Abstract group 448.1225: $D_8:D_{14}$

• Label: 448.1225
• Order: \( 2^{6} \cdot 7 \)
• Exponent: \( 2^{3} \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $15$
• Trans deg.: $56$
• Rank: $4$

Group information

Description:	$D_8:D_{14}$
Order:	\(448\)\(\medspace = 2^{6} \cdot 7 \)
Exponent:	\(56\)\(\medspace = 2^{3} \cdot 7 \)
Automorphism group:	$C_2\times D_4^2\times F_7$, of order \(5376\)\(\medspace = 2^{8} \cdot 3 \cdot 7 \)
Composition factors:	$C_2$ x 6, $C_7$
Derived length:	$2$

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

Group statistics

Order	1	2	4	7	8	14	28	56
Elements	1	127	64	6	64	90	48	48	448
Conjugacy classes	1	11	6	3	4	15	9	6	55
Divisions	1	11	6	1	4	5	3	2	33
Autjugacy classes	1	8	6	1	2	4	3	1	26

Dimension	1	2	4	6	8	12	24
Irr. complex chars.	16	28	8	0	3	0	0	55
Irr. rational chars.	16	4	2	8	0	2	1	33

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$56$
Rank:	$4$
Inequivalent generating quadruples:	$2298240$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	8	24
Arbitrary	6	6	10

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{2}=d^{56}=[a,b]=[a,c]=1, d^{a}=d^{41}, c^{b}=cd^{42}, d^{b}=d^{15}, d^{c}=d^{43} \rangle$
Permutation group:	Degree $15$ $\langle(10,11)(12,13)(14,15), (2,5)(6,8), (2,6)(3,7)(5,8), (1,2)(3,8)(4,5)(6,7), (1,3,4,7)(2,6,5,8), (1,4)(2,5)(3,7)(6,8), (9,10,12,14,15,13,11)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 27 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 7 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 41 & 8 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 28 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 41 & 0 \\ 0 & 13 \end{array}\right), \left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/56\Z)$
Direct product:	$D_7$ $\, \times\, $ $(D_8:C_2)$
Semidirect product:	$D_8$ $\,\rtimes\,$ $D_{14}$ (2)	$D_{56}$ $\,\rtimes\,$ $C_2^2$ (2)	$D_{28}$ $\,\rtimes\,$ $C_2^3$	$C_{56}$ $\,\rtimes\,$ $C_2^3$ (2)	all 45
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{28}$ . $C_2^3$ (2)	$C_{28}$ . $C_2^4$	$(C_4\times D_7)$ . $D_4$ (2)	$C_4$ . $(D_4\times D_7)$	all 21

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

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 1772 subgroups in 298 conjugacy classes, 101 normal (51 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_4\times D_{14}$
Commutator:	$G' \simeq$ $C_{28}$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_2^2\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $D_8:C_{14}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $D_8:D_{14}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{14}$	$G/\operatorname{soc} \simeq$ $C_2^2\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8:C_2^3$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 448: $D_8:D_{14}$

Order 224: $C_8:D_{14}$ x 2, $D_7\times \SD_{16}$ x 2, $C_8:D_{14}$ x 2, $D_7\times D_8$ x 2, $D_4:D_{14}$, $D_4\times D_{14}$, $D_8:C_{14}$, $D_4:D_{14}$, $D_{28}:C_2^2$, $C_8:D_{14}$, $D_7\times \OD_{16}$

Order 112: $D_4\times D_7$ x 8, $C_7:D_8$ x 4, $D_{56}$ x 2, $C_{56}:C_2$ x 2, $C_{56}:C_2$ x 2, $D_4:D_7$ x 2, $D_{28}:C_2$ x 2, $C_8\times D_7$ x 2, $C_4\times D_{14}$ x 2, $C_7\times \SD_{16}$ x 2, $C_7\times D_8$ x 2, $Q_8:D_7$ x 2, $C_7:\SD_{16}$ x 2, $C_7:\OD_{16}$, $C_2^2\times D_{14}$, $D_4:C_{14}$, $D_4\times C_{14}$, $C_{14}:D_4$, $D_{28}:C_2$, $Q_8\times D_7$, $C_2\times D_{28}$, $C_7\times \OD_{16}$

