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## G = C10×3- 1+2order 270 = 2·33·5

### Direct product of C10 and 3- 1+2

direct product, metacyclic, nilpotent (class 2), monomial, 3-elementary

Aliases: C10×3- 1+2, C90⋊C3, C18⋊C15, C92C30, C454C6, C32.C30, C30.2C32, (C3×C30).C3, (C3×C6).C15, (C3×C15).2C6, C15.5(C3×C6), C6.2(C3×C15), C3.2(C3×C30), SmallGroup(270,22)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C10×3- 1+2
 Chief series C1 — C3 — C15 — C3×C15 — C5×3- 1+2 — C10×3- 1+2
 Lower central C1 — C3 — C10×3- 1+2
 Upper central C1 — C30 — C10×3- 1+2

Generators and relations for C10×3- 1+2
G = < a,b,c | a10=b9=c3=1, ab=ba, ac=ca, cbc-1=b4 >

Smallest permutation representation of C10×3- 1+2
On 90 points
Generators in S90
(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 88 89 90)
(1 74 68 83 55 27 33 20 47)(2 75 69 84 56 28 34 11 48)(3 76 70 85 57 29 35 12 49)(4 77 61 86 58 30 36 13 50)(5 78 62 87 59 21 37 14 41)(6 79 63 88 60 22 38 15 42)(7 80 64 89 51 23 39 16 43)(8 71 65 90 52 24 40 17 44)(9 72 66 81 53 25 31 18 45)(10 73 67 82 54 26 32 19 46)
(11 56 75)(12 57 76)(13 58 77)(14 59 78)(15 60 79)(16 51 80)(17 52 71)(18 53 72)(19 54 73)(20 55 74)(21 41 62)(22 42 63)(23 43 64)(24 44 65)(25 45 66)(26 46 67)(27 47 68)(28 48 69)(29 49 70)(30 50 61)

G:=sub<Sym(90)| (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)(81,82,83,84,85,86,87,88,89,90), (1,74,68,83,55,27,33,20,47)(2,75,69,84,56,28,34,11,48)(3,76,70,85,57,29,35,12,49)(4,77,61,86,58,30,36,13,50)(5,78,62,87,59,21,37,14,41)(6,79,63,88,60,22,38,15,42)(7,80,64,89,51,23,39,16,43)(8,71,65,90,52,24,40,17,44)(9,72,66,81,53,25,31,18,45)(10,73,67,82,54,26,32,19,46), (11,56,75)(12,57,76)(13,58,77)(14,59,78)(15,60,79)(16,51,80)(17,52,71)(18,53,72)(19,54,73)(20,55,74)(21,41,62)(22,42,63)(23,43,64)(24,44,65)(25,45,66)(26,46,67)(27,47,68)(28,48,69)(29,49,70)(30,50,61)>;

G:=Group( (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)(81,82,83,84,85,86,87,88,89,90), (1,74,68,83,55,27,33,20,47)(2,75,69,84,56,28,34,11,48)(3,76,70,85,57,29,35,12,49)(4,77,61,86,58,30,36,13,50)(5,78,62,87,59,21,37,14,41)(6,79,63,88,60,22,38,15,42)(7,80,64,89,51,23,39,16,43)(8,71,65,90,52,24,40,17,44)(9,72,66,81,53,25,31,18,45)(10,73,67,82,54,26,32,19,46), (11,56,75)(12,57,76)(13,58,77)(14,59,78)(15,60,79)(16,51,80)(17,52,71)(18,53,72)(19,54,73)(20,55,74)(21,41,62)(22,42,63)(23,43,64)(24,44,65)(25,45,66)(26,46,67)(27,47,68)(28,48,69)(29,49,70)(30,50,61) );

G=PermutationGroup([[(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),(81,82,83,84,85,86,87,88,89,90)], [(1,74,68,83,55,27,33,20,47),(2,75,69,84,56,28,34,11,48),(3,76,70,85,57,29,35,12,49),(4,77,61,86,58,30,36,13,50),(5,78,62,87,59,21,37,14,41),(6,79,63,88,60,22,38,15,42),(7,80,64,89,51,23,39,16,43),(8,71,65,90,52,24,40,17,44),(9,72,66,81,53,25,31,18,45),(10,73,67,82,54,26,32,19,46)], [(11,56,75),(12,57,76),(13,58,77),(14,59,78),(15,60,79),(16,51,80),(17,52,71),(18,53,72),(19,54,73),(20,55,74),(21,41,62),(22,42,63),(23,43,64),(24,44,65),(25,45,66),(26,46,67),(27,47,68),(28,48,69),(29,49,70),(30,50,61)]])

110 conjugacy classes

 class 1 2 3A 3B 3C 3D 5A 5B 5C 5D 6A 6B 6C 6D 9A ··· 9F 10A 10B 10C 10D 15A ··· 15H 15I ··· 15P 18A ··· 18F 30A ··· 30H 30I ··· 30P 45A ··· 45X 90A ··· 90X order 1 2 3 3 3 3 5 5 5 5 6 6 6 6 9 ··· 9 10 10 10 10 15 ··· 15 15 ··· 15 18 ··· 18 30 ··· 30 30 ··· 30 45 ··· 45 90 ··· 90 size 1 1 1 1 3 3 1 1 1 1 1 1 3 3 3 ··· 3 1 1 1 1 1 ··· 1 3 ··· 3 3 ··· 3 1 ··· 1 3 ··· 3 3 ··· 3 3 ··· 3

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 type + + image C1 C2 C3 C3 C5 C6 C6 C10 C15 C15 C30 C30 3- 1+2 C2×3- 1+2 C5×3- 1+2 C10×3- 1+2 kernel C10×3- 1+2 C5×3- 1+2 C90 C3×C30 C2×3- 1+2 C45 C3×C15 3- 1+2 C18 C3×C6 C9 C32 C10 C5 C2 C1 # reps 1 1 6 2 4 6 2 4 24 8 24 8 2 2 8 8

Matrix representation of C10×3- 1+2 in GL3(𝔽181) generated by

 139 0 0 0 139 0 0 0 139
,
 0 1 0 180 133 97 180 133 48
,
 1 0 0 0 132 0 132 133 48
G:=sub<GL(3,GF(181))| [139,0,0,0,139,0,0,0,139],[0,180,180,1,133,133,0,97,48],[1,0,132,0,132,133,0,0,48] >;

C10×3- 1+2 in GAP, Magma, Sage, TeX

C_{10}\times 3_-^{1+2}
% in TeX

G:=Group("C10xES-(3,1)");
// GroupNames label

G:=SmallGroup(270,22);
// by ID

G=gap.SmallGroup(270,22);
# by ID

G:=PCGroup([5,-2,-3,-3,-5,-3,456,727]);
// Polycyclic

G:=Group<a,b,c|a^10=b^9=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

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