Finite Group Databasebeta

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C4×C3⋊(C3⋊#1043) #72229680

order3600
automorphism?
homologyAbelianization (H1) = C42 · Schur multiplier (H2) = C3×C4

Aliases

direct productC4×C3⋊(C3⋊#1043)

Splittings

FittingC32×C4×C52 ⋊ C4
derived, HallC32×C52 ⋊ C42
SylowC52 ⋊ C4×C3⋊Dic3
SylowC32 ⋊ C4×#1043

Subgroups

subgroupG/H
center, hypercenterC4C3⋊(C3⋊#1043)
commutator, hypocenterC32×C52C42
FrattiniC2C2×C3⋊(C3⋊#1043)
FittingC32×C4×C52C4
socleC2×C32×C52C2×C4
prime radical, cosocleC2×C32×C5⋊D5C22
LCSG ▷ C32×C52
UCSC1 ◁ C4

720 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 900

A minimal faithful permutation representation — 3 generators on 20 points:

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

Conjugacy classes & character table

264 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC4×C3⋊D3×#1043, C2×C4×C3⋊(C3⋊#1043), D4×C3⋊(C3⋊#1043), …
maximal normal subgroup ofC4×C3⋊D3×#1043, C2×C4×C3⋊(C3⋊#1043), D4×C3⋊(C3⋊#1043), …
maximal quotient ofC2×C4×C3⋊(C3⋊#1043)

GAP

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

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