C₂×C₃×#92404 #72244454

order	2916
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂×C₃⁵

Aliases

direct product

C₂×C₃×#92404

Splittings

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

Subgroups

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

6380 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 162

A minimal faithful permutation representation — 5 generators on 32 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72221874, C₂×C₃×#13091105, C₂×#72221877, …
maximal normal subgroup of	C₂×#72221874, C₂×C₃×#13091105, C₂×#72221877, …

GAP

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

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

C2×C3×#92404 #72244454