#89649

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	D₄×D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#107	C₂
socle, prime radical	C₂²×C₇	C₂×D₄
hypercenter	#107	D₇
hypocenter	C₇	#479
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

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

Polycyclic presentation:

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

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11736600, #11736604, #11736607, …
maximal normal subgroup of	#11736600, #11736604, #11736607, …
maximal quotient of	#11726543, #11726556, #11726563, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 411);