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C4×#11759817 #72229233

order4056
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC4×#11759817

Splittings

FittingC4×C132 ⋊ C2×C3
derived, SylowC132 ⋊ C2×C3×C4
HallC4×C13×D13 ⋊ C3
Hall#92788 ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC4#11759817
commutator, hypocenterC132C2×C3×C4
FrattiniC2C2×#11759817
FittingC4×C132C2×C3
socle, cosocleC2×C132C22×C3
prime radicalC2×#11759817C2
LCSG ▷ C132
UCSC1 ◁ C4

100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 104

A minimal faithful permutation representation — 2 generators on 30 points:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×#11759815⋊C2, #72229416, #72259361, …
maximal normal subgroup ofC4×#11759815⋊C2, #72229416, #72259361, …

GAP

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

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