C₃×#72212523 #72277707

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

Aliases

direct product

C₃×#72212523

Splittings

Fitting, Sylow	C₃×UT(4,𝔽₃) ⋊ C₂²
derived	UT(4,𝔽₃) ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃	#72212523
commutator, cosocle, hypocenter	UT(4,𝔽₃)	C₂²×C₃
Frattini	C₃³	C₃×D₃×C₃⋊D₃
Fitting	C₃×UT(4,𝔽₃)	C₂²
socle	C₃²	#11757730
LCS	G ▷ UT(4,𝔽₃)
UCS	C₁ ◁ C₃

2220 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 2220 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)(4,6,24)(5,16,30,22,21,27)(7,13,23)(8,9,18,11,25,20)(10,26,29,17,19,28)(12,15,14);(5,8)(6,12)(7,15)(9,18)(10,17)(11,22)(13,23)(14,24)(16,20)(19,27)(21,28)(25,30)(26,29);(4,5,8)(6,9,16)(7,11,17)(10,22,15)(12,20,18)(13,19,25)(14,21,26)(23,30,27)(24,29,28)

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72277755

GAP

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

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