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C32×#3297 #11770346

order1152
nilpotency class3
isoclinismisoclinic class of D4×D8
automorphism?
homologyAbelianization (H1) = C24×C32 · Schur multiplier (H2) = C24×C3

Aliases

direct productC32×#3297

Subgroups

subgroupG/H
center, socleC22×C32C22×D4
commutator, Frattini, cosocleC2×C4C24×C32
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C32 ◁ C32×C4×D4 ◁ G

1260 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 288

A minimal faithful permutation representation — 4 generators on 30 points:

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

Conjugacy classes & character table

315 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 4357);