Order 64: $C_8:C_2^3$

Order 56: $C_2\times D_{14}$ x 11, $C_7:D_4$ x 7, $C_4\times D_7$ x 7, $C_7\times D_4$ x 5, $D_{28}$ x 5, $C_2\times C_{28}$ x 2, $C_{14}:C_4$ x 2, $C_7:Q_8$ x 2, $C_{56}$ x 2, $C_7:C_8$ x 2, $C_2^2\times C_{14}$, $C_7\times Q_8$

Order 32: $D_8:C_2$ x 8, $C_2\times \SD_{16}$ x 2, $C_2\times D_8$ x 2, $D_4:C_2^2$, $C_2^2\times D_4$, $C_2\times \OD_{16}$

Order 28: $D_{14}$ x 19, $C_2\times C_{14}$ x 6, $C_{28}$ x 3, $C_7:C_4$ x 3

Order 16: $C_2\times D_4$ x 11, $\SD_{16}$ x 8, $D_8$ x 8, $D_4:C_2$ x 6, $\OD_{16}$ x 4, $C_2\times C_8$ x 2, $C_2^2\times C_4$ x 2, $C_2^4$, $C_2\times Q_8$

Order 14: $D_7$ x 6, $C_{14}$ x 5

Order 8: $D_4$ x 17, $C_2^3$ x 12, $C_2\times C_4$ x 11, $C_8$ x 4, $Q_8$ x 3

Order 7: $C_7$

Order 4: $C_2^2$ x 25, $C_4$ x 6

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 448: $D_8:D_{14}$

Order 224: $D_4:D_{14}$, $D_4\times D_{14}$, $D_8:C_{14}$, $D_4:D_{14}$, $D_{28}:C_2^2$, $C_8:D_{14}$, $D_7\times \SD_{16}$, $C_8:D_{14}$, $D_7\times D_8$, $C_8:D_{14}$, $D_7\times \OD_{16}$

Order 112: $D_4\times D_7$ x 6, $D_4:D_7$ x 2, $D_{28}:C_2$ x 2, $C_4\times D_{14}$ x 2, $C_7:D_8$ x 2, $C_7:\OD_{16}$, $D_{56}$, $C_{56}:C_2$, $C_2^2\times D_{14}$, $D_4:C_{14}$, $C_{56}:C_2$, $D_4\times C_{14}$, $C_{14}:D_4$, $D_{28}:C_2$, $Q_8\times D_7$, $C_8\times D_7$, $C_2\times D_{28}$, $C_7\times \SD_{16}$, $C_7\times D_8$, $C_7\times \OD_{16}$, $Q_8:D_7$, $C_7:\SD_{16}$

Order 64: $C_8:C_2^3$

Order 56: $C_4\times D_7$ x 7, $C_2\times D_{14}$ x 6, $C_7:D_4$ x 5, $C_7\times D_4$ x 4, $D_{28}$ x 4, $C_2\times C_{28}$ x 2, $C_{14}:C_4$ x 2, $C_7:Q_8$ x 2, $C_{56}$, $C_2^2\times C_{14}$, $C_7\times Q_8$, $C_7:C_8$

Order 32: $D_8:C_2$ x 6, $D_4:C_2^2$, $C_2^2\times D_4$, $C_2\times \SD_{16}$, $C_2\times D_8$, $C_2\times \OD_{16}$

Order 28: $D_{14}$ x 9, $C_2\times C_{14}$ x 4, $C_{28}$ x 3, $C_7:C_4$ x 3

Order 16: $C_2\times D_4$ x 9, $D_4:C_2$ x 6, $\SD_{16}$ x 4, $D_8$ x 4, $\OD_{16}$ x 3, $C_2^2\times C_4$ x 2, $C_2\times C_8$, $C_2^4$, $C_2\times Q_8$

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

Order 8: $D_4$ x 13, $C_2\times C_4$ x 11, $C_2^3$ x 7, $Q_8$ x 3, $C_8$ x 2

Order 7: $C_7$

Order 4: $C_2^2$ x 13, $C_4$ x 6

Order 2: $C_2$ x 8

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 448: $D_8:D_{14}$ ($C_1$)

