Abstract group 5488.l: $\He_7:D_8$

• Label: 5488.l
• Order: \( 2^{4} \cdot 7^{3} \)
• Exponent: \( 2^{3} \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3 \cdot 7^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 3 \)
• Perm deg.: $196$
• Trans deg.: $196$
• Rank: $2$

Group information

Description:	$\He_7:D_8$
Order:	\(5488\)\(\medspace = 2^{4} \cdot 7^{3} \)
Exponent:	\(56\)\(\medspace = 2^{3} \cdot 7 \)
Automorphism group:	$\He_7:(C_3\times \SD_{32})$, of order \(32928\)\(\medspace = 2^{5} \cdot 3 \cdot 7^{3} \)
Composition factors:	$C_2$ x 4, $C_7$ x 3
Derived length:	$4$

This group is nonabelian and solvable.

Group statistics

Order	1	2	4	7	8	14	28	56
Elements	1	441	98	342	196	2646	588	1176	5488
Conjugacy classes	1	3	1	9	2	9	6	12	43
Divisions	1	3	1	3	1	3	1	1	14
Autjugacy classes	1	2	1	2	2	2	1	2	13

Dimension	1	2	4	8	14	24	42	84	168
Irr. complex chars.	4	3	0	12	24	0	0	0	0	43
Irr. rational chars.	4	1	1	0	0	4	2	1	1	14

Minimal presentations

Permutation degree:	$196$
Transitive degree:	$196$
Rank:	$2$
Inequivalent generating pairs:	$336$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	14	14	42
Arbitrary	14	14	42

Constructions

Presentation:	${\langle a, b, c, d, e \mid c^{7}=d^{7}=e^{7}=[b,e]=[c,e]=[d,e]=1, a^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $196$ $\langle(2,10)(3,20)(5,24)(6,46)(7,53)(11,49)(12,84)(13,73)(15,102)(16,86)(17,76) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 3 & 6 & 0 & 0 \\ 2 & 0 & 6 & 0 \\ 0 & 5 & 3 & 1 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 2 & 6 \\ 2 & 5 & 4 & 1 \\ 5 & 0 & 6 & 2 \\ 5 & 4 & 4 & 4 \end{array}\right), \left(\begin{array}{rrrr} 4 & 5 & 0 & 0 \\ 0 & 0 & 6 & 3 \\ 6 & 3 & 4 & 2 \\ 1 & 3 & 2 & 6 \end{array}\right), \left(\begin{array}{rrrr} 3 & 1 & 1 & 3 \\ 3 & 6 & 5 & 1 \\ 3 & 5 & 3 & 6 \\ 6 & 3 & 4 & 6 \end{array}\right), \left(\begin{array}{rrrr} 1 & 6 & 4 & 4 \\ 4 & 5 & 3 & 4 \\ 6 & 2 & 4 & 1 \\ 2 & 6 & 3 & 1 \end{array}\right), \left(\begin{array}{rrrr} 4 & 1 & 4 & 1 \\ 3 & 6 & 1 & 3 \\ 5 & 0 & 1 & 1 \\ 5 & 0 & 2 & 1 \end{array}\right), \left(\begin{array}{rrrr} 4 & 5 & 3 & 2 \\ 1 & 5 & 1 & 3 \\ 3 & 5 & 4 & 2 \\ 6 & 3 & 6 & 5 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{7})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$\He_7$ $\,\rtimes\,$ $D_8$	$(\He_7:D_4)$ $\,\rtimes\,$ $C_2$ (2)	$(\He_7:C_8)$ $\,\rtimes\,$ $C_2$		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_7^2:D_7)$ . $D_4$	$C_7$ . $(C_7^2:D_8)$	$(\He_7:C_4)$ . $C_2^2$		more information

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

Homology

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

Subgroups

There are 4979 subgroups in 53 conjugacy classes, 9 normal (7 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $\He_7:D_8$
Commutator:	$G' \simeq$ $\He_7:C_4$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_7$	$G/\Phi \simeq$ $C_7^2:D_8$
Fitting:	$\operatorname{Fit} \simeq$ $\He_7$	$G/\operatorname{Fit} \simeq$ $D_8$
Radical:	$R \simeq$ $\He_7:D_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_7$	$G/\operatorname{soc} \simeq$ $C_7^2:D_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_8$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $\He_7$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 5488: $\He_7:D_8$

