C₃²×#3415 #11770464

order	1152
nilpotency class	4
isoclinism	isoclinic class of #3415
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃² · Schur multiplier (H₂) = C₂⁵×C₃

Aliases

direct product

C₃²×#3415

Subgroups

	subgroup	G/H
center, socle	C₂×C₃²	C₂²×D₈
commutator, Frattini, cosocle	C₈	C₂⁴×C₃²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₃²×#41 ◁ C₃²×#131 ◁ G

1110 conjugacy classes of subgroups; only the 540 normal subgroups are shown.

normal subgroups only

This group has 1110 conjugacy classes of subgroups — too many to draw. Only its 540 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)(9,14)(10,13)(11,24)(12,17)(15,21)(16,22)(18,28)(19,25)(20,34)(23,27)(26,35)(29,33)(30,32)(31,36)(37,38);(4,5,6)(8,13)(10,21)(11,22)(14,25)(15,16)(17,27)(19,33)(20,32)(23,35)(24,26)(28,36)(29,30)(31,38)(34,37);(7,9)(8,14)(10,19)(11,20)(12,18)(13,25)(15,29)(16,30)(17,28)(21,33)(22,32)(23,31)(24,34)(26,37)(27,36)(35,38);(9,18)(14,28)(19,31)(20,32)(25,36)(29,37)(30,34)(33,38)

Conjugacy classes & character table

288 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 4475);