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D23×#63271 #27160201

order12558
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productD23×#63271
unambiguous semidirect productC23⋊(C2×#63271)

Splittings

Fitting, socle, derived, HallC7×C13×C23 ⋊ C2×C3
HallC23×#63271 ⋊ C2
HallC7×C13×D23 ⋊ C3
HallC13×D23 ⋊ ΓL(1,𝔽8)
HallC7×D23 ⋊ C13⋊C3
HallC13×C23 ⋊ C2×ΓL(1,𝔽8)
HallC7×C23 ⋊ C2×C13⋊C3
HallC7×C13 ⋊ C3×D23
SylowC23 ⋊ C2×#63271
SylowC13 ⋊ ΓL(1,𝔽8)×D23
SylowC7 ⋊ C13⋊C3×D23

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC7×C13×C23C2×C3
LCSG ▷ C7×C13×C23
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 43
transitive degreeμt(G) = 2093

A minimal faithful permutation representation — 2 generators on 43 points:

(2,3,5)(4,7,10)(6,9,8)(11,13,12)(15,16,17)(18,20,19)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43);(1,2,4,3,6,10,7,11,13,5,8,12,9)(14,15,17,19,16,18,20)(21,22,24,26,28,30,32,34,36,38,40,42,43,41,39,37,35,33,31,29,27,25,23)

Conjugacy classes & character table

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

Relationships

maximal subgroup ofD23×#10587723, D23×#10587721, D23×#10587717, …
maximal normal subgroup ofD23×#10587723, D23×#10587721, D23×#10587717, …
maximal quotient ofDic23×#63271, C2×D23×#63271, D23×#11717942, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12558, 24);