#89601

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

Aliases

No aliases for this group.

Splittings

Sylow

C₇ ⋊ #479

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₂₈
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×C₄⋊C₈	C₂
socle, prime radical	C₂²×C₇	C₂×D₄
hypercenter	C₄⋊C₈	D₇
hypocenter	C₇	#479
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

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

Polycyclic presentation:

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

Conjugacy classes & character table

79 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11736470, #11736474, #11736475, …
maximal normal subgroup of	#11736470, #11736474, #11736475, …
maximal quotient of	#11726375, #11726377, #11726388, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 363);