C₇×((C₂²⋊C₉)⋊C₂)⋊C₃ #13094595

order	1512
isoclinism	isoclinic class of ((C₂²⋊C₉)⋊C₂)⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₇×((C₂²⋊C₉)⋊C₂)⋊C₃
unambiguous semidirect product	((C₂²⋊C₉)⋊C₂)⋊(C₃×C₇)
unambiguous semidirect product	#1094⋊(C₂×C₇)

	subgroup	G/H
center, hypercenter	C₇	((C₂²⋊C₉)⋊C₂)⋊C₃
commutator, cosocle, hypocenter	C₂²⋊C₉	C₂×C₃×C₇
Frattini	C₃	C₃×C₇×S₄
Fitting	C₂²×C₃²×C₇	D₃
socle	C₂²×C₃×C₇	C₃×D₃
LCS	G ▷ C₂²⋊C₉
UCS	C₁ ◁ C₇

78 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 126

A minimal faithful permutation representation — 2 generators on 20 points:

(1,2,3,4,5,6,7)(8,9,11,10)(13,17,20,15,16,19)(14,18);(9,10,11)(12,20,15,18,16,17,14,13,19)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1512, 492);