C₂²×D₃² #3849

order	144
isoclinism	isoclinic class of D₃²
automorphism	Aut(G) = #72241545 (order 6912) · Inn(G) = D₃² · Out(G) = #1020⋊C₂
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product	C₂²×D₃²
unambiguous semidirect product	(C₃×D₃)⋊C₂³
unambiguous semidirect product	(C₃⋊D₃)⋊C₂³

Fitting, socle	C₂²×C₃² ⋊ C₂²
derived, Sylow	C₃² ⋊ C₂⁴

	subgroup	G/H
center, hypercenter	C₂²	D₃²
commutator, cosocle, hypocenter	C₃²	C₂⁴
Frattini	C₁	G
Fitting, socle	C₂²×C₃²	C₂²
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂²

284 conjugacy classes of subgroups.

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 4 generators on 10 points:

(1,2)(3,4)(6,7)(8,9,10);(5,6,7)(9,10);(3,4);(1,2)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 192);