C₂³×AGL(1,ℤ/27) #72276264

order	3888
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₉ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂³×AGL(1,ℤ/27)

Splittings

Fitting	C₂³×#6875 ⋊ C₂
derived	C₂₇ ⋊ C₂⁴×C₉
Sylow	#6875 ⋊ C₂⁴

Subgroups

	subgroup	G/H
center, hypercenter	C₂³	AGL(1,ℤ/27)
commutator, hypocenter	C₂₇	C₂⁴×C₉
Frattini	C₃×C₉	C₂³×C₃×D₃
Fitting	C₂³×#6875	C₂
socle	C₂³×C₃	D₉⋊C₉
prime radical	C₂³×D₂₇⋊C₃	C₃
cosocle	C₂₇⋊C₃	C₂⁴×C₃
LCS	G ▷ C₂₇
UCS	C₁ ◁ C₂³

868 conjugacy classes of subgroups; only the 297 normal subgroups are shown.

normal subgroups only

This group has 868 conjugacy classes of subgroups — too many to draw. Only its 297 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 — 4 generators on 33 points:

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

Conjugacy classes & character table

248 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)(7,15,30,20,25,11,17,16,28,10,24,33,18,12,23,19,13,14,21,29,31,9,27,26,32,8,22), (8,11)(9,17)(10,21)(12,22)(13,23)(14,27)(15,31)(16,26)(18,19)(20,32)(24,33)(25,28)(29,30), (3,4), (1,2));