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D₇×#127 #90405

order	448
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

Splittings

Fitting	C₇×#127 ⋊ C₂
Sylow	C₇ ⋊ C₂×#127

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Fitting	C₇×#127	C₂
hypercenter	#127	D₇
hypocenter	C₇	C₂×#127
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #127

498 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 4 generators on 15 points:

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×#331, #11726137, #11726141, …
maximal normal subgroup of	D₇×#331, #11726137, #11726141, …
maximal quotient of	#11731764, #11731773, #11731779, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 1167);