C₃×#72243789 #72277486

order	8748
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×#72243789

Splittings

Fitting, Sylow	C₃×#10624754 ⋊ C₂²
derived	#10624754 ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃	#72243789
commutator, cosocle, hypocenter	#10624754	C₂²×C₃
Frattini	C₃³	C₃×D₃×C₃⋊D₃
Fitting	C₃×#10624754	C₂²
socle	C₃²	#11757730
LCS	G ▷ #10624754
UCS	C₁ ◁ C₃

2433 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 2433 conjugacy classes of subgroups — too many to draw. Only its 66 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)(5,10,12,18,9,8)(6,14)(7,17)(11,28,15,29,20,22)(13,24)(16,25)(19,23,21,26,27,30);(4,6,25,17,16,24)(5,19,30,10,15,29)(7,13,14)(8,11,28,9,27,26)(12,21,23,18,20,22);(4,9,8,17,12,10,7,5,18)(6,11,21,13,20,27,16,15,19)(14,23,28,25,30,29,24,26,22)

Conjugacy classes & character table

135 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72277755
automorphism group of	#6866, #6867

GAP

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

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