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G = C3×D29order 174 = 2·3·29

Direct product of C3 and D29

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

Aliases: C3×D29, C29⋊C6, C872C2, SmallGroup(174,2)

Series: Derived Chief Lower central Upper central

C1C29 — C3×D29
C1C29C87 — C3×D29
C29 — C3×D29
C1C3

Generators and relations for C3×D29
 G = < a,b,c | a3=b29=c2=1, ab=ba, ac=ca, cbc=b-1 >

29C2
29C6

Smallest permutation representation of C3×D29
On 87 points
Generators in S87
(1 61 31)(2 62 32)(3 63 33)(4 64 34)(5 65 35)(6 66 36)(7 67 37)(8 68 38)(9 69 39)(10 70 40)(11 71 41)(12 72 42)(13 73 43)(14 74 44)(15 75 45)(16 76 46)(17 77 47)(18 78 48)(19 79 49)(20 80 50)(21 81 51)(22 82 52)(23 83 53)(24 84 54)(25 85 55)(26 86 56)(27 87 57)(28 59 58)(29 60 30)
(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 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87)
(1 29)(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 20)(11 19)(12 18)(13 17)(14 16)(30 31)(32 58)(33 57)(34 56)(35 55)(36 54)(37 53)(38 52)(39 51)(40 50)(41 49)(42 48)(43 47)(44 46)(59 62)(60 61)(63 87)(64 86)(65 85)(66 84)(67 83)(68 82)(69 81)(70 80)(71 79)(72 78)(73 77)(74 76)

G:=sub<Sym(87)| (1,61,31)(2,62,32)(3,63,33)(4,64,34)(5,65,35)(6,66,36)(7,67,37)(8,68,38)(9,69,39)(10,70,40)(11,71,41)(12,72,42)(13,73,43)(14,74,44)(15,75,45)(16,76,46)(17,77,47)(18,78,48)(19,79,49)(20,80,50)(21,81,51)(22,82,52)(23,83,53)(24,84,54)(25,85,55)(26,86,56)(27,87,57)(28,59,58)(29,60,30), (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)(59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(30,31)(32,58)(33,57)(34,56)(35,55)(36,54)(37,53)(38,52)(39,51)(40,50)(41,49)(42,48)(43,47)(44,46)(59,62)(60,61)(63,87)(64,86)(65,85)(66,84)(67,83)(68,82)(69,81)(70,80)(71,79)(72,78)(73,77)(74,76)>;

G:=Group( (1,61,31)(2,62,32)(3,63,33)(4,64,34)(5,65,35)(6,66,36)(7,67,37)(8,68,38)(9,69,39)(10,70,40)(11,71,41)(12,72,42)(13,73,43)(14,74,44)(15,75,45)(16,76,46)(17,77,47)(18,78,48)(19,79,49)(20,80,50)(21,81,51)(22,82,52)(23,83,53)(24,84,54)(25,85,55)(26,86,56)(27,87,57)(28,59,58)(29,60,30), (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)(59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(30,31)(32,58)(33,57)(34,56)(35,55)(36,54)(37,53)(38,52)(39,51)(40,50)(41,49)(42,48)(43,47)(44,46)(59,62)(60,61)(63,87)(64,86)(65,85)(66,84)(67,83)(68,82)(69,81)(70,80)(71,79)(72,78)(73,77)(74,76) );

G=PermutationGroup([(1,61,31),(2,62,32),(3,63,33),(4,64,34),(5,65,35),(6,66,36),(7,67,37),(8,68,38),(9,69,39),(10,70,40),(11,71,41),(12,72,42),(13,73,43),(14,74,44),(15,75,45),(16,76,46),(17,77,47),(18,78,48),(19,79,49),(20,80,50),(21,81,51),(22,82,52),(23,83,53),(24,84,54),(25,85,55),(26,86,56),(27,87,57),(28,59,58),(29,60,30)], [(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),(59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87)], [(1,29),(2,28),(3,27),(4,26),(5,25),(6,24),(7,23),(8,22),(9,21),(10,20),(11,19),(12,18),(13,17),(14,16),(30,31),(32,58),(33,57),(34,56),(35,55),(36,54),(37,53),(38,52),(39,51),(40,50),(41,49),(42,48),(43,47),(44,46),(59,62),(60,61),(63,87),(64,86),(65,85),(66,84),(67,83),(68,82),(69,81),(70,80),(71,79),(72,78),(73,77),(74,76)])

C3×D29 is a maximal subgroup of   C87⋊C4

48 conjugacy classes

class 1  2 3A3B6A6B29A···29N87A···87AB
order12336629···2987···87
size1291129292···22···2

48 irreducible representations

dim111122
type+++
imageC1C2C3C6D29C3×D29
kernelC3×D29C87D29C29C3C1
# reps11221428

Matrix representation of C3×D29 in GL2(𝔽349) generated by

2260
0226
,
181
3480
,
01
10
G:=sub<GL(2,GF(349))| [226,0,0,226],[18,348,1,0],[0,1,1,0] >;

C3×D29 in GAP, Magma, Sage, TeX

C_3\times D_{29}
% in TeX

G:=Group("C3xD29");
// GroupNames label

G:=SmallGroup(174,2);
// by ID

G=gap.SmallGroup(174,2);
# by ID

G:=PCGroup([3,-2,-3,-29,1514]);
// Polycyclic

G:=Group<a,b,c|a^3=b^29=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D29 in TeX

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