C₃×#13164434 #72252991

order	5184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂³×C₃

solvable

Aliases

direct product

C₃×#13164434

Subgroups

	subgroup	G/H
center	C₂×C₃	C₂²×((C₂²⋊C₉)⋊C₂)⋊C₃
commutator, cosocle, hypocenter	Q₈⋊C₉	C₂³×C₃²
Frattini	C₂×C₃	C₂²×C₃²×S₄
Fitting	C₃³×H(5,𝔽₂)	D₃
socle	C₂×C₃²	C₂²×C₃×S₄
hypercenter	C₃×Q₈	((C₂²⋊C₉)⋊C₂)⋊C₃
LCS	G ▷ Q₈⋊C₉
UCS	C₁ ◁ C₂×C₃ ◁ C₃×Q₈

1838 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 1838 conjugacy classes of subgroups — too many to draw. Only its 184 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) = 44
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 3 generators on 44 points:

(1,16,18,4,21,9,7,10,5,2,6,15)(3,23,27,8,30,25,13,31,11,14,12,22)(17,19,20,24)(26,32,29,28)(33,34,35)(36,40,39,38,44,41,42,37,43);(1,3,7,13)(2,8,4,14)(5,11,18,27)(6,12,21,30)(9,22,15,25)(10,23,16,31)(17,26,20,29)(19,28,24,32)(37,40,44)(39,43,41);(1,2,7,4)(3,8,13,14)(5,15,18,9)(6,19,21,24)(10,20,16,17)(11,25,27,22)(12,28,30,32)(23,29,31,26)(37,39)(38,42)(40,43)(41,44)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	#72253023
maximal normal subgroup of	#72253023

GAP

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

gap> G := Group((1,16,18,4,21,9,7,10,5,2,6,15)(3,23,27,8,30,25,13,31,11,14,12,22)(17,19,20,24)(26,32,29,28)(33,34,35)(36,40,39,38,44,41,42,37,43), (1,3,7,13)(2,8,4,14)(5,11,18,27)(6,12,21,30)(9,22,15,25)(10,23,16,31)(17,26,20,29)(19,28,24,32)(37,40,44)(39,43,41), (1,2,7,4)(3,8,13,14)(5,15,18,9)(6,19,21,24)(10,20,16,17)(11,25,27,22)(12,28,30,32)(23,29,31,26)(37,39)(38,42)(40,43)(41,44));