Abstract group 448.336: $(Q_8\times D_7):C_4$

• Label: 448.336
• Order: \( 2^{6} \cdot 7 \)
• Exponent: \( 2^{3} \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \)
• Perm deg.: $27$
• Trans deg.: $224$
• Rank: $3$

Group information

Description:	$(Q_8\times D_7):C_4$
Order:	\(448\)\(\medspace = 2^{6} \cdot 7 \)
Exponent:	\(56\)\(\medspace = 2^{3} \cdot 7 \)
Automorphism group:	$(C_2^3\times F_7).C_2^5$, of order \(10752\)\(\medspace = 2^{9} \cdot 3 \cdot 7 \)
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	31	160	6	64	18	120	48	448
Conjugacy classes	1	5	12	3	4	9	18	12	64
Divisions	1	5	10	1	2	3	5	1	28
Autjugacy classes	1	4	7	1	2	3	4	1	23

Dimension	1	2	4	6	12	24
Irr. complex chars.	16	28	20	0	0	0	64
Irr. rational chars.	8	8	2	4	4	2	28

Minimal presentations

Permutation degree:	$27$
Transitive degree:	$224$
Rank:	$3$
Inequivalent generating triples:	$2688$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	5	10	16

Constructions

Presentation:	$\langle a, b, c, d \mid b^{2}=c^{56}=d^{2}=[a,d]=[b,d]=[c,d]=1, a^{2}=d, b^{a}=bc^{28}, c^{a}=c^{43}d, c^{b}=c^{13} \rangle$
Permutation group:	Degree $27$ $\langle(2,3)(4,5)(6,7)(14,20)(17,26)(21,27)(22,24), (8,9)(10,11)(12,13,16,19)(14,21,20,27) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(Q_8\times D_7)$ $\,\rtimes\,$ $C_4$ (4)	$(Q_8:C_4)$ $\,\rtimes\,$ $D_7$	$(Q_8:C_{28})$ $\,\rtimes\,$ $C_2$	$(C_4.D_{28})$ $\,\rtimes\,$ $C_2$	all 7
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$Q_8$ . $(C_4\times D_7)$ (2)	$(C_4\times D_7)$ . $D_4$ (2)	$C_4$ . $(D_4\times D_7)$	$(Q_8\times D_{14})$ . $C_2$	all 35

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

Homology

Abelianization:	$C_{2}^{2} \times C_{4} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 724 subgroups in 150 conjugacy classes, 55 normal (37 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $D_4\times D_7$
Commutator:	$G' \simeq$ $C_{28}$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $C_2\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $Q_8:C_{28}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $(Q_8\times D_7):C_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{14}$	$G/\operatorname{soc} \simeq$ $C_2\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3.D_4$
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

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 448: $(Q_8\times D_7):C_4$

Order 224: $C_8:D_{14}$, $(C_4\times D_7):C_4$, $Q_8:C_{28}$, $C_{14}.Q_{16}$, $C_{14}.Q_{16}$, $Q_8\times D_{14}$, $C_4.D_{28}$

Order 112: $Q_8\times D_7$ x 6, $C_{56}:C_2$ x 2, $C_4\times D_{14}$ x 2, $C_{14}:Q_8$ x 2, $C_{14}:C_8$, $Q_8\times C_{14}$, $C_2\times C_{56}$, $C_4:C_{28}$, $D_{14}:C_4$, $C_{28}:C_4$, $C_{28}:C_4$

Order 64: $C_2^3.D_4$

Order 56: $C_4\times D_7$ x 8, $C_7:Q_8$ x 7, $C_2\times C_{28}$ x 3, $C_{14}:C_4$ x 3, $C_7\times Q_8$ x 3, $C_{56}$, $C_2\times D_{14}$, $C_7:C_8$

Order 32: $Q_8:C_4$ x 4, $C_2^2\times Q_8$, $C_2\times \OD_{16}$, $C_4^2:C_2$

Order 28: $C_{28}$ x 5, $C_7:C_4$ x 5, $D_{14}$ x 4, $C_2\times C_{14}$

Order 16: $C_2\times Q_8$ x 9, $\OD_{16}$ x 2, $C_2\times C_8$ x 2, $C_4:C_4$ x 2, $C_2^2\times C_4$ x 2, $C_2^2:C_4$, $C_4^2$

Order 14: $C_{14}$ x 3, $D_7$ x 2

Order 8: $C_2\times C_4$ x 14, $Q_8$ x 10, $C_8$ x 2, $C_2^3$

Order 7: $C_7$

Order 4: $C_4$ x 10, $C_2^2$ x 5

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 448: $(Q_8\times D_7):C_4$

Order 224: $C_8:D_{14}$, $(C_4\times D_7):C_4$, $Q_8:C_{28}$, $C_{14}.Q_{16}$, $C_{14}.Q_{16}$, $Q_8\times D_{14}$, $C_4.D_{28}$

