D₃×#13091645 #72243750

order	8748
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃³ · Schur multiplier (H₂) = C₂×C₃³

Aliases

direct product

D₃×#13091645

Splittings

Fitting, Sylow

C₃×#10624930 ⋊ C₂²

Subgroups

	subgroup	G/H
center	C₃	C₃²×D₃×(C₃⋊D₃)⋊C₃
commutator, cosocle, hypocenter	C₃×H(3,𝔽₃)	C₂²×C₃³
Frattini	C₃²	C₃³×D₃²
Fitting	C₃×#10624930	C₂²
socle	C₃²	C₂×C₃²×(C₃⋊D₃)⋊C₃
hypercenter	H(3,𝔽₃)	D₃×(C₃⋊D₃)⋊C₃
LCS	G ▷ C₃×H(3,𝔽₃)
UCS	C₁ ◁ C₃ ◁ H(3,𝔽₃)

4645 conjugacy classes of subgroups; only the 301 normal subgroups are shown.

normal subgroups only

This group has 4645 conjugacy classes of subgroups — too many to draw. Only its 301 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	#72243756, D₃×#72221953
maximal normal subgroup of	D₃×#72221953

GAP

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

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