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D₇×H(5,𝔽₂) #90617

order	448
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

D₇×H(5,𝔽₂)

Splittings

Fitting	C₇×H(5,𝔽₂) ⋊ C₂
Sylow	C₇ ⋊ C₂×H(5,𝔽₂)

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂⁴×D₇
commutator, socle, cosocle	C₂×C₇	C₂⁵
Fitting	C₇×H(5,𝔽₂)	C₂
hypercenter	H(5,𝔽₂)	D₇
hypocenter	C₇	C₂×H(5,𝔽₂)
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ H(5,𝔽₂)

898 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 56

A minimal faithful permutation representation — 5 generators on 15 points:

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

Conjugacy classes & character table

85 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×#453, #11737723, D₇×#455, …
maximal normal subgroup of	D₇×#453, #11737723, D₇×#455, …
maximal quotient of	#11742925, #11742928, #11742929, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 1379);