#89292

order	448
isoclinism	isoclinic class of #6400
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting	C₇×#108 ⋊ C₂
Sylow	C₇ ⋊ #359

Subgroups

	subgroup	G/H
center	C₄	#1141
commutator	C₇×C₈	C₂×C₄
Frattini	C₂×C₈	C₂×D₇
Fitting	C₇×#108	C₂
socle	C₂×C₇	#102
prime radical	#6467	C₂
cosocle	C₂×C₇×C₈	C₂²
hypercenter	#108	D₇
hypocenter	C₇	#359
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #108

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

Polycyclic presentation:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×#359, #11726995, #11727012, …
maximal normal subgroup of	D₇×#359, #11726995, #11727012, …
maximal quotient of	#11725273, #11725274, #11725277, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 54);