Abstract group 56.12: $C_2\times D_{14}$

• Label: 56.12
• Order: \( 2^{3} \cdot 7 \)
• Exponent: \( 2 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3^{2} \)
• Perm deg.: $11$
• Trans deg.: $28$
• Rank: $3$

Group information

Description:	$C_2\times D_{14}$
Order:	\(56\)\(\medspace = 2^{3} \cdot 7 \)
Exponent:	\(14\)\(\medspace = 2 \cdot 7 \)
Automorphism group:	$S_4\times F_7$, of order \(1008\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7 \)
Composition factors:	$C_2$ x 3, $C_7$
Derived length:	$2$

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

Group statistics

Order	1	2	7	14
Elements	1	31	6	18	56
Conjugacy classes	1	7	3	9	20
Divisions	1	7	1	3	12
Autjugacy classes	1	2	1	1	5

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

Minimal presentations

Permutation degree:	$11$
Transitive degree:	$28$
Rank:	$3$
Inequivalent generating triples:	$56$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	3	3	7

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{2}=c^{14}=[a,b]=[a,c]=1, c^{b}=c^{13} \rangle$
Permutation group:	Degree $11$ $\langle(2,3)(4,5)(6,7)(8,9)(10,11), (8,9)(10,11), (8,10)(9,11), (1,2,4,6,7,5,3)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 6 & 0 \\ 1 & 0 & 6 \end{array}\right), \left(\begin{array}{rrr} 5 & 5 & 2 \\ 3 & 5 & 1 \\ 5 & 3 & 3 \end{array}\right), \left(\begin{array}{rrr} 6 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & 6 \end{array}\right), \left(\begin{array}{rrr} 3 & 1 & 6 \\ 0 & 6 & 0 \\ 2 & 4 & 2 \end{array}\right) \right\rangle \subseteq \GL_{3}(\F_{7})$
Transitive group:	28T9				more information
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $D_7$
Semidirect product:	$C_7$ $\,\rtimes\,$ $C_2^3$	$C_{14}$ $\,\rtimes\,$ $C_2^2$ (3)	$(C_2\times C_{14})$ $\,\rtimes\,$ $C_2$		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 98 subgroups in 32 conjugacy classes, 21 normal (5 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $D_7$
Commutator:	$G' \simeq$ $C_7$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{14}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_2\times D_{14}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{14}$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3$
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

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

Subgroup information Click on a subgroup in the diagram to see information about it.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 56: $C_2\times D_{14}$

Order 28: $D_{14}$ x 6, $C_2\times C_{14}$

Order 14: $D_7$ x 4, $C_{14}$ x 3

Order 8: $C_2^3$

Order 7: $C_7$

Order 4: $C_2^2$ x 7

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 56: $C_2\times D_{14}$

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

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

Order 8: $C_2^3$

Order 7: $C_7$

Order 4: $C_2^2$ x 2

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 56: $C_2\times D_{14}$ ($C_1$)

Order 28: $D_{14}$ ($C_2$) x 6, $C_2\times C_{14}$ ($C_2$)

Order 14: $D_7$ ($C_2^2$) x 4, $C_{14}$ ($C_2^2$) x 3

Order 7: $C_7$ ($C_2^3$)

Order 4: $C_2^2$ ($D_7$)

Order 2: $C_2$ ($D_{14}$) x 3

Order 1: $C_1$ ($C_2\times D_{14}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 56: $C_2\times D_{14}$ ($C_1$)

Order 28: $C_2\times C_{14}$ ($C_2$), $D_{14}$ ($C_2$)

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

Order 7: $C_7$ ($C_2^3$)

Order 4: $C_2^2$ ($D_7$)

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

Order 1: $C_1$ ($C_2\times D_{14}$)

Series

Derived series	$C_2\times D_{14}$	$\rhd$	$C_7$	$\rhd$	$C_1$
Chief series	$C_2\times D_{14}$	$\rhd$	$D_{14}$	$\rhd$	$C_{14}$	$\rhd$	$C_7$	$\rhd$	$C_1$
Lower central series	$C_2\times D_{14}$	$\rhd$	$C_7$
Upper central series	$C_1$	$\lhd$	$C_2^2$

Supergroups

This group is a maximal subgroup of 44 larger groups in the database.

This group is a maximal quotient of 46 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	2D	2E	2F	2G	7A	14A	14B	14C
	Size	1	1	1	1	7	7	7	7	6	6	6	6
	2 P	1A	1A	1A	1A	1A	1A	1A	1A	7A	7A	7A	7A
	7 P	1A	2A	2B	2C	2D	2E	2F	2G	7A	14A	14B	14C
56.12.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
56.12.1b		$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$
56.12.1c		$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$
56.12.1d		$1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$
56.12.1e		$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$
56.12.1f		$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$1$
56.12.1g		$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$
56.12.1h		$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$1$
56.12.2a		$6$	$6$	$6$	$6$	$0$	$0$	$0$	$0$	$-1$	$-1$	$-1$	$-1$
56.12.2b		$6$	$-6$	$-6$	$6$	$0$	$0$	$0$	$0$	$-1$	$-1$	$1$	$1$
56.12.2c		$6$	$-6$	$6$	$-6$	$0$	$0$	$0$	$0$	$-1$	$1$	$1$	$1$
56.12.2d		$6$	$6$	$-6$	$-6$	$0$	$0$	$0$	$0$	$-1$	$1$	$-1$	$-1$