#89310

order	448
isoclinism	isoclinic class of #1141
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting	C₇×M₃₂ ⋊ C₂
Sylow	C₇ ⋊ #350

Subgroups

	subgroup	G/H
center	C₈	C₂²⋊D₇
commutator	C₄×C₇	C₂×C₈
Frattini	C₂×C₈	C₂×D₇
Fitting	C₇×M₃₂	C₂
socle	C₂×C₇	C₂²⋊C₈
prime radical	#1157	C₄
cosocle	C₂×C₇×C₈	C₂²
hypercenter	M₃₂	D₇
hypocenter	C₇	#350
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ M₃₂

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 2 generators on 39 points:

(1,2,4,6,7,5,3)(8,10,11,20,13,21,24,35,14,22,25,36,12,27,23,37)(9,15,16,28,17,29,31,38,18,30,32,39,26,19,33,34);(2,3)(4,5)(6,7)(8,9)(10,19)(11,16)(12,26)(13,17)(14,18)(15,21)(20,34)(22,29)(23,33)(24,31)(25,32)(27,30)(28,35)(36,38)(37,39)

Polycyclic presentation:

2^6,7|b^2=c,c^2=e,d^2=f,e^2=f,b^a=bd,d^a=df,g^a=g^6,d^b=df

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×#350, #11726879, #11727012, …
maximal normal subgroup of	D₇×#350, #11726879, #11727012, …
maximal quotient of	#11725321, #11725324, #11725329, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 72);