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C2×C3×#19149853 #45686337

order33300
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3×C5

Aliases

direct productC2×C3×#19149853

Splittings

Fitting, socleC2×C3×C52×C37 ⋊ C2×C3
derived, HallC52×C37 ⋊ C22×C32
HallC3×C37×C52⋊C3 ⋊ C22
HallC2×C5⋊D185 ⋊ C32
HallC3×C52⋊C3 ⋊ C2×D37
SylowC37 ⋊ C2×C3×(C5⋊D5)⋊C3
SylowC52 ⋊ C2×C32×D37

Subgroups

subgroupG/H
center, hypercenterC2×C3#19149853
commutator, cosocle, hypocenterC52×C37C22×C32
FrattiniC1G
Fitting, socleC2×C3×C52×C37C2×C3
LCSG ▷ C52×C37
UCSC1 ◁ C2×C3

160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 57
transitive degreeμt(G) = 3330

A minimal faithful permutation representation — 2 generators on 57 points:

(6,7,9)(8,18,14,11,12,20)(10,15,19,16,17,13)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57);(1,2)(3,4,5)(6,8,13,19,11)(7,10,12,18,16)(9,14,15,17,20)(21,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23)

Conjugacy classes & character table

1248 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33300, 610);