#89874

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

Aliases

No aliases for this group.

Splittings

Sylow

C₇ ⋊ #424

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₇
commutator, socle	C₂²×C₇	C₂²×C₄
Fitting	C₂²×C₇×C₈	C₂
prime radical	C₂³×C₇	C₂×C₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₂²×C₈	D₇
hypocenter	C₇	#424
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈

114 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 27 points:

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

Polycyclic presentation:

2^6,7|a^2=e,c^2=d,d^2=f,b^a=bf,c^a=ce,g^a=g^6

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11725331, #11726399, #11726448, …
maximal normal subgroup of	#11725331, #11726399, #11726448, …
maximal quotient of	#11727509, #11727540, #11727663, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 636);