Abstract group 448.1086: $(Q_8\times D_{14}):C_2$

• Label: 448.1086
• Order: \( 2^{6} \cdot 7 \)
• Exponent: \( 2^{2} \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3 \)
• Perm deg.: $19$
• Trans deg.: $224$
• Rank: $4$

Group information

Description:	$(Q_8\times D_{14}):C_2$
Order:	\(448\)\(\medspace = 2^{6} \cdot 7 \)
Exponent:	\(28\)\(\medspace = 2^{2} \cdot 7 \)
Automorphism group:	$C_{14}.(C_6\times D_4).C_2^5$, of order \(21504\)\(\medspace = 2^{10} \cdot 3 \cdot 7 \)
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	14	28
Elements	1	91	164	6	42	144	448
Conjugacy classes	1	8	16	3	15	24	67
Divisions	1	8	14	1	4	7	35
Autjugacy classes	1	6	8	1	4	4	24

Dimension	1	2	4	6	12	24
Irr. complex chars.	16	32	19	0	0	0	67
Irr. rational chars.	16	4	3	8	2	2	35

Minimal presentations

Permutation degree:	$19$
Transitive degree:	$224$
Rank:	$4$
Inequivalent generating quadruples:	$574560$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	6	8	12

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{28}=d^{4}=[a,d]=[b,d]=1, b^{a}=bc^{14}, c^{a}=c^{15}d^{2}, c^{b}=c^{27}, d^{c}=d^{3} \rangle$
Permutation group:	Degree $19$ $\langle(1,2)(3,5)(4,6)(7,8)(9,10)(11,12)(14,15)(16,17)(18,19), (1,3,2,5)(4,8,6,7) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 64 & 0 \\ 0 & 64 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 21 & 97 \end{array}\right), \left(\begin{array}{rr} 71 & 0 \\ 0 & 71 \end{array}\right), \left(\begin{array}{rr} 1 & 15 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 83 & 77 \\ 56 & 97 \end{array}\right), \left(\begin{array}{rr} 97 & 70 \\ 0 & 62 \end{array}\right), \left(\begin{array}{rr} 34 & 90 \\ 70 & 29 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/105\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_7:Q_8)$ $\,\rtimes\,$ $D_4$ (4)	$C_7$ $\,\rtimes\,$ $(Q_8:D_4)$	$(Q_8\times D_{14})$ $\,\rtimes\,$ $C_2$	$(D_{14}:Q_8)$ $\,\rtimes\,$ $C_2$ (2)	all 15
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_4$ . $(D_4\times D_7)$ (2)	$(C_2\times Q_8)$ . $D_{14}$	$C_2$ . $(Q_8.D_{14})$	$C_2$ . $(D_4\times D_{14})$	all 33

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 1404 subgroups in 290 conjugacy classes, 105 normal (43 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2^2\times D_{14}$
Commutator:	$G' \simeq$ $C_2\times C_{14}$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_2^2\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{28}.D_4$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $(Q_8\times D_{14}):C_2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{14}$	$G/\operatorname{soc} \simeq$ $C_2^4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_8:D_4$
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: $(Q_8\times D_{14}):C_2$

Order 224: $D_{14}:Q_8$ x 2, $D_{28}:C_4$ x 2, $C_2^3.D_{14}$ x 2, $D_{14}:D_4$ x 2, $C_4:D_{28}$, $C_7:(C_4\times Q_8)$, $Q_8\times D_{14}$, $D_{28}:C_2^2$, $C_{28}.D_4$, $C_{28}.D_4$, $C_{28}:D_4$

Order 112: $D_{14}:C_4$ x 7, $C_4\times D_{14}$ x 5, $Q_8\times D_7$ x 4, $D_{28}:C_2$ x 4, $C_{14}:D_4$ x 3, $C_2\times D_{28}$ x 3, $C_{14}:Q_8$ x 3, $C_4:C_{28}$ x 3, $C_{14}.D_4$ x 3, $C_{28}:C_4$ x 3, $C_2^2:C_{28}$ x 2, $Q_8\times C_{14}$, $C_2^2\times C_{28}$, $C_{14}.D_4$

Order 64: $Q_8:D_4$

Order 56: $C_4\times D_7$ x 10, $C_2\times C_{28}$ x 8, $C_7:Q_8$ x 8, $C_7:D_4$ x 6, $D_{28}$ x 6, $C_{14}:C_4$ x 5, $C_2\times D_{14}$ x 3, $C_7\times Q_8$ x 2, $C_2^2\times C_{14}$

