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G = C3×C29⋊C4order 348 = 22·3·29

Direct product of C3 and C29⋊C4

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

Aliases: C3×C29⋊C4, C29⋊C12, C872C4, D29.C6, (C3×D29).2C2, SmallGroup(348,5)

Series: Derived Chief Lower central Upper central

C1C29 — C3×C29⋊C4
C1C29D29C3×D29 — C3×C29⋊C4
C29 — C3×C29⋊C4
C1C3

Generators and relations for C3×C29⋊C4
 G = < a,b,c | a3=b29=c4=1, ab=ba, ac=ca, cbc-1=b17 >

29C2
29C4
29C6
29C12

Smallest permutation representation of C3×C29⋊C4
On 87 points
Generators in S87
(1 59 30)(2 60 31)(3 61 32)(4 62 33)(5 63 34)(6 64 35)(7 65 36)(8 66 37)(9 67 38)(10 68 39)(11 69 40)(12 70 41)(13 71 42)(14 72 43)(15 73 44)(16 74 45)(17 75 46)(18 76 47)(19 77 48)(20 78 49)(21 79 50)(22 80 51)(23 81 52)(24 82 53)(25 83 54)(26 84 55)(27 85 56)(28 86 57)(29 87 58)
(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)
(2 13 29 18)(3 25 28 6)(4 8 27 23)(5 20 26 11)(7 15 24 16)(9 10 22 21)(12 17 19 14)(31 42 58 47)(32 54 57 35)(33 37 56 52)(34 49 55 40)(36 44 53 45)(38 39 51 50)(41 46 48 43)(60 71 87 76)(61 83 86 64)(62 66 85 81)(63 78 84 69)(65 73 82 74)(67 68 80 79)(70 75 77 72)

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

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

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

33 conjugacy classes

class 1  2 3A3B4A4B6A6B12A12B12C12D29A···29G87A···87N
order123344661212121229···2987···87
size1291129292929292929294···44···4

33 irreducible representations

dim11111144
type+++
imageC1C2C3C4C6C12C29⋊C4C3×C29⋊C4
kernelC3×C29⋊C4C3×D29C29⋊C4C87D29C29C3C1
# reps112224714

Matrix representation of C3×C29⋊C4 in GL4(𝔽349) generated by

122000
012200
001220
000122
,
301129301348
1000
0100
0010
,
1000
312632255
4820942339
16730110300
G:=sub<GL(4,GF(349))| [122,0,0,0,0,122,0,0,0,0,122,0,0,0,0,122],[301,1,0,0,129,0,1,0,301,0,0,1,348,0,0,0],[1,312,48,167,0,6,209,301,0,322,42,10,0,55,339,300] >;

C3×C29⋊C4 in GAP, Magma, Sage, TeX

C_3\times C_{29}\rtimes C_4
% in TeX

G:=Group("C3xC29:C4");
// GroupNames label

G:=SmallGroup(348,5);
// by ID

G=gap.SmallGroup(348,5);
# by ID

G:=PCGroup([4,-2,-3,-2,-29,24,2307,907]);
// Polycyclic

G:=Group<a,b,c|a^3=b^29=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^17>;
// generators/relations

Export

Subgroup lattice of C3×C29⋊C4 in TeX

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