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Cayley diagram

Dic37⋊C3 #89217

order444
isoclinismisoclinic class of D37⋊C3
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productDic37⋊C3

Splittings

derived, SylowC37 ⋊ C3×C4
HallDic37 ⋊ C3
HallC37⋊C3 ⋊ C4

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D37⋊C3
commutator, hypocenterC37C3×C4
Fitting, socle, cosocleC2×C37C2×C3
prime radicalC2×C37⋊C3C2
LCSG ▷ C37
UCSC1 ◁ C2

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 41
transitive degreeμt(G) = 148

A minimal faithful permutation representation — 2 generators on 41 points:

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

Polycyclic presentation:

2,3,2,37|a^2=c,d^a=d^36,d^b=d^26

Conjugacy classes & character table

24 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#64438⋊C3, (C37⋊C3)⋊Q8, C4×D37⋊C3, …
maximal normal subgroup of#64438⋊C3, (C37⋊C3)⋊Q8, C4×D37⋊C3, …
maximal quotient of#64436⋊C3, C2×Dic37⋊C3, #13050607, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(444, 1);