#89864

order	448
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₇ ⋊ #494

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	#494
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

122 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 448

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)(9,11)(12,13,16,19)(14,20,22,27)(15,23,25,18)(17,24,26,21)

Polycyclic presentation:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11725359, #11726195, #11726203, …
maximal normal subgroup of	#11725359, #11726195, #11726203, …
maximal quotient of	#11727331, #11727333, #11727344, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 626);