Abstract group 784.135: $D_{14}:D_{14}$

• Label: 784.135
• Order: \( 2^{4} \cdot 7^{2} \)
• Exponent: \( 2^{2} \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{2} \cdot 7^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3^{2} \)
• Perm deg.: $18$
• Trans deg.: $28$
• Rank: $3$

Group information

Description:	$D_{14}:D_{14}$
Order:	\(784\)\(\medspace = 2^{4} \cdot 7^{2} \)
Exponent:	\(28\)\(\medspace = 2^{2} \cdot 7 \)
Automorphism group:	$C_7^2.C_6^2.C_2^4$, of order \(28224\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 7^{2} \)
Composition factors:	$C_2$ x 4, $C_7$ x 2
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.

Group statistics

Order	1	2	4	7	14	28
Elements	1	227	28	48	312	168	784
Conjugacy classes	1	7	2	15	45	6	76
Divisions	1	7	2	5	15	2	32
Autjugacy classes	1	5	1	2	5	1	15

Dimension	1	2	4	6	12
Irr. complex chars.	8	26	42	0	0	76
Irr. rational chars.	8	2	0	8	14	32

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$28$
Rank:	$3$
Inequivalent generating triples:	$5376$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	4	12
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{14}=d^{14}=[a,b]=[b,d]=[c,d]=1, c^{a}=cd^{7}, d^{a}=d^{13}, c^{b}=c^{13}d^{7} \rangle$
Permutation group:	Degree $18$ $\langle(2,3)(4,5)(6,7)(9,11), (9,11)(13,14)(15,16)(17,18), (8,9)(10,11), (8,10)(9,11), (12,13,15,17,18,16,14), (1,2,4,6,7,5,3)\rangle$
Transitive group:	28T91				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$D_{14}$ $\,\rtimes\,$ $D_{14}$	$C_2^2$ $\,\rtimes\,$ $D_7^2$	$C_{14}^2$ $\,\rtimes\,$ $C_2^2$	$(C_7:D_7)$ $\,\rtimes\,$ $D_4$	all 17
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2$ . $(D_7\times D_{14})$	$C_{14}$ . $(C_2\times D_{14})$	$(C_7:D_{14})$ . $C_2^2$	$(C_7\times C_{14})$ . $C_2^3$	more information

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 1952 subgroups in 156 conjugacy classes, 34 normal (12 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_7\times D_{14}$
Commutator:	$G' \simeq$ $C_7\times C_{14}$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $D_7\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{14}^2$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $D_{14}:D_{14}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_7\times C_{14}$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Subgroup information

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

Classes of subgroups up to conjugation

Order 784: $D_{14}:D_{14}$

Order 392: $C_{14}\wr C_2$ x 2, $C_7:D_{28}$ x 2, $C_{14}:D_{14}$, $D_7\times D_{14}$, $C_{14}.D_{14}$

Order 196: $C_7:D_{14}$ x 4, $D_7^2$ x 2, $C_7:C_{28}$ x 2, $C_7\times D_{14}$ x 2, $C_{14}^2$

Order 112: $D_4\times D_7$ x 2

Order 98: $C_7:D_7$ x 3, $C_7\times C_{14}$ x 2, $C_7\times D_7$ x 2

Order 56: $C_2\times D_{14}$ x 7, $C_7:D_4$ x 4, $C_7\times D_4$ x 2, $D_{28}$ x 2, $C_4\times D_7$ x 2

Order 49: $C_7^2$

Order 28: $D_{14}$ x 32, $C_2\times C_{14}$ x 7, $C_{28}$ x 2, $C_7:C_4$ x 2

Order 16: $C_2\times D_4$

Order 14: $D_7$ x 20, $C_{14}$ x 15

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

Order 7: $C_7$ x 5

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

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 784: $D_{14}:D_{14}$

Order 392: $C_{14}:D_{14}$, $D_7\times D_{14}$, $C_{14}\wr C_2$, $C_7:D_{28}$, $C_{14}.D_{14}$

Order 196: $C_7:D_{14}$ x 3, $D_7^2$, $C_7:C_{28}$, $C_{14}^2$, $C_7\times D_{14}$

Order 112: $D_4\times D_7$

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

Order 56: $C_2\times D_{14}$ x 3, $C_7:D_4$ x 2, $C_7\times D_4$, $D_{28}$, $C_4\times D_7$

Order 49: $C_7^2$

Order 28: $D_{14}$ x 8, $C_2\times C_{14}$ x 3, $C_{28}$, $C_7:C_4$

Order 16: $C_2\times D_4$

Order 14: $C_{14}$ x 5, $D_7$ x 5

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

Order 7: $C_7$ x 2

Order 4: $C_2^2$ x 6, $C_4$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 784: $D_{14}:D_{14}$ ($C_1$)

Order 392: $C_{14}\wr C_2$ ($C_2$) x 2, $C_7:D_{28}$ ($C_2$) x 2, $C_{14}:D_{14}$ ($C_2$), $D_7\times D_{14}$ ($C_2$), $C_{14}.D_{14}$ ($C_2$)

Order 196: $C_7:C_{28}$ ($C_2^2$) x 2, $C_7:D_{14}$ ($C_2^2$) x 2, $C_7\times D_{14}$ ($C_2^2$) x 2, $C_{14}^2$ ($C_2^2$)

Order 98: $C_7:D_7$ ($D_4$) x 2, $C_7\times C_{14}$ ($C_2^3$)

Order 56: $C_7:D_4$ ($D_7$) x 2

Order 49: $C_7^2$ ($C_2\times D_4$)

Order 28: $C_2\times C_{14}$ ($D_{14}$) x 2, $D_{14}$ ($D_{14}$) x 2, $C_7:C_4$ ($D_{14}$) x 2

Order 14: $C_{14}$ ($C_2\times D_{14}$) x 2

Order 7: $C_7$ ($D_4\times D_7$) x 2

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

Order 2: $C_2$ ($D_7\times D_{14}$)

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

Normal subgroups up to automorphism (quotient in parentheses)

Order 784: $D_{14}:D_{14}$ ($C_1$)

Order 392: $C_{14}:D_{14}$ ($C_2$), $D_7\times D_{14}$ ($C_2$), $C_{14}\wr C_2$ ($C_2$), $C_7:D_{28}$ ($C_2$), $C_{14}.D_{14}$ ($C_2$)

Order 196: $C_7:D_{14}$ ($C_2^2$) x 2, $C_7:C_{28}$ ($C_2^2$), $C_{14}^2$ ($C_2^2$), $C_7\times D_{14}$ ($C_2^2$)

Order 98: $C_7\times C_{14}$ ($C_2^3$), $C_7:D_7$ ($D_4$)

Order 56: $C_7:D_4$ ($D_7$)

Order 49: $C_7^2$ ($C_2\times D_4$)

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

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

Order 7: $C_7$ ($D_4\times D_7$)

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

Order 2: $C_2$ ($D_7\times D_{14}$)

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

Series

Derived series

$D_{14}:D_{14}$

$\rhd$

$C_7\times C_{14}$

$\rhd$

$C_1$

Chief series

$D_{14}:D_{14}$

$\rhd$

$D_7\times D_{14}$

$\rhd$

$C_7\times D_{14}$

$\rhd$

$C_7\times C_{14}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$D_{14}:D_{14}$

$\rhd$

$C_7\times C_{14}$

$\rhd$

$C_7^2$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 20 larger groups in the database.

This group is a maximal quotient of 42 larger groups in the database.

Character theory

Complex character table

See the $76 \times 76$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $32 \times 32$ rational character table.