D₃×#72209832 #72233506

order	18432
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁷

Aliases

Fitting	C₃×#6499874 ⋊ C₂×D₃
derived	C₃×#11710730 ⋊ C₂³

	subgroup	G/H
center	C₂	D₃×#72209653
commutator, cosocle, hypocenter	C₃×#11710730	C₂³
Frattini	C₂⁴	C₂×D₃×#1020
Fitting	C₃×#6499874	C₂×D₃
socle	C₂×C₃	C₂×#72209653
hypercenter	C₂²	D₃×#87588⋊C₂
LCS	G ▷ C₃×#11710730
UCS	C₁ ◁ C₂ ◁ C₂²

At least 40791 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has at least 40791 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 3 generators on 19 points:

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

automorphism group of	C₂²×Dic₆, C₂²×D₁₂
holomorph of	C₂²×C₃×C₄

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((3,4,9)(6,14,12)(7,16,13)(10,11,15)(18,19), (1,6,5,13)(2,10,8,3)(4,12,11,16)(7,15,14,9)(17,18,19), (3,4)(6,7)(10,11)(13,14));