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G = C3xD23order 138 = 2·3·23

Direct product of C3 and D23

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

Aliases: C3xD23, C23:C6, C69:2C2, SmallGroup(138,2)

Series: Derived Chief Lower central Upper central

C1C23 — C3xD23
C1C23C69 — C3xD23
C23 — C3xD23
C1C3

Generators and relations for C3xD23
 G = < a,b,c | a3=b23=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 52 in 8 conjugacy classes, 6 normal (all characteristic)
Quotients: C1, C2, C3, C6, D23, C3xD23
23C2
23C6

Smallest permutation representation of C3xD23
On 69 points
Generators in S69
(1 54 38)(2 55 39)(3 56 40)(4 57 41)(5 58 42)(6 59 43)(7 60 44)(8 61 45)(9 62 46)(10 63 24)(11 64 25)(12 65 26)(13 66 27)(14 67 28)(15 68 29)(16 69 30)(17 47 31)(18 48 32)(19 49 33)(20 50 34)(21 51 35)(22 52 36)(23 53 37)
(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 28)(25 27)(29 46)(30 45)(31 44)(32 43)(33 42)(34 41)(35 40)(36 39)(37 38)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 54)(61 69)(62 68)(63 67)(64 66)

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

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

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

39 conjugacy classes

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

39 irreducible representations

dim111122
type+++
imageC1C2C3C6D23C3xD23
kernelC3xD23C69D23C23C3C1
# reps11221122

Matrix representation of C3xD23 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] >;

C3xD23 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 C3xD23 in TeX

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