_{42}:C_6$

Order 441: $C_7^2:C_3^2$

Order 392: $D_7\times D_{14}$ x 2

Order 343: $\He_7$

Order 336: $C_6:D_{28}$, $D_{28}:C_6$

Order 294: $C_7:F_7$ x 6, $C_7^2:C_6$ x 3, $C_7\times F_7$ x 3, $C_7:F_7$ x 3, $C_7^2:C_6$ x 2, $C_7:C_{42}$ x 2, $C_7:F_7$ x 2, $C_7:F_7$ x 2, $C_7^2:C_6$, $C_7:F_7$

Order 252: $C_6\times F_7$ x 7, $C_{42}:C_6$ x 3, $C_{42}:C_6$ x 2, $C_{21}:C_{12}$

Order 196: $D_7^2$ x 4, $C_7\times D_{14}$ x 4, $C_7:D_{14}$ x 2

Order 168: $C_2^2\times F_7$ x 7, $C_6\times D_{14}$ x 2, $C_{28}:C_6$, $C_2\times D_{42}$, $C_6:C_{28}$, $C_{14}:C_{12}$, $C_3:D_{28}$

Order 147: $C_7^2:C_3$ x 2, $C_7:C_{21}$ x 2, $C_7^2:C_3$

Order 144: $C_6^2:C_2^2$

Order 126: $C_{21}:C_6$ x 5, $C_3\times F_7$ x 4, $C_{21}:C_6$

Order 112: $C_2\times D_{28}$

Order 98: $C_7\times D_7$ x 6, $C_7\times C_{14}$ x 2, $C_7:D_7$ x 2

Order 84: $C_2\times F_7$ x 26, $C_3\times D_{14}$ x 7, $C_{14}:C_6$ x 6, $D_{42}$ x 3, $C_2\times C_{42}$ x 2, $C_3:C_{28}$, $C_7:C_{12}$

Order 72: $C_2\times C_6^2$, $C_6\times D_6$, $C_6\wr C_2$, $C_6:C_{12}$

Order 63: $C_{21}:C_3$ x 2

Order 56: $C_2\times D_{14}$ x 4, $C_2\times C_{28}$, $D_{28}$

Order 49: $C_7^2$ x 2

Order 48: $C_6\times D_4$, $C_6:D_4$

Order 42: $F_7$ x 16, $C_7:C_6$ x 14, $C_{42}$ x 5, $C_3\times D_7$ x 4, $D_{21}$

Order 36: $C_6^2$ x 4, $C_6\times S_3$ x 3, $C_3:C_{12}$

Order 28: $D_{14}$ x 14, $C_2\times C_{14}$ x 3, $C_{28}$

Order 24: $C_2^2\times C_6$ x 4, $C_2\times C_{12}$, $C_3:D_4$, $C_6:C_4$, $C_2\times D_6$, $C_3\times D_4$

Order 21: $C_7:C_3$ x 6, $C_{21}$ x 2

Order 18: $C_3\times C_6$ x 4, $C_3\times S_3$

Order 16: $C_2\times D_4$

Order 14: $D_7$ x 8, $C_{14}$ x 7

Order 12: $C_2\times C_6$ x 16, $D_6$ x 3, $C_{12}$, $C_3:C_4$

Order 9: $C_3^2$

Order 8: $C_2^3$ x 2, $D_4$, $C_2\times C_4$

Order 7: $C_7$ x 3

Order 6: $C_6$ x 14, $S_3$

Order 4: $C_2^2$ x 7, $C_4$

Order 3: $C_3$ x 3

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 49392: $\He_7:(C_6^2:C_2^2)$ ($C_1$)

Order 24696: $\He_7.C_6.C_6.C_2$ ($C_2$) x 3, $C_2\times \He_7:(C_6\times S_3)$ ($C_2$), $\He_7.C_6^2.C_2$ ($C_2$), $C_2\times \He_7:(C_3\times C_3:C_4)$ ($C_2$), $C_2\times \He_7:C_6^2$ ($C_2$)

Order 16464: $C_2\times \He_7:(C_3:D_4)$ ($C_3$)

Order 12348: $\He_7:(C_3\times C_3:C_4)$ ($C_2^2$) x 2, $\He_7:(C_6\times S_3)$ ($C_2^2$) x 2, $\He_7:C_6^2$ ($C_2^2$) x 2, $C_2\times \He_7:(C_3\times C_6)$ ($C_2^2$)

Order 8232: $\He_7:(C_3:D_4)$ ($C_6$) x 4, $\He_7:(C_2\times C_3:C_4)$ ($C_6$), $C_2\times \He_7:(C_2\times C_6)$ ($C_6$), $C_2\times \He_7:D_6$ ($C_6$), $C_2\times \He_7:(C_2\times C_6)$ ($S_3$)

Order 6174: $\He_7:(C_3\times C_6)$ ($C_2^3$), $\He_7:(C_3\times C_6)$ ($D_4$), $C_2\times \He_7:C_3^2$ ($D_4$)

Order 4116: $\He_7:(C_2\times C_6)$ ($D_6$) x 2, $\He_7:D_6$ ($C_2\times C_6$) x 2, $\He_7:(C_2\times C_6)$ ($C_2\times C_6$) x 2, $\He_7:(C_3:C_4)$ ($C_2\times C_6$) x 2, $\He_7:(C_2\times C_6)$ ($C_2\times C_6$), $C_2\times \He_7:C_6$ ($D_6$)

