#89774

order	448
isoclinism	isoclinic class of #1141
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₇×#117 ⋊ C₂
Sylow	C₇ ⋊ #416

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₇
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#117	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂×#1157	C₂
cosocle	C₂×C₄×C₇	C₂³
hypercenter	#117	D₇
hypocenter	C₇	#416
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #117

158 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)(8,9,10,11)(13,18)(15,23)(17,24)(19,20)(21,27)(25,26);(12,14,16,22)(13,17,20,26)(15,21,23,27)(18,24,19,25);(2,3)(4,5)(6,7)(12,13)(14,17)(15,18)(16,20)(19,23)(21,24)(22,26)(25,27)

Polycyclic presentation:

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

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11726452, #11726454, #11726493, …
maximal normal subgroup of	#11726452, #11726454, #11726493, …
maximal quotient of	#11727185, #11727197, C₄×#6383, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 536);