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D5×#13046196 #27165477

order12810
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productD5×#13046196
unambiguous semidirect productC5⋊(C2×#13046196)

Splittings

Fitting, socle, derived, HallC5×C7×C61 ⋊ C2×C3
HallC5×#13046196 ⋊ C2
HallC7×D5×C61 ⋊ C3
HallD5×C61 ⋊ ΓL(1,𝔽8)
HallC7×C61 ⋊ C3×D5
HallC5×C61 ⋊ C2×ΓL(1,𝔽8)
HallC7×D5 ⋊ C61⋊C3
SylowC61 ⋊ D5×ΓL(1,𝔽8)
HallC5×C7 ⋊ C2×C61⋊C3
SylowC7 ⋊ D5×C61⋊C3
SylowC5 ⋊ C2×#13046196

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C7×C61C2×C3
LCSG ▷ C5×C7×C61
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 73
transitive degreeμt(G) = 2135

A minimal faithful permutation representation — 2 generators on 73 points:

(2,3,5)(4,7,12)(6,10,9)(8,14,22)(11,17,19)(13,20,16)(15,24,25)(18,27,32)(21,30,29)(23,33,26)(28,38,36)(31,41,43)(34,44,40)(35,39,37)(42,49,47)(45,52,51)(46,50,48)(53,57,55)(54,58,56)(59,61,60)(63,64,65)(66,68,67)(70,71)(72,73);(1,2,4,8,15,25,36,47,55,60,57,41,17,3,6,11,18,28,39,22,32,43,51,56,49,27,7,13,21,31,42,50,33,12,19,29,40,48,38,14,23,34,45,53,58,44,20,5,9,16,26,37,24,35,46,54,59,61,52,30,10)(62,63,64,66,65,67,68)(69,70,72,73,71)

Conjugacy classes & character table

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

Relationships

maximal subgroup of#38785066, #38785070, #38785078, …
maximal normal subgroup of#38785066, #38785070, #38785078, …
maximal quotient ofDic5×#13046196, C2×D5×#13046196, D5×#16838635, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12810, 48);