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Dic11×C61⋊C5 #27211829

order13420
automorphism?
homologyAbelianization (H1) = C4×C5 · Schur multiplier (H2) = C1

Aliases

direct productDic11×C61⋊C5

Splittings

derived, HallC11×C61 ⋊ C4×C5
HallC11×C61⋊C5 ⋊ C4
HallDic11×C61 ⋊ C5
SylowC61 ⋊ C5×Dic11
SylowC11 ⋊ C4×C61⋊C5

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D11×C61⋊C5
commutator, hypocenterC11×C61C4×C5
Fitting, socle, cosocleC2×C11×C61C2×C5
prime radicalC2×C11×C61⋊C5C2
LCSG ▷ C11×C61
UCSC1 ◁ C2

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 76
transitive degreeμt(G) = 2684

A minimal faithful permutation representation — 2 generators on 76 points:

(2,3,5,9,17)(4,7,13,25,35)(6,11,21,38,32)(8,15,28,48,26)(10,19,34,18,33)(12,23,40,29,49)(14,27,46,16,30)(20,36,43,44,52)(22,39,55,54,51)(24,42,57,61,45)(31,50,41,56,58)(37,53,60,59,47)(63,64)(65,66)(67,68)(69,70)(71,72)(73,74,75,76);(1,2,4,8,16,31,19,35,49,3,6,12,24,43,46,36,52,21,7,14,5,10,20,37,53,61,39,15,29,11,22,38,54,60,17,32,30,40,23,41,13,26,45,33,51,50,56,42,58,27,47,55,48,59,34,25,44,57,9,18,28)(62,63,65,67,69,71,72,70,68,66,64)

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13420, 21);