D₃×#10624930 #72243651

order	4374
automorphism	?
homology	Abelianization (H₁) = C₂×C₃⁴ · Schur multiplier (H₂) = C₃⁶

Aliases

direct product

D₃×#10624930

Splittings

Fitting, Sylow

C₃×#10624930 ⋊ C₂

Subgroups

	subgroup	G/H
center	C₃	C₃²×D₃×H(3,𝔽₃)
commutator, cosocle	C₃³	C₂×C₃⁴
Frattini	C₃²	C₃⁴×D₃
Fitting	C₃×#10624930	C₂
socle	C₃²	C₂×C₃²×H(3,𝔽₃)
hypercenter	#10624930	D₃
hypocenter	C₃	C₂×#10624930
LCS	G ▷ C₃³ ▷ C₃² ▷ C₃
UCS	C₁ ◁ C₃ ◁ C₃×H(3,𝔽₃) ◁ #10624930

5272 conjugacy classes of subgroups; only the 705 normal subgroups are shown.

normal subgroups only

This group has 5272 conjugacy classes of subgroups — too many to draw. Only its 705 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 — 4 generators on 30 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	#72243652, D₃×#13091645, D₃×#13091647, …
maximal normal subgroup of	D₃×#13091645, D₃×#13091647, D₃×#13091644

GAP

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

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