Order 224: $C_8:D_{14}$ ($C_2$) x 2, $D_7\times \SD_{16}$ ($C_2$) x 2, $C_8:D_{14}$ ($C_2$) x 2, $D_7\times D_8$ ($C_2$) x 2, $D_4:D_{14}$ ($C_2$), $D_4\times D_{14}$ ($C_2$), $D_8:C_{14}$ ($C_2$), $D_4:D_{14}$ ($C_2$), $D_{28}:C_2^2$ ($C_2$), $C_8:D_{14}$ ($C_2$), $D_7\times \OD_{16}$ ($C_2$)

Order 112: $D_4\times D_7$ ($C_2^2$) x 5, $C_7:D_8$ ($C_2^2$) x 4, $D_{56}$ ($C_2^2$) x 2, $C_{56}:C_2$ ($C_2^2$) x 2, $C_{56}:C_2$ ($C_2^2$) x 2, $C_8\times D_7$ ($C_2^2$) x 2, $C_7\times \SD_{16}$ ($C_2^2$) x 2, $C_7\times D_8$ ($C_2^2$) x 2, $Q_8:D_7$ ($C_2^2$) x 2, $C_7:\SD_{16}$ ($C_2^2$) x 2, $C_7:\OD_{16}$ ($C_2^2$), $D_4:C_{14}$ ($C_2^2$), $D_4\times C_{14}$ ($C_2^2$), $D_{28}:C_2$ ($C_2^2$), $Q_8\times D_7$ ($C_2^2$), $D_4:D_7$ ($C_2^2$), $D_{28}:C_2$ ($C_2^2$), $C_2\times D_{28}$ ($C_2^2$), $C_4\times D_{14}$ ($C_2^2$), $C_7\times \OD_{16}$ ($C_2^2$)

Order 56: $C_7\times D_4$ ($C_2^3$) x 3, $D_{28}$ ($C_2^3$) x 3, $C_4\times D_7$ ($C_2^3$) x 2, $C_4\times D_7$ ($D_4$) x 2, $C_{56}$ ($C_2^3$) x 2, $C_7:C_8$ ($C_2^3$) x 2, $C_2\times C_{28}$ ($C_2^3$), $C_{14}:C_4$ ($D_4$), $C_7:Q_8$ ($C_2^3$), $C_2\times D_{14}$ ($D_4$), $C_7\times Q_8$ ($C_2^3$)

Order 32: $D_8:C_2$ ($D_7$)

Order 28: $D_{14}$ ($C_2\times D_4$) x 2, $C_7:C_4$ ($C_2\times D_4$) x 2, $C_2\times C_{14}$ ($C_2\times D_4$), $C_{28}$ ($C_2^4$), $C_{28}$ ($C_2\times D_4$)

Order 16: $\SD_{16}$ ($D_{14}$) x 2, $D_8$ ($D_{14}$) x 2, $\OD_{16}$ ($D_{14}$), $D_4:C_2$ ($D_{14}$), $C_2\times D_4$ ($D_{14}$)

Order 14: $D_7$ ($D_8:C_2$) x 2, $C_{14}$ ($C_2^2\times D_4$)

Order 8: $D_4$ ($C_2\times D_{14}$) x 3, $C_8$ ($C_2\times D_{14}$) x 2, $Q_8$ ($C_2\times D_{14}$), $C_2\times C_4$ ($C_2\times D_{14}$)

Order 7: $C_7$ ($C_8:C_2^3$)

Order 4: $C_2^2$ ($D_4\times D_7$), $C_4$ ($C_2^2\times D_{14}$), $C_4$ ($D_4\times D_7$)

Order 2: $C_2$ ($D_4\times D_{14}$)

Order 1: $C_1$ ($D_8:D_{14}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 448: $D_8:D_{14}$ ($C_1$)

