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## G = C15×3- 1+2order 405 = 34·5

### Direct product of C15 and 3- 1+2

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

Aliases: C15×3- 1+2, C45⋊C32, C15.2C33, C33.2C15, C9⋊(C3×C15), (C3×C45)⋊4C3, (C3×C9)⋊4C15, C32.5(C3×C15), (C32×C15).2C3, C3.2(C32×C15), (C3×C15).8C32, SmallGroup(405,13)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C15×3- 1+2
 Chief series C1 — C3 — C15 — C45 — C5×3- 1+2 — C15×3- 1+2
 Lower central C1 — C3 — C15×3- 1+2
 Upper central C1 — C3×C15 — C15×3- 1+2

Generators and relations for C15×3- 1+2
G = < a,b,c | a15=b9=c3=1, ab=ba, ac=ca, cbc-1=b4 >

Subgroups: 100 in 76 conjugacy classes, 64 normal (10 characteristic)
C1, C3, C3, C3, C5, C9, C32, C32, C32, C15, C15, C15, C3×C9, 3- 1+2, C33, C45, C3×C15, C3×C15, C3×C15, C3×3- 1+2, C3×C45, C5×3- 1+2, C32×C15, C15×3- 1+2
Quotients: C1, C3, C5, C32, C15, 3- 1+2, C33, C3×C15, C3×3- 1+2, C5×3- 1+2, C32×C15, C15×3- 1+2

Smallest permutation representation of C15×3- 1+2
On 135 points
Generators in S135
(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)(91 92 93 94 95 96 97 98 99 100 101 102 103 104 105)(106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135)
(1 106 62 124 100 47 82 44 24)(2 107 63 125 101 48 83 45 25)(3 108 64 126 102 49 84 31 26)(4 109 65 127 103 50 85 32 27)(5 110 66 128 104 51 86 33 28)(6 111 67 129 105 52 87 34 29)(7 112 68 130 91 53 88 35 30)(8 113 69 131 92 54 89 36 16)(9 114 70 132 93 55 90 37 17)(10 115 71 133 94 56 76 38 18)(11 116 72 134 95 57 77 39 19)(12 117 73 135 96 58 78 40 20)(13 118 74 121 97 59 79 41 21)(14 119 75 122 98 60 80 42 22)(15 120 61 123 99 46 81 43 23)
(1 134 87)(2 135 88)(3 121 89)(4 122 90)(5 123 76)(6 124 77)(7 125 78)(8 126 79)(9 127 80)(10 128 81)(11 129 82)(12 130 83)(13 131 84)(14 132 85)(15 133 86)(16 49 74)(17 50 75)(18 51 61)(19 52 62)(20 53 63)(21 54 64)(22 55 65)(23 56 66)(24 57 67)(25 58 68)(26 59 69)(27 60 70)(28 46 71)(29 47 72)(30 48 73)(31 41 36)(32 42 37)(33 43 38)(34 44 39)(35 45 40)(91 101 96)(92 102 97)(93 103 98)(94 104 99)(95 105 100)(106 116 111)(107 117 112)(108 118 113)(109 119 114)(110 120 115)

G:=sub<Sym(135)| (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)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135), (1,106,62,124,100,47,82,44,24)(2,107,63,125,101,48,83,45,25)(3,108,64,126,102,49,84,31,26)(4,109,65,127,103,50,85,32,27)(5,110,66,128,104,51,86,33,28)(6,111,67,129,105,52,87,34,29)(7,112,68,130,91,53,88,35,30)(8,113,69,131,92,54,89,36,16)(9,114,70,132,93,55,90,37,17)(10,115,71,133,94,56,76,38,18)(11,116,72,134,95,57,77,39,19)(12,117,73,135,96,58,78,40,20)(13,118,74,121,97,59,79,41,21)(14,119,75,122,98,60,80,42,22)(15,120,61,123,99,46,81,43,23), (1,134,87)(2,135,88)(3,121,89)(4,122,90)(5,123,76)(6,124,77)(7,125,78)(8,126,79)(9,127,80)(10,128,81)(11,129,82)(12,130,83)(13,131,84)(14,132,85)(15,133,86)(16,49,74)(17,50,75)(18,51,61)(19,52,62)(20,53,63)(21,54,64)(22,55,65)(23,56,66)(24,57,67)(25,58,68)(26,59,69)(27,60,70)(28,46,71)(29,47,72)(30,48,73)(31,41,36)(32,42,37)(33,43,38)(34,44,39)(35,45,40)(91,101,96)(92,102,97)(93,103,98)(94,104,99)(95,105,100)(106,116,111)(107,117,112)(108,118,113)(109,119,114)(110,120,115)>;

