C₇⋊Dic₇ #5995

order	196
isoclinism	isoclinic class of C₇⋊D₇
automorphism	Aut(G) order 197568
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₇

Aliases

C₇⋊Dic₇

C₇² ⋊ C₄

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

30 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 196

A minimal faithful permutation representation — 3 generators on 18 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(196, 6);