Order 2744: $\He_7:D_4$ x 2, $\He_7:C_8$

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

Order 686: $C_7^2:C_{14}$ x 2, $C_7^2:D_7$

Order 343: $\He_7$

Order 196: $D_7^2$ x 2

Order 112: $D_{56}$

Order 98: $C_7\times D_7$ x 4, $C_7:D_7$ x 2

Order 56: $D_{28}$ x 2, $C_{56}$

Order 49: $C_7^2$ x 2

Order 28: $D_{14}$ x 4, $C_{28}$

Order 16: $D_8$

Order 14: $D_7$ x 6, $C_{14}$ x 3

Order 8: $D_4$ x 2, $C_8$

Order 7: $C_7$ x 3

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 5488: $\He_7:D_8$

Order 2744: $\He_7:C_8$, $\He_7:D_4$

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

Order 686: $C_7^2:D_7$, $C_7^2:C_{14}$

Order 343: $\He_7$

Order 196: $D_7^2$

Order 112: $D_{56}$

Order 98: $C_7\times D_7$ x 2, $C_7:D_7$

Order 56: $D_{28}$, $C_{56}$

Order 49: $C_7^2$

Order 28: $D_{14}$ x 2, $C_{28}$

Order 16: $D_8$

Order 14: $D_7$ x 3, $C_{14}$ x 2

Order 8: $D_4$, $C_8$

Order 7: $C_7$ x 2

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 5488: $\He_7:D_8$ ($C_1$)

Order 2744: $\He_7:D_4$ ($C_2$) x 2, $\He_7:C_8$ ($C_2$)

Order 1372: $\He_7:C_4$ ($C_2^2$)

Order 686: $C_7^2:D_7$ ($D_4$)

Order 343: $\He_7$ ($D_8$)

Order 7: $C_7$ ($C_7^2:D_8$)

Order 1: $C_1$ ($\He_7:D_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 5488: $\He_7:D_8$ ($C_1$)

Order 2744: $\He_7:C_8$ ($C_2$), $\He_7:D_4$ ($C_2$)

Order 1372: $\He_7:C_4$ ($C_2^2$)

Order 686: $C_7^2:D_7$ ($D_4$)

Order 343: $\He_7$ ($D_8$)

Order 7: $C_7$ ($C_7^2:D_8$)

Order 1: $C_1$ ($\He_7:D_8$)

Series

Derived series

$\He_7:D_8$

$\rhd$

$\He_7:C_4$

$\rhd$

$\He_7$

$\rhd$

$C_7$

$\rhd$

$C_1$

Chief series

$\He_7:D_8$

$\rhd$

$\He_7:D_4$

$\rhd$

$\He_7:C_4$

$\rhd$

$C_7^2:D_7$

$\rhd$

$\He_7$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$\He_7:D_8$

$\rhd$

$\He_7:C_4$

$\rhd$

$C_7^2:D_7$

$\rhd$

$\He_7$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 4 larger groups in the database.

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

Character theory

Complex character table

See the $43 \times 43$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	2C	4A	7A	7B	7C	8A	14A	14B	14C	28A	56A
	Size	1	49	196	196	98	6	168	168	196	294	1176	1176	588	1176
	2 P	1A	1A	1A	1A	2A	7A	7B	7C	4A	7A	7B	7C	14A	28A
	7 P	1A	2A	2B	2C	4A	7A	7B	7C	8A	14A	14B	14C	28A	56A
5488.l.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
5488.l.1b		$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$
5488.l.1c		$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$-1$
5488.l.1d		$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$
5488.l.2a		$2$	$2$	$0$	$0$	$-2$	$2$	$2$	$2$	$0$	$2$	$0$	$0$	$-2$	$0$
5488.l.2b		$4$	$-4$	$0$	$0$	$0$	$4$	$4$	$4$	$0$	$-4$	$0$	$0$	$0$	$0$
5488.l.8a		$24$	$0$	$0$	$6$	$0$	$24$	$-4$	$3$	$0$	$0$	$-1$	$0$	$0$	$0$
5488.l.8b		$24$	$0$	$6$	$0$	$0$	$24$	$3$	$-4$	$0$	$0$	$0$	$-1$	$0$	$0$
5488.l.8c		$24$	$0$	$-6$	$0$	$0$	$24$	$3$	$-4$	$0$	$0$	$0$	$1$	$0$	$0$
5488.l.8d		$24$	$0$	$0$	$-6$	$0$	$24$	$-4$	$3$	$0$	$0$	$1$	$0$	$0$	$0$
5488.l.14a		$42$	$-6$	$0$	$0$	$-6$	$-7$	$0$	$0$	$6$	$1$	$0$	$0$	$1$	$-1$
5488.l.14b		$42$	$-6$	$0$	$0$	$-6$	$-7$	$0$	$0$	$-6$	$1$	$0$	$0$	$1$	$1$
5488.l.14c		$84$	$-12$	$0$	$0$	$12$	$-14$	$0$	$0$	$0$	$2$	$0$	$0$	$-2$	$0$
5488.l.14d		$168$	$24$	$0$	$0$	$0$	$-28$	$0$	$0$	$0$	$-4$	$0$	$0$	$0$	$0$