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D₇×#108 #89664

order	448
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

Splittings

Fitting	C₇×#108 ⋊ C₂
Sylow	C₇ ⋊ C₂×#108

Subgroups

	subgroup	G/H
center	C₄	D₄×D₇
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#108	C₂
socle	C₂×C₇	C₂×C₄⋊C₄
prime radical	C₂²×D₇	Q₈
cosocle	C₂×C₄×C₇	C₂³
hypercenter	#108	D₇
hypocenter	C₇	C₂×#108
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

106 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:

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×#359, #11726995, D₇×#361, …
maximal normal subgroup of	D₇×#359, #11726995, D₇×#361, …
maximal quotient of	D₇×#334, #11726281, #11726283, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 426);