Order 224: $D_4:D_{14}$ ($C_2$), $D_4\times D_{14}$ ($C_2$), $D_8:C_{14}$ ($C_2$), $D_4:D_{14}$ ($C_2$), $D_{28}:C_2^2$ ($C_2$), $C_8:D_{14}$ ($C_2$), $D_7\times \SD_{16}$ ($C_2$), $C_8:D_{14}$ ($C_2$), $D_7\times D_8$ ($C_2$), $C_8:D_{14}$ ($C_2$), $D_7\times \OD_{16}$ ($C_2$)

Order 112: $D_4\times D_7$ ($C_2^2$) x 3, $C_7:D_8$ ($C_2^2$) x 2, $C_7:\OD_{16}$ ($C_2^2$), $D_{56}$ ($C_2^2$), $C_{56}:C_2$ ($C_2^2$), $D_4:C_{14}$ ($C_2^2$), $C_{56}:C_2$ ($C_2^2$), $D_4\times C_{14}$ ($C_2^2$), $D_{28}:C_2$ ($C_2^2$), $Q_8\times D_7$ ($C_2^2$), $D_4:D_7$ ($C_2^2$), $D_{28}:C_2$ ($C_2^2$), $C_8\times D_7$ ($C_2^2$), $C_2\times D_{28}$ ($C_2^2$), $C_4\times D_{14}$ ($C_2^2$), $C_7\times \SD_{16}$ ($C_2^2$), $C_7\times D_8$ ($C_2^2$), $C_7\times \OD_{16}$ ($C_2^2$), $Q_8:D_7$ ($C_2^2$), $C_7:\SD_{16}$ ($C_2^2$)

Order 56: $C_7\times D_4$ ($C_2^3$) x 2, $D_{28}$ ($C_2^3$) x 2, $C_4\times D_7$ ($C_2^3$) x 2, $C_4\times D_7$ ($D_4$) x 2, $C_2\times C_{28}$ ($C_2^3$), $C_{14}:C_4$ ($D_4$), $C_7:Q_8$ ($C_2^3$), $C_{56}$ ($C_2^3$), $C_2\times D_{14}$ ($D_4$), $C_7\times Q_8$ ($C_2^3$), $C_7:C_8$ ($C_2^3$)

Order 32: $D_8:C_2$ ($D_7$)

Order 28: $D_{14}$ ($C_2\times D_4$) x 2, $C_7:C_4$ ($C_2\times D_4$) x 2, $C_2\times C_{14}$ ($C_2\times D_4$), $C_{28}$ ($C_2^4$), $C_{28}$ ($C_2\times D_4$)

Order 16: $\SD_{16}$ ($D_{14}$), $D_8$ ($D_{14}$), $\OD_{16}$ ($D_{14}$), $D_4:C_2$ ($D_{14}$), $C_2\times D_4$ ($D_{14}$)

Order 14: $C_{14}$ ($C_2^2\times D_4$), $D_7$ ($D_8:C_2$)

Order 8: $D_4$ ($C_2\times D_{14}$) x 2, $Q_8$ ($C_2\times D_{14}$), $C_2\times C_4$ ($C_2\times D_{14}$), $C_8$ ($C_2\times D_{14}$)

Order 7: $C_7$ ($C_8:C_2^3$)

Order 4: $C_2^2$ ($D_4\times D_7$), $C_4$ ($C_2^2\times D_{14}$), $C_4$ ($D_4\times D_7$)

Order 2: $C_2$ ($D_4\times D_{14}$)

Order 1: $C_1$ ($D_8:D_{14}$)

Series

Derived series

$D_8:D_{14}$

$\rhd$

$C_{28}$

$\rhd$

$C_1$

Chief series

$D_8:D_{14}$

$\rhd$

$D_7\times D_8$

$\rhd$

$D_4\times D_7$

$\rhd$

$C_4\times D_7$

$\rhd$

$C_{28}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$D_8:D_{14}$

$\rhd$

$C_{28}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2\times C_4$

$\lhd$

$D_8:C_2$

Supergroups

This group is a maximal subgroup of 15 larger groups in the database.

This group is a maximal quotient of 12 larger groups in the database.

Character theory

Complex character table

See the $55 \times 55$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $33 \times 33$ rational character table.