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## G = C15×He3order 405 = 34·5

### Direct product of C15 and He3

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

Aliases: C15×He3, C332C15, C15.1C33, C32⋊(C3×C15), (C3×C15)⋊C32, (C32×C15)⋊2C3, C3.1(C32×C15), SmallGroup(405,12)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C15×He3
 Chief series C1 — C3 — C15 — C3×C15 — C5×He3 — C15×He3
 Lower central C1 — C3 — C15×He3
 Upper central C1 — C3×C15 — C15×He3

Generators and relations for C15×He3
G = < a,b,c,d | a15=b3=c3=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, cd=dc >

Subgroups: 208 in 112 conjugacy classes, 64 normal (8 characteristic)
C1, C3, C3, C3, C5, C32, C32, C32, C15, C15, C15, He3, C33, C3×C15, C3×C15, C3×C15, C3×He3, C5×He3, C32×C15, C15×He3
Quotients: C1, C3, C5, C32, C15, He3, C33, C3×C15, C3×He3, C5×He3, C32×C15, C15×He3

Smallest permutation representation of C15×He3
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 86 105)(2 87 91)(3 88 92)(4 89 93)(5 90 94)(6 76 95)(7 77 96)(8 78 97)(9 79 98)(10 80 99)(11 81 100)(12 82 101)(13 83 102)(14 84 103)(15 85 104)(16 33 126)(17 34 127)(18 35 128)(19 36 129)(20 37 130)(21 38 131)(22 39 132)(23 40 133)(24 41 134)(25 42 135)(26 43 121)(27 44 122)(28 45 123)(29 31 124)(30 32 125)(46 116 66)(47 117 67)(48 118 68)(49 119 69)(50 120 70)(51 106 71)(52 107 72)(53 108 73)(54 109 74)(55 110 75)(56 111 61)(57 112 62)(58 113 63)(59 114 64)(60 115 65)
(1 42 62)(2 43 63)(3 44 64)(4 45 65)(5 31 66)(6 32 67)(7 33 68)(8 34 69)(9 35 70)(10 36 71)(11 37 72)(12 38 73)(13 39 74)(14 40 75)(15 41 61)(16 118 96)(17 119 97)(18 120 98)(19 106 99)(20 107 100)(21 108 101)(22 109 102)(23 110 103)(24 111 104)(25 112 105)(26 113 91)(27 114 92)(28 115 93)(29 116 94)(30 117 95)(46 90 124)(47 76 125)(48 77 126)(49 78 127)(50 79 128)(51 80 129)(52 81 130)(53 82 131)(54 83 132)(55 84 133)(56 85 134)(57 86 135)(58 87 121)(59 88 122)(60 89 123)
(1 37 67)(2 38 68)(3 39 69)(4 40 70)(5 41 71)(6 42 72)(7 43 73)(8 44 74)(9 45 75)(10 31 61)(11 32 62)(12 33 63)(13 34 64)(14 35 65)(15 36 66)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)(46 134 80)(47 135 81)(48 121 82)(49 122 83)(50 123 84)(51 124 85)(52 125 86)(53 126 87)(54 127 88)(55 128 89)(56 129 90)(57 130 76)(58 131 77)(59 132 78)(60 133 79)(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), (1,86,105)(2,87,91)(3,88,92)(4,89,93)(5,90,94)(6,76,95)(7,77,96)(8,78,97)(9,79,98)(10,80,99)(11,81,100)(12,82,101)(13,83,102)(14,84,103)(15,85,104)(16,33,126)(17,34,127)(18,35,128)(19,36,129)(20,37,130)(21,38,131)(22,39,132)(23,40,133)(24,41,134)(25,42,135)(26,43,121)(27,44,122)(28,45,123)(29,31,124)(30,32,125)(46,116,66)(47,117,67)(48,118,68)(49,119,69)(50,120,70)(51,106,71)(52,107,72)(53,108,73)(54,109,74)(55,110,75)(56,111,61)(57,112,62)(58,113,63)(59,114,64)(60,115,65), (1,42,62)(2,43,63)(3,44,64)(4,45,65)(5,31,66)(6,32,67)(7,33,68)(8,34,69)(9,35,70)(10,36,71)(11,37,72)(12,38,73)(13,39,74)(14,40,75)(15,41,61)(16,118,96)(17,119,97)(18,120,98)(19,106,99)(20,107,100)(21,108,101)(22,109,102)(23,110,103)(24,111,104)(25,112,105)(26,113,91)(27,114,92)(28,115,93)(29,116,94)(30,117,95)(46,90,124)(47,76,125)(48,77,126)(49,78,127)(50,79,128)(51,80,129)(52,81,130)(53,82,131)(54,83,132)(55,84,133)(56,85,134)(57,86,135)(58,87,121)(59,88,122)(60,89,123), (1,37,67)(2,38,68)(3,39,69)(4,40,70)(5,41,71)(6,42,72)(7,43,73)(8,44,74)(9,45,75)(10,31,61)(11,32,62)(12,33,63)(13,34,64)(14,35,65)(15,36,66)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25)(46,134,80)(47,135,81)(48,121,82)(49,122,83)(50,123,84)(51,124,85)(52,125,86)(53,126,87)(54,127,88)(55,128,89)(56,129,90)(57,130,76)(58,131,77)(59,132,78)(60,133,79)(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), (1,86,105)(2,87,91)(3,88,92)(4,89,93)(5,90,94)(6,76,95)(7,77,96)(8,78,97)(9,79,98)(10,80,99)(11,81,100)(12,82,101)(13,83,102)(14,84,103)(15,85,104)(16,33,126)(17,34,127)(18,35,128)(19,36,129)(20,37,130)(21,38,131)(22,39,132)(23,40,133)(24,41,134)(25,42,135)(26,43,121)(27,44,122)(28,45,123)(29,31,124)(30,32,125)(46,116,66)(47,117,67)(48,118,68)(49,119,69)(50,120,70)(51,106,71)(52,107,72)(53,108,73)(54,109,74)(55,110,75)(56,111,61)(57,112,62)(58,113,63)(59,114,64)(60,115,65), (1,42,62)(2,43,63)(3,44,64)(4,45,65)(5,31,66)(6,32,67)(7,33,68)(8,34,69)(9,35,70)(10,36,71)(11,37,72)(12,38,73)(13,39,74)(14,40,75)(15,41,61)(16,118,96)(17,119,97)(18,120,98)(19,106,99)(20,107,100)(21,108,101)(22,109,102)(23,110,103)(24,111,104)(25,112,105)(26,113,91)(27,114,92)(28,115,93)(29,116,94)(30,117,95)(46,90,124)(47,76,125)(48,77,126)(49,78,127)(50,79,128)(51,80,129)(52,81,130)(53,82,131)(54,83,132)(55,84,133)(56,85,134)(57,86,135)(58,87,121)(59,88,122)(60,89,123), (1,37,67)(2,38,68)(3,39,69)(4,40,70)(5,41,71)(6,42,72)(7,43,73)(8,44,74)(9,45,75)(10,31,61)(11,32,62)(12,33,63)(13,34,64)(14,35,65)(15,36,66)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25)(46,134,80)(47,135,81)(48,121,82)(49,122,83)(50,123,84)(51,124,85)(52,125,86)(53,126,87)(54,127,88)(55,128,89)(56,129,90)(57,130,76)(58,131,77)(59,132,78)(60,133,79)(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)], [(1,86,105),(2,87,91),(3,88,92),(4,89,93),(5,90,94),(6,76,95),(7,77,96),(8,78,97),(9,79,98),(10,80,99),(11,81,100),(12,82,101),(13,83,102),(14,84,103),(15,85,104),(16,33,126),(17,34,127),(18,35,128),(19,36,129),(20,37,130),(21,38,131),(22,39,132),(23,40,133),(24,41,134),(25,42,135),(26,43,121),(27,44,122),(28,45,123),(29,31,124),(30,32,125),(46,116,66),(47,117,67),(48,118,68),(49,119,69),(50,120,70),(51,106,71),(52,107,72),(53,108,73),(54,109,74),(55,110,75),(56,111,61),(57,112,62),(58,113,63),(59,114,64),(60,115,65)], [(1,42,62),(2,43,63),(3,44,64),(4,45,65),(5,31,66),(6,32,67),(7,33,68),(8,34,69),(9,35,70),(10,36,71),(11,37,72),(12,38,73),(13,39,74),(14,40,75),(15,41,61),(16,118,96),(17,119,97),(18,120,98),(19,106,99),(20,107,100),(21,108,101),(22,109,102),(23,110,103),(24,111,104),(25,112,105),(26,113,91),(27,114,92),(28,115,93),(29,116,94),(30,117,95),(46,90,124),(47,76,125),(48,77,126),(49,78,127),(50,79,128),(51,80,129),(52,81,130),(53,82,131),(54,83,132),(55,84,133),(56,85,134),(57,86,135),(58,87,121),(59,88,122),(60,89,123)], [(1,37,67),(2,38,68),(3,39,69),(4,40,70),(5,41,71),(6,42,72),(7,43,73),(8,44,74),(9,45,75),(10,31,61),(11,32,62),(12,33,63),(13,34,64),(14,35,65),(15,36,66),(16,26,21),(17,27,22),(18,28,23),(19,29,24),(20,30,25),(46,134,80),(47,135,81),(48,121,82),(49,122,83),(50,123,84),(51,124,85),(52,125,86),(53,126,87),(54,127,88),(55,128,89),(56,129,90),(57,130,76),(58,131,77),(59,132,78),(60,133,79),(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 ··· 3AF 5A 5B 5C 5D 15A ··· 15AF 15AG ··· 15DX order 1 3 ··· 3 3 ··· 3 5 5 5 5 15 ··· 15 15 ··· 15 size 1 1 ··· 1 3 ··· 3 1 1 1 1 1 ··· 1 3 ··· 3