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)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135), (1,106,62,124,100,47,82,44,24)(2,107,63,125,101,48,83,45,25)(3,108,64,126,102,49,84,31,26)(4,109,65,127,103,50,85,32,27)(5,110,66,128,104,51,86,33,28)(6,111,67,129,105,52,87,34,29)(7,112,68,130,91,53,88,35,30)(8,113,69,131,92,54,89,36,16)(9,114,70,132,93,55,90,37,17)(10,115,71,133,94,56,76,38,18)(11,116,72,134,95,57,77,39,19)(12,117,73,135,96,58,78,40,20)(13,118,74,121,97,59,79,41,21)(14,119,75,122,98,60,80,42,22)(15,120,61,123,99,46,81,43,23), (1,134,87)(2,135,88)(3,121,89)(4,122,90)(5,123,76)(6,124,77)(7,125,78)(8,126,79)(9,127,80)(10,128,81)(11,129,82)(12,130,83)(13,131,84)(14,132,85)(15,133,86)(16,49,74)(17,50,75)(18,51,61)(19,52,62)(20,53,63)(21,54,64)(22,55,65)(23,56,66)(24,57,67)(25,58,68)(26,59,69)(27,60,70)(28,46,71)(29,47,72)(30,48,73)(31,41,36)(32,42,37)(33,43,38)(34,44,39)(35,45,40)(91,101,96)(92,102,97)(93,103,98)(94,104,99)(95,105,100)(106,116,111)(107,117,112)(108,118,113)(109,119,114)(110,120,115) );

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),(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105),(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135)], [(1,106,62,124,100,47,82,44,24),(2,107,63,125,101,48,83,45,25),(3,108,64,126,102,49,84,31,26),(4,109,65,127,103,50,85,32,27),(5,110,66,128,104,51,86,33,28),(6,111,67,129,105,52,87,34,29),(7,112,68,130,91,53,88,35,30),(8,113,69,131,92,54,89,36,16),(9,114,70,132,93,55,90,37,17),(10,115,71,133,94,56,76,38,18),(11,116,72,134,95,57,77,39,19),(12,117,73,135,96,58,78,40,20),(13,118,74,121,97,59,79,41,21),(14,119,75,122,98,60,80,42,22),(15,120,61,123,99,46,81,43,23)], [(1,134,87),(2,135,88),(3,121,89),(4,122,90),(5,123,76),(6,124,77),(7,125,78),(8,126,79),(9,127,80),(10,128,81),(11,129,82),(12,130,83),(13,131,84),(14,132,85),(15,133,86),(16,49,74),(17,50,75),(18,51,61),(19,52,62),(20,53,63),(21,54,64),(22,55,65),(23,56,66),(24,57,67),(25,58,68),(26,59,69),(27,60,70),(28,46,71),(29,47,72),(30,48,73),(31,41,36),(32,42,37),(33,43,38),(34,44,39),(35,45,40),(91,101,96),(92,102,97),(93,103,98),(94,104,99),(95,105,100),(106,116,111),(107,117,112),(108,118,113),(109,119,114),(110,120,115)]])

165 conjugacy classes

 class 1 3A ··· 3H 3I ··· 3N 5A 5B 5C 5D 9A ··· 9R 15A ··· 15AF 15AG ··· 15BD 45A ··· 45BT order 1 3 ··· 3 3 ··· 3 5 5 5 5 9 ··· 9 15 ··· 15 15 ··· 15 45 ··· 45 size 1 1 ··· 1 3 ··· 3 1 1 1 1 3 ··· 3 1 ··· 1 3 ··· 3 3 ··· 3

165 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 type + image C1 C3 C3 C3 C5 C15 C15 C15 3- 1+2 C5×3- 1+2 kernel C15×3- 1+2 C3×C45 C5×3- 1+2 C32×C15 C3×3- 1+2 C3×C9 3- 1+2 C33 C15 C3 # reps 1 6 18 2 4 24 72 8 6 24

Matrix representation of C15×3- 1+2 in GL4(𝔽181) generated by

 48 0 0 0 0 117 0 0 0 0 117 0 0 0 0 117
,
 132 0 0 0 0 37 15 83 0 0 0 132 0 134 85 144
,
 132 0 0 0 0 132 179 147 0 0 1 0 0 0 0 48
G:=sub<GL(4,GF(181))| [48,0,0,0,0,117,0,0,0,0,117,0,0,0,0,117],[132,0,0,0,0,37,0,134,0,15,0,85,0,83,132,144],[132,0,0,0,0,132,0,0,0,179,1,0,0,147,0,48] >;

C15×3- 1+2 in GAP, Magma, Sage, TeX

C_{15}\times 3_-^{1+2}
% in TeX

G:=Group("C15xES-(3,1)");
// GroupNames label

G:=SmallGroup(405,13);
// by ID

G=gap.SmallGroup(405,13);
# by ID

G:=PCGroup([5,-3,-3,-3,-5,-3,675,1381]);
// Polycyclic

G:=Group<a,b,c|a^15=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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