#89492

order	448
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₇ ⋊ #433

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₇
commutator, socle	C₂²×C₇	C₂²×C₄
Fitting	C₇×C₂²⋊C₈	C₂
prime radical	C₂³×C₇	C₂×C₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₂²⋊C₈	D₇
hypocenter	C₇	#433
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

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

Polycyclic presentation:

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

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#89492, #11736029, #11736049, …
maximal normal subgroup of	C₂×#89492, #11736029, #11736049, …
maximal quotient of	#11726352, #11726358, #11726360, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 254);