Order 112: $Q_8\times D_7$ x 2, $C_4\times D_{14}$ x 2, $C_{14}:Q_8$ x 2, $C_{14}:C_8$, $C_{56}:C_2$, $Q_8\times C_{14}$, $C_2\times C_{56}$, $C_4:C_{28}$, $D_{14}:C_4$, $C_{28}:C_4$, $C_{28}:C_4$

Order 64: $C_2^3.D_4$

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

Order 32: $Q_8:C_4$ x 4, $C_2^2\times Q_8$, $C_2\times \OD_{16}$, $C_4^2:C_2$

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

Order 16: $C_2\times Q_8$ x 5, $C_2\times C_8$ x 2, $C_4:C_4$ x 2, $C_2^2\times C_4$ x 2, $\OD_{16}$, $C_2^2:C_4$, $C_4^2$

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

Order 8: $C_2\times C_4$ x 9, $Q_8$ x 5, $C_8$ x 2, $C_2^3$

Order 7: $C_7$

Order 4: $C_4$ x 7, $C_2^2$ x 4

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 448: $(Q_8\times D_7):C_4$ ($C_1$)

Order 224: $C_8:D_{14}$ ($C_2$), $(C_4\times D_7):C_4$ ($C_2$), $Q_8:C_{28}$ ($C_2$), $C_{14}.Q_{16}$ ($C_2$), $C_{14}.Q_{16}$ ($C_2$), $Q_8\times D_{14}$ ($C_2$), $C_4.D_{28}$ ($C_2$)

Order 112: $Q_8\times D_7$ ($C_4$) x 4, $C_{14}:C_8$ ($C_2^2$), $Q_8\times C_{14}$ ($C_2^2$), $C_4\times D_{14}$ ($C_2^2$), $C_{14}:Q_8$ ($C_2^2$), $C_2\times C_{56}$ ($C_2^2$), $C_4:C_{28}$ ($C_2^2$), $C_{28}:C_4$ ($C_2^2$)

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

Order 32: $Q_8:C_4$ ($D_7$)

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

Order 16: $C_2\times C_8$ ($D_{14}$), $C_4:C_4$ ($D_{14}$), $C_2\times Q_8$ ($D_{14}$)

Order 14: $C_{14}$ ($Q_{16}:C_2$) x 2, $C_{14}$ ($C_2^3:C_4$)

Order 8: $Q_8$ ($C_4\times D_7$) x 2, $C_2\times C_4$ ($C_2\times D_{14}$)

Order 7: $C_7$ ($C_2^3.D_4$)

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

Order 2: $C_2$ ($D_{14}.D_4$), $C_2$ ($Q_{16}:D_7$), $C_2$ ($C_8.D_{14}$)

Order 1: $C_1$ ($(Q_8\times D_7):C_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 448: $(Q_8\times D_7):C_4$ ($C_1$)

Order 224: $C_8:D_{14}$ ($C_2$), $(C_4\times D_7):C_4$ ($C_2$), $Q_8:C_{28}$ ($C_2$), $C_{14}.Q_{16}$ ($C_2$), $C_{14}.Q_{16}$ ($C_2$), $Q_8\times D_{14}$ ($C_2$), $C_4.D_{28}$ ($C_2$)

Order 112: $C_{14}:C_8$ ($C_2^2$), $Q_8\times C_{14}$ ($C_2^2$), $Q_8\times D_7$ ($C_4$), $C_4\times D_{14}$ ($C_2^2$), $C_{14}:Q_8$ ($C_2^2$), $C_2\times C_{56}$ ($C_2^2$), $C_4:C_{28}$ ($C_2^2$), $C_{28}:C_4$ ($C_2^2$)

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

Order 32: $Q_8:C_4$ ($D_7$)

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

Order 16: $C_2\times C_8$ ($D_{14}$), $C_4:C_4$ ($D_{14}$), $C_2\times Q_8$ ($D_{14}$)

Order 14: $C_{14}$ ($Q_{16}:C_2$) x 2, $C_{14}$ ($C_2^3:C_4$)

Order 8: $Q_8$ ($C_4\times D_7$), $C_2\times C_4$ ($C_2\times D_{14}$)

Order 7: $C_7$ ($C_2^3.D_4$)

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

Order 2: $C_2$ ($D_{14}.D_4$), $C_2$ ($Q_{16}:D_7$), $C_2$ ($C_8.D_{14}$)

Order 1: $C_1$ ($(Q_8\times D_7):C_4$)

Series

Derived series

$(Q_8\times D_7):C_4$

$\rhd$

$C_{28}$

$\rhd$

$C_1$

Chief series

$(Q_8\times D_7):C_4$

$\rhd$

$(C_4\times D_7):C_4$

$\rhd$

$C_4:C_{28}$

$\rhd$

$C_2\times C_{28}$

$\rhd$

$C_2\times C_{14}$

$\rhd$

$C_{14}$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$(Q_8\times D_7):C_4$

$\rhd$

$C_{28}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times C_4$

$\lhd$

$Q_8:C_4$

Supergroups

This group is a maximal subgroup of 10 larger groups in the database.

This group is a maximal quotient of 8 larger groups in the database.

Character theory

Complex character table

See the $64 \times 64$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $28 \times 28$ rational character table.