C₂²×C₃×C₂²⋊Q₈ #87317

order	384
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃ · Schur multiplier (H₂) = C₂⁹

Aliases

C₂²×C₃×C₂²⋊Q₈

	subgroup	G/H
center, socle	C₂⁴×C₃	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃
prime radical	C₂⁵×C₃	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ G

1840 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 5 generators on 19 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 19893);