Abstract group 28.3: $D_{14}$

• Label: 28.3
• Order: \( 2^{2} \cdot 7 \)
• Exponent: \( 2 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{2} \cdot 3 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 3 \)
• Perm deg.: $9$
• Trans deg.: $14$
• Rank: $2$

Group information

Description:	$D_{14}$
Order:	\(28\)\(\medspace = 2^{2} \cdot 7 \)
Exponent:	\(14\)\(\medspace = 2 \cdot 7 \)
Automorphism group:	$C_2\times F_7$, of order \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Composition factors:	$C_2$ x 2, $C_7$
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	7	14
Elements	1	15	6	6	28
Conjugacy classes	1	3	3	3	10
Divisions	1	3	1	1	6
Autjugacy classes	1	2	1	1	5

Dimension	1	2	6
Irr. complex chars.	4	6	0	10
Irr. rational chars.	4	0	2	6

Minimal presentations

Permutation degree:	$9$
Transitive degree:	$14$
Rank:	$2$
Inequivalent generating pairs:	$3$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	2	6
Arbitrary	2	2	6

Constructions

Presentation:	$\langle a, b \mid a^{2}=b^{14}=1, b^{a}=b^{13} \rangle$
Permutation group:	$\langle(2,3)(4,5)(6,7), (8,9), (1,2,4,6,7,5,3)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & -1 & 0 & 1 & 1 & 1 \\ -1 & 0 & 1 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrrrr} -1 & 1 & 1 & 0 & -1 & -1 \\ -1 & 1 & 0 & 0 & -1 & 0 \\ 1 & -1 & 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & -1 & -1 & 0 & 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 6 \end{array}\right), \left(\begin{array}{rr} 6 & 0 \\ 0 & 6 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{7})$
Transitive group:	14T3	28T4			more information
Direct product:	$C_2$ $\, \times\, $ $D_7$
Semidirect product:	$C_{14}$ $\,\rtimes\,$ $C_2$	$C_7$ $\,\rtimes\,$ $C_2^2$			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}^{2} $
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 28 subgroups in 10 conjugacy classes, 7 normal (5 characteristic).

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

Special subgroups

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

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

Order 7: $C_7$

Order 4: $C_2^2$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 28: $D_{14}$

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

Order 7: $C_7$

Order 4: $C_2^2$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 28: $D_{14}$ ($C_1$)

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

Order 7: $C_7$ ($C_2^2$)

Order 2: $C_2$ ($D_7$)

Order 1: $C_1$ ($D_{14}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 28: $D_{14}$ ($C_1$)

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

Order 7: $C_7$ ($C_2^2$)

Order 2: $C_2$ ($D_7$)

Order 1: $C_1$ ($D_{14}$)

Series

Derived series	$D_{14}$	$\rhd$	$C_7$	$\rhd$	$C_1$
Chief series	$D_{14}$	$\rhd$	$D_7$	$\rhd$	$C_7$	$\rhd$	$C_1$
Lower central series	$D_{14}$	$\rhd$	$C_7$
Upper central series	$C_1$	$\lhd$	$C_2$

Supergroups

This group is a maximal subgroup of 75 larger groups in the database.

This group is a maximal quotient of 81 larger groups in the database.

Character theory

Complex character table

		1A	2A	2B	2C	7A1	7A2	7A3	14A1	14A3	14A5
	Size	1	1	7	7	2	2	2	2	2	2
	2 P	1A	1A	1A	1A	7A2	7A3	7A1	7A1	7A3	7A2
	7 P	1A	2A	2B	2C	7A3	7A1	7A2	14A3	14A5	14A1
	Type
28.3.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
28.3.1b	R	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$
28.3.1c	R	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$
28.3.1d	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$
28.3.2a1	R	$2$	$2$	$0$	$0$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-3}+{\zeta }_7^3$
28.3.2a2	R	$2$	$2$	$0$	$0$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-2}+{\zeta }_7^2$
28.3.2a3	R	$2$	$2$	$0$	$0$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-1}+{\zeta }_7$
28.3.2b1	R	$2$	$-2$	$0$	$0$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-2}+{\zeta }_7^2$	$-{\zeta }_7^{-2}-{\zeta }_7^2$	$-{\zeta }_7^{-1}-{\zeta }_7$	$-{\zeta }_7^{-3}-{\zeta }_7^3$
28.3.2b2	R	$2$	$-2$	$0$	$0$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-1}+{\zeta }_7$	$-{\zeta }_7^{-1}-{\zeta }_7$	$-{\zeta }_7^{-3}-{\zeta }_7^3$	$-{\zeta }_7^{-2}-{\zeta }_7^2$
28.3.2b3	R	$2$	$-2$	$0$	$0$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7^3$	$-{\zeta }_7^{-3}-{\zeta }_7^3$	$-{\zeta }_7^{-2}-{\zeta }_7^2$	$-{\zeta }_7^{-1}-{\zeta }_7$

Rational character table

		1A	2A	2B	2C	7A	14A
	Size	1	1	7	7	6	6
	2 P	1A	1A	1A	1A	7A	7A
	7 P	1A	2A	2B	2C	7A	14A
28.3.1a		$1$	$1$	$1$	$1$	$1$	$1$
28.3.1b		$1$	$-1$	$-1$	$1$	$1$	$-1$
28.3.1c		$1$	$-1$	$1$	$-1$	$1$	$-1$
28.3.1d		$1$	$1$	$-1$	$-1$	$1$	$1$
28.3.2a		$6$	$6$	$0$	$0$	$-1$	$-1$
28.3.2b		$6$	$-6$	$0$	$0$	$-1$	$1$