Finite Group Databasebeta

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C2×#10710962 #72248115

order1536
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C28

Aliases

direct productC2×#10710962

Splittings

FittingC2×C3×#2120 ⋊ C2
SylowC3 ⋊ C2×#12755

Subgroups

subgroupG/H
centerC22#70033
commutatorC2×C3×D4C23×C4
FrattiniC2×#41C23×D3
FittingC2×C3×#2120C2
socleC22×C3#1790
prime radicalC2×#86052C2
cosocleC2×C3×#41C24
hypercenterC2×#2120D3
hypocenterC3C2×#12755
LCSG ▷ C2×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#2120

5598 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 4 generators on 21 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72248142, #72254161, C2×D3×#12755
maximal normal subgroup of#72248142, #72254161, C2×D3×#12755

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2)(19,20,21), (3,5)(4,10)(6,8)(7,15)(9,17)(11,12)(13,18)(14,16)(20,21), (3,4,8,13)(5,11)(6,9)(7,12)(10,14,18,15)(16,17), (4,9)(10,12)(11,13)(17,18));