C₂×C₃⋊#92392 #72253996

order	2916
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂×C₃³

Aliases

direct product

C₂×C₃⋊#92392

Splittings

Fitting	C₂×C₃×H(5,𝔽₃) ⋊ C₂
Sylow	C₃×H(5,𝔽₃) ⋊ C₂²

Subgroups

	subgroup	G/H
center	C₂×C₃	C₃²×C₃⋊(C₃⋊D₃)
commutator, cosocle, hypocenter	C₃×H(3,𝔽₃)	C₂²×C₃²
Frattini	C₃	C₂×C₃²×C₃⋊(C₃⋊D₃)
Fitting	C₂×C₃×H(5,𝔽₃)	C₂
socle	C₂×C₃²	C₃²×C₃⋊D₃
hypercenter	C₂×H(3,𝔽₃)	C₃⋊(C₃⋊D₃)
LCS	G ▷ C₃×H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₂×H(3,𝔽₃)

4420 conjugacy classes of subgroups; only the 358 normal subgroups are shown.

normal subgroups only

This group has 4420 conjugacy classes of subgroups — too many to draw. Only its 358 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) = 32
transitive degree	μ_t(G) = 108

A minimal faithful permutation representation — 4 generators on 32 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72221874, C₂×#72221880
maximal normal subgroup of	C₂×#72221880

GAP

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

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