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D₇×C₄⋊Q₈ #90414

order	448
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

D₇×C₄⋊Q₈

Splittings

Fitting	C₇×C₄⋊Q₈ ⋊ C₂
Sylow	C₇ ⋊ C₂×C₄⋊Q₈

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Fitting	C₇×C₄⋊Q₈	C₂
prime radical	C₂²×D₇	C₂³
hypercenter	C₄⋊Q₈	D₇
hypocenter	C₇	C₂×C₄⋊Q₈
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

290 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 4 generators on 19 points:

(1,2,4,6,7,5,3)(8,9,10,11);(12,14,15,19)(13,16,17,18);(9,11)(12,13,15,17)(14,18,19,16);(2,3)(4,5)(6,7)

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×#332, #11726174, #11726176, …
maximal normal subgroup of	D₇×#332, #11726174, #11726176, …
maximal quotient of	#11731765, #11731771, #11731774, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 1176);