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G = C3×D23order 138 = 2·3·23

Direct product of C3 and D23

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C3×D23, C23⋊C6, C692C2, SmallGroup(138,2)

Series: Derived Chief Lower central Upper central

C1C23 — C3×D23
C1C23C69 — C3×D23
C23 — C3×D23
C1C3

Generators and relations for C3×D23
 G = < a,b,c | a3=b23=c2=1, ab=ba, ac=ca, cbc=b-1 >

23C2
23C6

Smallest permutation representation of C3×D23
On 69 points
Generators in S69
(1 68 29)(2 69 30)(3 47 31)(4 48 32)(5 49 33)(6 50 34)(7 51 35)(8 52 36)(9 53 37)(10 54 38)(11 55 39)(12 56 40)(13 57 41)(14 58 42)(15 59 43)(16 60 44)(17 61 45)(18 62 46)(19 63 24)(20 64 25)(21 65 26)(22 66 27)(23 67 28)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23)(24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46)(47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69)
(1 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 15)(10 14)(11 13)(24 33)(25 32)(26 31)(27 30)(28 29)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(47 65)(48 64)(49 63)(50 62)(51 61)(52 60)(53 59)(54 58)(55 57)(66 69)(67 68)

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

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

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

39 conjugacy classes

class 1  2 3A3B6A6B23A···23K69A···69V
order12336623···2369···69
size1231123232···22···2

39 irreducible representations

dim111122
type+++
imageC1C2C3C6D23C3×D23
kernelC3×D23C69D23C23C3C1
# reps11221122

Matrix representation of C3×D23 in GL2(𝔽139) generated by

420
042
,
01
138123
,
01
10
G:=sub<GL(2,GF(139))| [42,0,0,42],[0,138,1,123],[0,1,1,0] >;

C3×D23 in GAP, Magma, Sage, TeX

C_3\times D_{23}
% in TeX

G:=Group("C3xD23");
// GroupNames label

G:=SmallGroup(138,2);
// by ID

G=gap.SmallGroup(138,2);
# by ID

G:=PCGroup([3,-2,-3,-23,1190]);
// Polycyclic

G:=Group<a,b,c|a^3=b^23=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D23 in TeX

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