Finite Group Databasebeta

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C2×C32×#443 #11770002

order1152
nilpotency class3
isoclinismisoclinic class of D8
automorphism?
homologyAbelianization (H1) = C23×C32×C4 · Schur multiplier (H2) = C25×C3

Aliases

direct productC2×C32×#443

Subgroups

subgroupG/H
centerC2×C32×C8D4
commutatorC4C23×C32×C4
Frattini, cosocleC2×C4C24×C32
socleC22×C32C4×D4
prime radicalC2×C32×#135C2
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C32×C8 ◁ C22×C32×C8 ◁ G

1416 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 288

A minimal faithful permutation representation — 4 generators on 24 points:

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

Conjugacy classes & character table

504 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC2×C3⋊D3×#443, C2×#11797868
maximal subgroup ofC2×C32×#2179, C2×C3⋊D3×#443, C2×#11797868
maximal normal subgroup ofC2×C32×#2179, C2×C3⋊D3×#443, C2×#11797868

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 4013);