C₂²×D₃×#92314 #72229065

order	11664
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁶×C₃

Aliases

direct product

C₂²×D₃×#92314

Splittings

Fitting	C₂²×C₃×#6900 ⋊ C₂²
derived	C₃×C₉² ⋊ C₂⁴×C₃
Sylow	C₃×#6900 ⋊ C₂⁴

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	D₃×#92314
commutator, cosocle, hypocenter	C₃×C₉²	C₂⁴×C₃
Frattini	C₃²	C₂²×C₃×D₃×C₃⋊D₃
Fitting	C₂²×C₃×#6900	C₂²
socle	C₂²×C₃³	C₂×C₃×C₃⋊D₃
LCS	G ▷ C₃×C₉²
UCS	C₁ ◁ C₂²

8436 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 648

A minimal faithful permutation representation — 4 generators on 25 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	D₃×#72213202
maximal normal subgroup of	D₃×#72213202

GAP

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

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

C22×D3×#92314 #72229065