Order 32: $C_4^2:C_2$ x 3, $C_2^2:Q_8$ x 3, $C_4:D_4$ x 3, $C_4\times D_4$ x 3, $D_4:C_2^2$, $C_2^2\times Q_8$, $C_4\times Q_8$

Order 28: $D_{14}$ x 10, $C_{28}$ x 7, $C_7:C_4$ x 7, $C_2\times C_{14}$ x 4

Order 16: $C_2^2:C_4$ x 10, $C_2\times Q_8$ x 8, $C_4:C_4$ x 6, $C_2\times D_4$ x 6, $C_2^2\times C_4$ x 6, $D_4:C_2$ x 4, $C_4^2$ x 3

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

Order 8: $C_2\times C_4$ x 23, $D_4$ x 12, $Q_8$ x 10, $C_2^3$ x 4

Order 7: $C_7$

Order 4: $C_2^2$ x 14, $C_4$ x 14

Order 2: $C_2$ x 8

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 448: $(Q_8\times D_{14}):C_2$

Order 224: $D_{14}:Q_8$, $C_4:D_{28}$, $D_{28}:C_4$, $C_7:(C_4\times Q_8)$, $C_2^3.D_{14}$, $D_{14}:D_4$, $Q_8\times D_{14}$, $D_{28}:C_2^2$, $C_{28}.D_4$, $C_{28}.D_4$, $C_{28}:D_4$

Order 112: $D_{14}:C_4$ x 4, $C_4\times D_{14}$ x 3, $C_{14}:D_4$ x 2, $C_2\times D_{28}$ x 2, $C_{14}:Q_8$ x 2, $C_4:C_{28}$ x 2, $C_{14}.D_4$ x 2, $C_{28}:C_4$ x 2, $Q_8\times C_{14}$, $C_2^2\times C_{28}$, $Q_8\times D_7$, $D_{28}:C_2$, $C_2^2:C_{28}$, $C_{14}.D_4$

Order 64: $Q_8:D_4$

Order 56: $C_2\times C_{28}$ x 5, $C_{14}:C_4$ x 4, $C_4\times D_7$ x 3, $C_7:D_4$ x 2, $D_{28}$ x 2, $C_7:Q_8$ x 2, $C_2\times D_{14}$ x 2, $C_2^2\times C_{14}$, $C_7\times Q_8$

Order 32: $C_4^2:C_2$ x 2, $C_2^2:Q_8$ x 2, $C_4:D_4$ x 2, $C_4\times D_4$ x 2, $D_4:C_2^2$, $C_2^2\times Q_8$, $C_4\times Q_8$

Order 28: $D_{14}$ x 6, $C_2\times C_{14}$ x 4, $C_{28}$ x 4, $C_7:C_4$ x 4

Order 16: $C_2^2:C_4$ x 6, $C_4:C_4$ x 4, $C_2\times Q_8$ x 4, $C_2\times D_4$ x 4, $C_2^2\times C_4$ x 4, $C_4^2$ x 2, $D_4:C_2$

Order 14: $C_{14}$ x 4, $D_7$ x 2

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

Order 7: $C_7$

Order 4: $C_2^2$ x 10, $C_4$ x 8

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 448: $(Q_8\times D_{14}):C_2$ ($C_1$)

Order 224: $D_{14}:Q_8$ ($C_2$) x 2, $D_{28}:C_4$ ($C_2$) x 2, $C_2^3.D_{14}$ ($C_2$) x 2, $D_{14}:D_4$ ($C_2$) x 2, $C_4:D_{28}$ ($C_2$), $C_7:(C_4\times Q_8)$ ($C_2$), $Q_8\times D_{14}$ ($C_2$), $D_{28}:C_2^2$ ($C_2$), $C_{28}.D_4$ ($C_2$), $C_{28}.D_4$ ($C_2$), $C_{28}:D_4$ ($C_2$)

Order 112: $D_{14}:C_4$ ($C_2^2$) x 7, $C_4\times D_{14}$ ($C_2^2$) x 5, $C_{14}:D_4$ ($C_2^2$) x 3, $C_2\times D_{28}$ ($C_2^2$) x 3, $C_{14}:Q_8$ ($C_2^2$) x 3, $C_4:C_{28}$ ($C_2^2$) x 3, $C_{14}.D_4$ ($C_2^2$) x 3, $C_{28}:C_4$ ($C_2^2$) x 3, $C_2^2:C_{28}$ ($C_2^2$) x 2, $Q_8\times C_{14}$ ($C_2^2$), $C_2^2\times C_{28}$ ($C_2^2$), $C_{14}.D_4$ ($C_2^2$)

