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D₇×M₃₂ #89678

order	448
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

D₇×M₃₂

Splittings

Fitting	C₇×M₃₂ ⋊ C₂
Sylow	C₇ ⋊ C₂×M₃₂

Subgroups

	subgroup	G/H
center, Frattini	C₈	C₂²×D₇
commutator, socle	C₂×C₇	C₂²×C₈
Fitting	C₇×M₃₂	C₂
prime radical	C₂²×D₇	C₈
cosocle	C₇×C₈	C₂³
hypercenter	M₃₂	D₇
hypocenter	C₇	C₂×M₃₂
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₈ ◁ M₃₂

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×#347, #11726841, D₇×#349, …
maximal normal subgroup of	D₇×#347, #11726841, D₇×#349, …
maximal quotient of	#11726822, D₇×#346, #11726825, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 440);