#89356

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

Aliases

No aliases for this group.

Splittings

Sylow

C₇ ⋊ #350

Subgroups

	subgroup	G/H
center	C₈	C₂²⋊D₇
commutator	C₄×C₇	C₂×C₈
Frattini	C₂×C₈	C₂×D₇
Fitting	C₇×#131	C₂
socle	C₂×C₇	C₂²⋊C₈
prime radical	C₇×#41	C₄
cosocle	C₂×C₇×C₈	C₂²
hypercenter	#131	D₇
hypocenter	C₇	#350
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #131

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

Polycyclic presentation:

2^6,7|a^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, #11728677, …
maximal normal subgroup of	D₇×#350, #11726879, #11728677, …
maximal quotient of	#11725477, #11725495, #11725524, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 118);