C₃²×#3028 #11770077

order	1152
nilpotency class	3
isoclinism	isoclinic class of #3026
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃² · Schur multiplier (H₂) = C₂⁶×C₃

Aliases

direct product

C₃²×#3028

Subgroups

	subgroup	G/H
center, socle	C₂×C₃²	C₂×#120
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃²
prime radical	C₃²×#585	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂²×C₃²×C₄ ◁ G

2148 conjugacy classes of subgroups; only the 636 normal subgroups are shown.

normal subgroups only

This group has 2148 conjugacy classes of subgroups — too many to draw. Only its 636 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) = 38
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 4 generators on 38 points:

(1,2,3)(7,8,12,16)(9,13,19,26)(10,14,21,28)(11,22,23,15)(17,29,31,24)(18,32,33,25)(20,34,35,27)(30,36,38,37);(4,5,6)(7,9)(8,13)(10,17)(11,18)(12,19)(14,24)(15,25)(16,26)(20,30)(21,31)(22,32)(23,33)(27,36)(28,29)(34,37)(35,38);(7,10)(8,14)(9,17)(11,20)(12,21)(13,24)(15,27)(16,28)(18,30)(19,31)(22,34)(23,35)(25,36)(26,29)(32,37)(33,38);(7,11,12,23)(8,15,16,22)(9,18,19,33)(10,20,21,35)(13,25,26,32)(14,27,28,34)(17,30,31,38)(24,36,29,37)

Conjugacy classes & character table

261 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 4088);