C₃⋊D₃×3₋¹⁺⁴ #72244288

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

Aliases

direct product

C₃⋊D₃×3₋¹⁺⁴

Splittings

Fitting, Sylow

C₃²×3₋¹⁺⁴ ⋊ C₂

Subgroups

	subgroup	G/H
center, Frattini	C₃	C₃⁴×C₃⋊D₃
commutator, socle, cosocle	C₃³	C₂×C₃⁴
Fitting	C₃²×3₋¹⁺⁴	C₂
prime radical	C₃×C₃⋊D₃×H(3,𝔽₃)	C₃
hypercenter	3₋¹⁺⁴	C₃⋊D₃
hypocenter	C₃²	C₂×3₋¹⁺⁴
LCS	G ▷ C₃³ ▷ C₃²
UCS	C₁ ◁ C₃ ◁ 3₋¹⁺⁴

31144 conjugacy classes of subgroups; only the 1491 normal subgroups are shown.

normal subgroups only

This group has 31144 conjugacy classes of subgroups — too many to draw. Only its 1491 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) = 33
transitive degree	μ_t(G) = 162

A minimal faithful permutation representation — 4 generators on 33 points:

(2,3)(5,6)(9,17,16)(12,22,24)(15,26,25)(19,28,31)(21,29,30)(27,32,33);(4,5,6)(8,13,14)(9,16,17)(11,20,23)(12,22,24)(15,25,26)(27,32,33);(1,2,3)(7,17,26,10,16,15,18,9,25)(8,22,29,14,24,21,13,12,30)(11,28,32,20,31,27,23,19,33);(7,8,11)(9,12,19)(10,14,20)(13,23,18)(15,21,27)(16,24,31)(17,22,28)(25,30,33)(26,29,32)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃⋊D₃×#10624749
maximal normal subgroup of	C₃⋊D₃×#10624749

GAP

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

gap> G := Group((2,3)(5,6)(9,17,16)(12,22,24)(15,26,25)(19,28,31)(21,29,30)(27,32,33), (4,5,6)(8,13,14)(9,16,17)(11,20,23)(12,22,24)(15,25,26)(27,32,33), (1,2,3)(7,17,26,10,16,15,18,9,25)(8,22,29,14,24,21,13,12,30)(11,28,32,20,31,27,23,19,33), (7,8,11)(9,12,19)(10,14,20)(13,23,18)(15,21,27)(16,24,31)(17,22,28)(25,30,33)(26,29,32));