C₇⋊#261 #67245

order	364
isoclinism	isoclinic class of C₇⋊#261
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₇×C₁₃	C₄
prime radical, cosocle	C₇×D₁₃	C₂
LCS	G ▷ C₇×C₁₃
UCS	C₁

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 91

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

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

Polycyclic presentation:

2^2,7,13|a^2=b,c^a=c^6,d^a=d^5,d^b=d^12

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(364, 6);