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D₇×SD₃₂ #89685

order	448
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

D₇×SD₃₂

Splittings

Fitting	C₇×SD₃₂ ⋊ C₂
Sylow	C₇ ⋊ C₂×SD₃₂

Subgroups

	subgroup	G/H
center	C₂	D₇×D₈
commutator, cosocle	C₇×C₈	C₂³
Frattini	C₈	C₂²×D₇
Fitting	C₇×SD₃₂	C₂
socle	C₂×C₇	C₂×D₈
prime radical	D₇×D₈	C₂
hypercenter	SD₃₂	D₇
hypocenter	C₇	C₂×SD₃₂
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ SD₃₂

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 3 generators on 23 points:

(2,6)(3,10)(4,11)(7,8)(9,15)(12,16)(13,14)(17,18,20,22,23,21,19);(1,2,5,9)(3,6,12,15)(4,13,11,8)(7,16,14,10);(18,19)(20,21)(22,23)

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×D₇×SD₃₂, #11739469, D₇×#508, …
maximal normal subgroup of	C₂×D₇×SD₃₂, #11739469, D₇×#508, …
maximal quotient of	#11726929, #11726933, #11726937, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 447);