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D4×#10621728 #72229348

order5408
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

direct productD4×#10621728

Splittings

FittingD4×C132 ⋊ C4
SylowC132 ⋊ C4×D4

Subgroups

subgroupG/H
center, FrattiniC2C22×#10621728
commutator, socleC2×C132C22×C4
FittingD4×C132C4
prime radicalD4×C13⋊D13C2
cosocleC2×C13⋊D13C23
hypercenterD4#10621728
hypocenterC132C4×D4
LCSG ▷ C2×C132 ▷ C132
UCSC1 ◁ C2 ◁ D4

752 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 676

A minimal faithful permutation representation — 3 generators on 30 points:

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

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD4×#14523144, D4×#14523150, D4×#14523145, …
maximal normal subgroup ofD4×#14523144, D4×#14523150, D4×#14523145, …
maximal quotient ofC2×D4×#10621728

GAP

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

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