#89768

order	448
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₇ ⋊ #485

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#117	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂³×C₇	D₄
hypercenter	#117	D₇
hypocenter	C₇	#485
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #117

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 31 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

79 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11736292, #11736293, #11736294, …
maximal normal subgroup of	#11736292, #11736293, #11736294, …
maximal quotient of	#11727212, #11727213, #11727218, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 530);