D₃×#11402 #72225845

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

D₃×#11402

Splittings

Fitting	C₃×#11402 ⋊ C₂
Sylow	C₃ ⋊ C₂×#11402

Subgroups

	subgroup	G/H
center, Frattini	C₄×C₈	C₂³×D₃
commutator, socle	C₂²×C₃	C₂²×C₄×C₈
Fitting	C₃×#11402	C₂
prime radical	C₂³×D₃	C₄×C₈
cosocle	C₃×C₄×C₈	C₂⁴
hypercenter	#11402	D₃
hypocenter	C₃	C₂×#11402
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₈ ◁ #11402

1532 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 384

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

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

Conjugacy classes & character table

336 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃×#126544, D₃×#122220
maximal normal subgroup of	D₃×#126544, D₃×#122220
maximal quotient of	D₃×#122220

GAP

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

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

D3×#11402 #72225845