#89303

order	448
isoclinism	isoclinic class of #1132
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₇×C₁₆ ⋊ C₂
Sylow	C₇ ⋊ #350

Subgroups

	subgroup	G/H
center	C₈	D₂₈
commutator	C₄×C₇	C₂×C₈
Frattini	C₂×C₈	C₂×D₇
Fitting	C₂×C₇×C₁₆	C₂
socle	C₂×C₇	C₂²⋊C₈
prime radical	#1157	C₄
cosocle	C₂×C₇×C₈	C₂²
hypercenter	C₂×C₁₆	D₇
hypocenter	C₇	#350
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂×C₁₆

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

Polycyclic presentation:

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

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11726810, #11726830, #11726876, …
maximal normal subgroup of	#11726810, #11726830, #11726876, …
maximal quotient of	#11725309, #11725312, #11725329, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 65);