#89769

order	448
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₇ ⋊ #405

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₇
commutator	C₂×C₇	C₂×C₄²
Fitting	C₇×#117	C₂
socle	C₂²×C₇	C₂²×C₄
prime radical	C₂³×C₇	C₂×C₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	#117	D₇
hypocenter	C₇	#405
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #117

130 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)(8,9,10,11)(14,20)(17,23)(21,26)(24,27);(12,14)(13,17)(15,21)(16,20)(18,24)(19,23)(22,26)(25,27);(2,3)(4,5)(6,7)(12,13,15,18,16,19,22,25)(14,17,21,24,20,23,26,27)

Polycyclic presentation:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11726451, #11726453, #11726493, …
maximal normal subgroup of	#11726451, #11726453, #11726493, …
maximal quotient of	#11727188, #11727189, #11727190, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 531);