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C2×C32×#368 #11769209

order1152
nilpotency class4
isoclinismisoclinic class of D16
automorphism?
homologyAbelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C22×C3

Aliases

direct productC2×C32×#368

Subgroups

subgroupG/H
centerC2×C32×C4D8
commutatorC8C22×C32×C4
Frattini, cosocleC2×C8C23×C32
socleC22×C32#107
prime radicalC23×C32Q16
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C32×C4 ◁ C22×C32×C4 ◁ C22×C32×C8 ◁ G

432 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 40
transitive degreeμt(G) = 576

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2)(35,36,37);(3,4,7,12,9,13,8,23)(5,14,17,29,19,30,18,33)(6,20,21,27,22,34,15,11)(10,24,25,32,26,31,28,16)(38,39,40);(3,5,15,31,8,18,22,32,9,19,21,24,7,17,6,16)(4,10,20,29,12,25,27,30,13,26,34,33,23,28,11,14)

Conjugacy classes & character table

396 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C32×#2179
maximal normal subgroup ofC2×C32×#2179

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 3220);