#89794

order	448
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₇ ⋊ #479

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₄×C₇×Q₈	C₂
socle, prime radical	C₂²×C₇	C₂×D₄
hypercenter	C₄×Q₈	D₇
hypocenter	C₇	#479
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 31 points:

(1,2,4,6,7,5,3)(16,18,20,26)(17,22,21,30)(19,28,29,25)(23,31,27,24);(8,10,11,15)(9,12,13,14)(17,21)(22,30)(23,27)(24,31);(2,3)(4,5)(6,7)(8,9,11,13)(10,14,15,12)(16,17,20,21)(18,24,26,31)(19,27,29,23)(22,28,30,25)

Polycyclic presentation:

2^6,7|a^2=e,b^2=ef,c^2=f,d^2=f,b^a=be,c^a=cd,d^a=df,g^a=g^6,d^c=df

Conjugacy classes & character table

79 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11737065, #11737075, #11737078, …
maximal normal subgroup of	#11737065, #11737075, #11737078, …
maximal quotient of	#11727212, #11727214, #11727222, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 556);