Order 3087: $\He_7:C_3^2$ ($C_2\times D_4$)

Order 2744: $C_2\times \He_7:C_2^2$ ($C_3\times S_3$)

Order 2058: $\He_7:C_6$ ($C_3:D_4$), $\He_7:C_6$ ($C_2\times D_6$), $C_2\times \He_7:C_3$ ($C_3:D_4$), $\He_7:C_6$ ($C_2^2\times C_6$), $\He_7:C_6$ ($C_3\times D_4$), $\He_7:C_6$ ($C_3\times D_4$)

Order 1372: $C_7^2:D_{14}$ ($C_6\times S_3$) x 2, $C_7^2:D_{14}$ ($C_6\times S_3$)

Order 1029: $\He_7:C_3$ ($C_6\times D_4$), $\He_7:C_3$ ($C_6:D_4$)

Order 686: $C_7^2:D_7$ ($C_6\times D_6$), $C_7^2:D_7$ ($C_6\wr C_2$), $C_2\times \He_7$ ($C_6\wr C_2$)

Order 343: $\He_7$ ($C_6^2:C_2^2$)

Order 14: $C_{14}$ ($C_7^2:(C_3\times C_3:D_4)$)

Order 7: $C_7$ ($C_7^2.(C_6\times C_3:D_4)$)

Order 2: $C_2$ ($\He_7.C_6.C_6.C_2$)

Order 1: $C_1$ ($\He_7:(C_6^2:C_2^2)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 49392: $\He_7:(C_6^2:C_2^2)$ ($C_1$)

Order 24696: $C_2\times \He_7:(C_6\times S_3)$ ($C_2$), $\He_7.C_6.C_6.C_2$ ($C_2$), $C_2\times \He_7:(C_3\times C_3:C_4)$ ($C_2$), $C_2\times \He_7:C_6^2$ ($C_2$)

Order 16464: $C_2\times \He_7:(C_3:D_4)$ ($C_3$)

Order 12348: $\He_7:(C_3\times C_3:C_4)$ ($C_2^2$), $\He_7:(C_6\times S_3)$ ($C_2^2$), $\He_7:C_6^2$ ($C_2^2$), $C_2\times \He_7:(C_3\times C_6)$ ($C_2^2$)

Order 8232: $\He_7:(C_2\times C_3:C_4)$ ($C_6$), $C_2\times \He_7:(C_2\times C_6)$ ($C_6$), $\He_7:(C_3:D_4)$ ($C_6$), $C_2\times \He_7:D_6$ ($C_6$), $C_2\times \He_7:(C_2\times C_6)$ ($S_3$)

Order 6174: $\He_7:(C_3\times C_6)$ ($C_2^3$), $\He_7:(C_3\times C_6)$ ($D_4$), $C_2\times \He_7:C_3^2$ ($D_4$)

Order 4116: $\He_7:(C_2\times C_6)$ ($C_2\times C_6$), $C_2\times \He_7:C_6$ ($D_6$), $\He_7:(C_2\times C_6)$ ($D_6$), $\He_7:D_6$ ($C_2\times C_6$), $\He_7:(C_2\times C_6)$ ($C_2\times C_6$), $\He_7:(C_3:C_4)$ ($C_2\times C_6$)

Order 3087: $\He_7:C_3^2$ ($C_2\times D_4$)

Order 2744: $C_2\times \He_7:C_2^2$ ($C_3\times S_3$)

Order 2058: $\He_7:C_6$ ($C_3:D_4$), $\He_7:C_6$ ($C_2\times D_6$), $C_2\times \He_7:C_3$ ($C_3:D_4$), $\He_7:C_6$ ($C_2^2\times C_6$), $\He_7:C_6$ ($C_3\times D_4$), $\He_7:C_6$ ($C_3\times D_4$)

Order 1372: $C_7^2:D_{14}$ ($C_6\times S_3$), $C_7^2:D_{14}$ ($C_6\times S_3$)

Order 1029: $\He_7:C_3$ ($C_6\times D_4$), $\He_7:C_3$ ($C_6:D_4$)

Order 686: $C_7^2:D_7$ ($C_6\times D_6$), $C_7^2:D_7$ ($C_6\wr C_2$), $C_2\times \He_7$ ($C_6\wr C_2$)

Order 343: $\He_7$ ($C_6^2:C_2^2$)

Order 14: $C_{14}$ ($C_7^2:(C_3\times C_3:D_4)$)

Order 7: $C_7$ ($C_7^2.(C_6\times C_3:D_4)$)

Order 2: $C_2$ ($\He_7.C_6.C_6.C_2$)

Order 1: $C_1$ ($\He_7:(C_6^2:C_2^2)$)

Series

Derived series

$\He_7:(C_6^2:C_2^2)$

$\rhd$

$\He_7:C_6$

$\rhd$

$\He_7$

$\rhd$

$C_7$

$\rhd$

$C_1$

Chief series

$\He_7:(C_6^2:C_2^2)$

$\rhd$

$\He_7.C_6.C_6.C_2$

$\rhd$

$\He_7:(C_6\times S_3)$

$\rhd$

$\He_7:(C_3\times C_6)$

$\rhd$

$\He_7:C_6$

$\rhd$

$\He_7:C_3$

$\rhd$

$\He_7$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$\He_7:(C_6^2:C_2^2)$

$\rhd$

$\He_7:C_6$

$\rhd$

$\He_7:C_3$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

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

Character theory

Complex character table

See the $82 \times 82$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $56 \times 56$ rational character table.