C₇⋊(C₇⋊C₈) #87635

order	392
isoclinism	isoclinic class of C₇⋊D₇
automorphism	?
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₇

Aliases

C₇⋊(C₇⋊C₈)

C₇² ⋊ C₈

	subgroup	G/H
center, Frattini, hypercenter	C₄	C₇⋊D₇
commutator, hypocenter	C₇²	C₈
Fitting, cosocle	C₄×C₇²	C₂
socle, prime radical	C₂×C₇²	C₄
LCS	G ▷ C₇²
UCS	C₁ ◁ C₄

40 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 392

A minimal faithful permutation representation — 3 generators on 22 points:

(1,2,3,5,4,6,7,8)(10,11)(12,13)(14,15)(17,18)(19,20)(21,22);(16,17,19,21,22,20,18);(9,10,12,14,15,13,11)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(392, 15);