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D₇×#101 #89522

order	448
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

Splittings

Fitting	C₇×#101 ⋊ C₂
Sylow	C₇ ⋊ C₂×#101

Subgroups

	subgroup	G/H
center	C₂	D₇×C₂²⋊C₄
commutator	C₂²×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#101	C₂
socle	C₂×C₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₇	C₂×C₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	#101	D₇
hypocenter	C₇	C₂×#101
LCS	G ▷ C₂²×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

146 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:

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

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×#355, #11726918, D₇×#356, …
maximal normal subgroup of	D₇×#355, #11726918, D₇×#356, …
maximal quotient of	#11725879, #11725881, #11725882, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 284);