165 irreducible representations

 dim 1 1 1 1 1 1 3 3 type + image C1 C3 C3 C5 C15 C15 He3 C5×He3 kernel C15×He3 C5×He3 C32×C15 C3×He3 He3 C33 C15 C3 # reps 1 18 8 4 72 32 6 24

Matrix representation of C15×He3 in GL4(𝔽31) generated by

 18 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 25 0 0 0 0 25 0 0 0 0 0 25 0 11 6 6
,
 1 0 0 0 0 5 0 0 0 0 5 0 0 0 0 5
,
 1 0 0 0 0 5 5 6 0 0 1 0 0 0 0 25
G:=sub<GL(4,GF(31))| [18,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[25,0,0,0,0,25,0,11,0,0,0,6,0,0,25,6],[1,0,0,0,0,5,0,0,0,0,5,0,0,0,0,5],[1,0,0,0,0,5,0,0,0,5,1,0,0,6,0,25] >;

C15×He3 in GAP, Magma, Sage, TeX

C_{15}\times {\rm He}_3
% in TeX

G:=Group("C15xHe3");
// GroupNames label

G:=SmallGroup(405,12);
// by ID

G=gap.SmallGroup(405,12);
# by ID

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

G:=Group<a,b,c,d|a^15=b^3=c^3=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^-1,c*d=d*c>;
// generators/relations

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