#89817

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×C₂²⋊Q₈	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂³×C₇	D₄
hypercenter	C₂²⋊Q₈	D₇
hypocenter	C₇	#485
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊Q₈

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

Polycyclic presentation:

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

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11737581, #11737582, #11737599, …
maximal normal subgroup of	#11737581, #11737582, #11737599, …
maximal quotient of	#11727257, #11727260, #11727272, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 579);