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Cayley diagram

(C31⋊C3)⋊C4 #67308

order372
isoclinismisoclinic class of D31⋊C3
automorphismAut(G) = C2×AGL(1,𝔽31) (order 1860) · Inn(G) = D31⋊C3 · Out(G) = C2×C5
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect product(C31⋊C3)⋊C4
unambiguous semidirect productDic31⋊C3

Splittings

derived, SylowC31 ⋊ C3×C4
HallDic31 ⋊ C3
HallC31⋊C3 ⋊ C4

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D31⋊C3
commutator, hypocenterC31C3×C4
Fitting, socle, cosocleC2×C31C2×C3
prime radicalC2×C31⋊C3C2
LCSG ▷ C31
UCSC1 ◁ C2

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 35
transitive degreeμt(G) = 124

A minimal faithful permutation representation — 2 generators on 35 points:

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

Polycyclic presentation:

2,3,2,31|a^2=c,d^a=d^30,d^b=d^25

Conjugacy classes & character table

22 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C31⋊C3)⋊Q8, C4×D31⋊C3, C2×(C31⋊C3)⋊C4, …
maximal normal subgroup of(C31⋊C3)⋊Q8, C4×D31⋊C3, C2×(C31⋊C3)⋊C4, …
maximal quotient of(C31⋊C3)⋊C8, C2×(C31⋊C3)⋊C4, Dic31⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(372, 1);