C₃×#13091644 #72243674

order	4374
automorphism	?
homology	Abelianization (H₁) = C₂×C₃³ · Schur multiplier (H₂) = C₃⁵

Aliases

direct product

C₃×#13091644

Splittings

Fitting, Sylow	C₃×#10624930 ⋊ C₂
derived	C₃⁴ ⋊ C₂×C₃³

Subgroups

	subgroup	G/H
center, hypercenter	C₃	#13091644
commutator, cosocle, hypocenter	C₃⁴	C₂×C₃³
Frattini	C₃²	C₃³×C₃⋊D₃
Fitting	C₃×#10624930	C₂
socle	C₃²	C₃×C₃⋊((C₃⋊D₃)⋊C₃)
LCS	G ▷ C₃⁴
UCS	C₁ ◁ C₃

4252 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 4252 conjugacy classes of subgroups — too many to draw. Only its 222 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 54

A minimal faithful permutation representation — 3 generators on 30 points:

(1,2,3)(7,15)(10,11)(13,14)(17,21)(19,20)(22,23)(25,26)(27,28)(29,30);(4,13,22,15,14,23,7,6,12)(5,20,18,10,9,27,11,19,28)(8,16,29,21,25,24,17,26,30);(4,5,8)(6,19,26)(7,11,17)(9,25,14)(10,21,15)(12,18,24)(13,20,16)(22,27,30)(23,28,29)

Conjugacy classes & character table

198 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72243695, C₃×#72221953, D₃×#13091644, …
maximal normal subgroup of	C₃×#72221953, D₃×#13091644, #72243995

GAP

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

gap> G := Group((1,2,3)(7,15)(10,11)(13,14)(17,21)(19,20)(22,23)(25,26)(27,28)(29,30), (4,13,22,15,14,23,7,6,12)(5,20,18,10,9,27,11,19,28)(8,16,29,21,25,24,17,26,30), (4,5,8)(6,19,26)(7,11,17)(9,25,14)(10,21,15)(12,18,24)(13,20,16)(22,27,30)(23,28,29));