#89854

order	448
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₇ ⋊ #491

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₇	C₂×D₄
hypercenter	C₄⋊Q₈	D₇
hypocenter	C₇	#491
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 448

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

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

Polycyclic presentation:

2^6,7|a^2=e,b^2=d,c^2=f,e^2=f,b^a=bf,c^a=cde,g^a=g^6,c^b=cd,e^c=ef

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11725357, #11729000, #11729001, …
maximal normal subgroup of	#11725357, #11729000, #11729001, …
maximal quotient of	#11727325, #11727329, #11727337, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 616);