C₂×C₃⋊#92391 #72244358

order	2916
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂×C₃⁴

Aliases

direct product

C₂×C₃⋊#92391

Splittings

Fitting	C₂×C₃×H(5,𝔽₃) ⋊ C₂
derived	C₃⁴ ⋊ C₂²×C₃²
Sylow	C₃×H(5,𝔽₃) ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₂	C₃⋊#92391
commutator, cosocle, hypocenter	C₃⁴	C₂²×C₃²
Frattini	C₃	C₂×C₃²×C₃⋊(C₃⋊D₃)
Fitting	C₂×C₃×H(5,𝔽₃)	C₂
socle	C₂×C₃²	C₃²×C₃⋊D₃
LCS	G ▷ C₃⁴
UCS	C₁ ◁ C₂

4396 conjugacy classes of subgroups; only the 362 normal subgroups are shown.

normal subgroups only

This group has 4396 conjugacy classes of subgroups — too many to draw. Only its 362 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) = 162

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72221887, C₂×#72221892
maximal normal subgroup of	C₂×#72221892

GAP

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

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