Abstract group 56.4: $C_4\times D_7$

• Label: 56.4
• Order: \( 2^{3} \cdot 7 \)
• Exponent: \( 2^{2} \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{3} \cdot 3 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \)
• Perm deg.: $11$
• Trans deg.: $28$
• Rank: $2$

Group information

Description:	$C_4\times D_7$
Order:	\(56\)\(\medspace = 2^{3} \cdot 7 \)
Exponent:	\(28\)\(\medspace = 2^{2} \cdot 7 \)
Automorphism group:	$C_2^2\times F_7$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \)
Composition factors:	$C_2$ x 3, $C_7$
Derived length:	$2$

This group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.

Group statistics

Order	1	2	4	7	14	28
Elements	1	15	16	6	6	12	56
Conjugacy classes	1	3	4	3	3	6	20
Divisions	1	3	2	1	1	1	9
Autjugacy classes	1	2	2	1	1	1	8

Dimension	1	2	6	12
Irr. complex chars.	8	12	0	0	20
Irr. rational chars.	4	2	2	1	9

Minimal presentations

Permutation degree:	$11$
Transitive degree:	$28$
Rank:	$2$
Inequivalent generating pairs:	$6$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	4	12
Arbitrary	2	4	8

Constructions

Presentation:	$\langle a, b \mid a^{2}=b^{28}=1, b^{a}=b^{13} \rangle$
Permutation group:	Degree $11$ $\langle(2,3)(4,5)(6,7)(8,9)(10,11), (8,10,9,11), (8,9)(10,11), (1,2,4,6,7,5,3)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 0 \\ 0 & 12 \end{array}\right), \left(\begin{array}{rr} 11 & 6 \\ 3 & 11 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{13})$
Transitive group:	28T8				more information
Direct product:	$C_4$ $\, \times\, $ $D_7$
Semidirect product:	$C_{28}$ $\,\rtimes\,$ $C_2$	$(C_7:C_4)$ $\,\rtimes\,$ $C_2$	$C_7$ $\,\rtimes\,$ $(C_2\times C_4)$		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{14}$ . $C_2$	$C_2$ . $D_{14}$	$C_{14}$ . $C_2^2$		more information

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{2} \times C_{4} $
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 46 subgroups in 16 conjugacy classes, 11 normal (9 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_4$	$G/Z \simeq$ $D_7$
Commutator:	$G' \simeq$ $C_7$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{28}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_4\times D_7$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{14}$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_4$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

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 56: $C_4\times D_7$

Order 28: $D_{14}$, $C_{28}$, $C_7:C_4$

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

Order 8: $C_2\times C_4$

Order 7: $C_7$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 56: $C_4\times D_7$

Order 28: $D_{14}$, $C_{28}$, $C_7:C_4$

Order 14: $C_{14}$, $D_7$

Order 8: $C_2\times C_4$

Order 7: $C_7$

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 56: $C_4\times D_7$ ($C_1$)

Order 28: $D_{14}$ ($C_2$), $C_{28}$ ($C_2$), $C_7:C_4$ ($C_2$)

Order 14: $D_7$ ($C_4$) x 2, $C_{14}$ ($C_2^2$)

Order 7: $C_7$ ($C_2\times C_4$)

Order 4: $C_4$ ($D_7$)

Order 2: $C_2$ ($D_{14}$)

Order 1: $C_1$ ($C_4\times D_7$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 56: $C_4\times D_7$ ($C_1$)

Order 28: $D_{14}$ ($C_2$), $C_{28}$ ($C_2$), $C_7:C_4$ ($C_2$)

Order 14: $C_{14}$ ($C_2^2$), $D_7$ ($C_4$)

Order 7: $C_7$ ($C_2\times C_4$)

Order 4: $C_4$ ($D_7$)

Order 2: $C_2$ ($D_{14}$)

Order 1: $C_1$ ($C_4\times D_7$)

Series

Derived series	$C_4\times D_7$	$\rhd$	$C_7$	$\rhd$	$C_1$
Chief series	$C_4\times D_7$	$\rhd$	$D_{14}$	$\rhd$	$C_{14}$	$\rhd$	$C_7$	$\rhd$	$C_1$
Lower central series	$C_4\times D_7$	$\rhd$	$C_7$
Upper central series	$C_1$	$\lhd$	$C_4$

Supergroups

This group is a maximal subgroup of 63 larger groups in the database.

This group is a maximal quotient of 57 larger groups in the database.

Character theory

Complex character table

See the $20 \times 20$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	2C	4A	4B	7A	14A	28A
	Size	1	1	7	7	2	14	6	6	12
	2 P	1A	1A	1A	1A	2A	2A	7A	7A	14A
	7 P	1A	2A	2B	2C	4A	4B	7A	14A	28A
56.4.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
56.4.1b		$1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$-1$
56.4.1c		$1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$1$
56.4.1d		$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$
56.4.1e		$2$	$-2$	$-2$	$2$	$0$	$0$	$2$	$-2$	$0$
56.4.1f		$2$	$-2$	$2$	$-2$	$0$	$0$	$2$	$-2$	$0$
56.4.2a		$6$	$6$	$0$	$0$	$6$	$0$	$-1$	$-1$	$-1$
56.4.2b		$6$	$6$	$0$	$0$	$-6$	$0$	$-1$	$-1$	$1$
56.4.2c		$12$	$-12$	$0$	$0$	$0$	$0$	$-2$	$2$	$0$