#89822

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

Aliases

No aliases for this group.

Splittings

Sylow

C₇ ⋊ #451

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	#451
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊Q₈

148 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 27 points:

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

Polycyclic presentation:

2^6,7|a^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	#11726072, #11726073, #11726080, …
maximal normal subgroup of	#11726072, #11726073, #11726080, …
maximal quotient of	#11727258, #11727279, #11727281, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 584);