Order 56: $C_2\times C_{28}$ ($C_2^3$) x 6, $C_{14}:C_4$ ($C_2^3$) x 5, $C_7:Q_8$ ($D_4$) x 4, $C_2\times D_{14}$ ($C_2^3$) x 3, $C_2^2\times C_{14}$ ($C_2^3$)

Order 32: $C_2^2:Q_8$ ($D_7$)

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

Order 16: $C_4:C_4$ ($D_{14}$) x 3, $C_2^2:C_4$ ($D_{14}$) x 2, $C_2\times Q_8$ ($D_{14}$), $C_2^2\times C_4$ ($D_{14}$)

Order 14: $C_{14}$ ($C_4.C_2^3$), $C_{14}$ ($D_4:C_2^2$), $C_{14}$ ($C_2^2\times D_4$)

Order 8: $C_2\times C_4$ ($C_2\times D_{14}$) x 6, $C_2^3$ ($C_2\times D_{14}$)

Order 7: $C_7$ ($Q_8:D_4$)

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

Order 2: $C_2$ ($D_4:D_{14}$), $C_2$ ($Q_8.D_{14}$), $C_2$ ($D_4\times D_{14}$)

Order 1: $C_1$ ($(Q_8\times D_{14}):C_2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 448: $(Q_8\times D_{14}):C_2$ ($C_1$)

Order 224: $D_{14}:Q_8$ ($C_2$), $C_4:D_{28}$ ($C_2$), $D_{28}:C_4$ ($C_2$), $C_7:(C_4\times Q_8)$ ($C_2$), $C_2^3.D_{14}$ ($C_2$), $D_{14}:D_4$ ($C_2$), $Q_8\times D_{14}$ ($C_2$), $D_{28}:C_2^2$ ($C_2$), $C_{28}.D_4$ ($C_2$), $C_{28}.D_4$ ($C_2$), $C_{28}:D_4$ ($C_2$)

Order 112: $D_{14}:C_4$ ($C_2^2$) x 4, $C_4\times D_{14}$ ($C_2^2$) x 3, $C_{14}:D_4$ ($C_2^2$) x 2, $C_2\times D_{28}$ ($C_2^2$) x 2, $C_{14}:Q_8$ ($C_2^2$) x 2, $C_4:C_{28}$ ($C_2^2$) x 2, $C_{14}.D_4$ ($C_2^2$) x 2, $C_{28}:C_4$ ($C_2^2$) x 2, $Q_8\times C_{14}$ ($C_2^2$), $C_2^2\times C_{28}$ ($C_2^2$), $C_2^2:C_{28}$ ($C_2^2$), $C_{14}.D_4$ ($C_2^2$)

Order 56: $C_2\times C_{28}$ ($C_2^3$) x 4, $C_{14}:C_4$ ($C_2^3$) x 4, $C_2\times D_{14}$ ($C_2^3$) x 2, $C_7:Q_8$ ($D_4$), $C_2^2\times C_{14}$ ($C_2^3$)

Order 32: $C_2^2:Q_8$ ($D_7$)

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

Order 16: $C_4:C_4$ ($D_{14}$) x 2, $C_2^2:C_4$ ($D_{14}$), $C_2\times Q_8$ ($D_{14}$), $C_2^2\times C_4$ ($D_{14}$)

Order 14: $C_{14}$ ($C_4.C_2^3$), $C_{14}$ ($D_4:C_2^2$), $C_{14}$ ($C_2^2\times D_4$)

Order 8: $C_2\times C_4$ ($C_2\times D_{14}$) x 4, $C_2^3$ ($C_2\times D_{14}$)

Order 7: $C_7$ ($Q_8:D_4$)

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

Order 2: $C_2$ ($D_4:D_{14}$), $C_2$ ($Q_8.D_{14}$), $C_2$ ($D_4\times D_{14}$)

Order 1: $C_1$ ($(Q_8\times D_{14}):C_2$)

Series

Derived series

$(Q_8\times D_{14}):C_2$

$\rhd$

$C_2\times C_{14}$

$\rhd$

$C_1$

Chief series

$(Q_8\times D_{14}):C_2$

$\rhd$

$C_{28}:D_4$

$\rhd$

$C_2^2\times C_{28}$

$\rhd$

$C_2\times C_{28}$

$\rhd$

$C_{28}$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$(Q_8\times D_{14}):C_2$

$\rhd$

$C_2\times C_{14}$

$\rhd$

$C_7$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2^2:Q_8$

Supergroups

This group is a maximal subgroup of 13 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 $67 \times 67$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $35 \times 35$ rational character table.