C₇⋊((C₂²⋊C₉)⋊C₂) #92629

order	504
isoclinism	isoclinic class of C₇⋊((C₂²⋊C₉)⋊C₂)
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product	C₇⋊((C₂²⋊C₉)⋊C₂)
unambiguous semidirect product	(C₂²⋊C₉)⋊D₇

derived	C₇×C₂²⋊C₉ ⋊ C₂
Sylow	C₇ ⋊ (C₂²⋊C₉)⋊C₂

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

36 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:

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

Polycyclic presentation:

2,3,7,3,2^2|b^3=d^2,b^a=b^2d,c^a=c^6,d^a=d^2,e^a=f,f^a=e,e^b=f,f^b=ef

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 55);