#89802

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

Aliases

No aliases for this group.

Splittings

Sylow

C₇ ⋊ #441

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₇
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₄×C₇×Q₈	C₂
socle, prime radical	C₂²×C₇	C₂×D₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₄×Q₈	D₇
hypocenter	C₇	#441
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

108 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)(8,9,10,11)(14,21)(17,26)(20,27)(22,24);(12,14,16,21)(13,17,19,26)(15,24,25,22)(18,27,23,20);(2,3)(4,5)(6,7)(12,13,16,19)(14,20,21,27)(15,23,25,18)(17,24,26,22)

Polycyclic presentation:

2^6,7|a^2=f,b^2=f,c^2=d,e^2=f,b^a=be,e^a=ef,g^a=g^6,c^b=cf,e^b=ef

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11737093, #11737095, #11737102, …
maximal normal subgroup of	#11737093, #11737095, #11737102, …
maximal quotient of	#11727212, #11727218, #11727229, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 564);