C₇×#24700 #13201916

order	1792
nilpotency class	3
isoclinism	isoclinic class of #24378
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁵

Aliases

	subgroup	G/H
center, socle	C₂³×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂×C₇×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂²×C₄²×C₇ ◁ G

932 conjugacy classes of subgroups.

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 4 generators on 27 points:

(1,2,3,4,5,6,7)(12,13,15,16)(14,18,19,17)(20,21,22,25,23,26,27,24);(8,9)(10,11)(13,16)(17,18);(9,11)(21,24)(22,27)(25,26);(12,14,15,19)(13,17,16,18)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 17688);