C₂³×#6249⋊C₂ #72276414

order	3456
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

solvable

Aliases

direct product

C₂³×#6249⋊C₂

Splittings

Fitting	C₂³×H(3,𝔽₃) ⋊ SD₁₆
Sylow	H(3,𝔽₃) ⋊ C₂³×SD₁₆

Subgroups

	subgroup	G/H
center, hypercenter	C₂³	#6249⋊C₂
commutator, cosocle	#1088	C₂⁵
Frattini	C₃	C₂³×AΓL(1,𝔽₉)
Fitting	C₂³×H(3,𝔽₃)	SD₁₆
socle	C₂³×C₃	AΓL(1,𝔽₉)
prime radical	C₂³×#1088⋊C₂	C₂
hypocenter	H(3,𝔽₃)	C₂³×SD₁₆
LCS	G ▷ #1088 ▷ #272 ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂³

3328 conjugacy classes of subgroups; only the 508 normal subgroups are shown.

normal subgroups only

This group has 3328 conjugacy classes of subgroups — too many to draw. Only its 508 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) = 216

A minimal faithful permutation representation — 5 generators on 33 points:

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

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

GAP

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

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