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## G = C5×C3.He3order 405 = 34·5

### Direct product of C5 and C3.He3

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

Aliases: C5×C3.He3, C15.5He3, 3- 1+2.C15, (C3×C45).2C3, (C3×C9).2C15, C3.5(C5×He3), C32.4(C3×C15), (C3×C15).4C32, (C5×3- 1+2).C3, SmallGroup(405,10)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C5×C3.He3
 Chief series C1 — C3 — C32 — C3×C15 — C5×3- 1+2 — C5×C3.He3
 Lower central C1 — C3 — C32 — C5×C3.He3
 Upper central C1 — C15 — C3×C15 — C5×C3.He3

Generators and relations for C5×C3.He3
G = < a,b,c,d,e | a5=b3=d3=1, c3=b-1, e3=b, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=bcd-1, ede-1=b-1d >

Smallest permutation representation of C5×C3.He3
On 135 points
Generators in S135
(1 71 40 25 47)(2 72 41 26 48)(3 64 42 27 49)(4 65 43 19 50)(5 66 44 20 51)(6 67 45 21 52)(7 68 37 22 53)(8 69 38 23 54)(9 70 39 24 46)(10 126 34 133 106)(11 118 35 134 107)(12 119 36 135 108)(13 120 28 127 100)(14 121 29 128 101)(15 122 30 129 102)(16 123 31 130 103)(17 124 32 131 104)(18 125 33 132 105)(55 91 80 116 86)(56 92 81 117 87)(57 93 73 109 88)(58 94 74 110 89)(59 95 75 111 90)(60 96 76 112 82)(61 97 77 113 83)(62 98 78 114 84)(63 99 79 115 85)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 52 49)(47 53 50)(48 54 51)(55 61 58)(56 62 59)(57 63 60)(64 70 67)(65 71 68)(66 72 69)(73 79 76)(74 80 77)(75 81 78)(82 88 85)(83 89 86)(84 90 87)(91 97 94)(92 98 95)(93 99 96)(100 106 103)(101 107 104)(102 108 105)(109 115 112)(110 116 113)(111 117 114)(118 124 121)(119 125 122)(120 126 123)(127 133 130)(128 134 131)(129 135 132)
(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)
(10 13 16)(11 14 17)(12 15 18)(28 31 34)(29 32 35)(30 33 36)(55 61 58)(56 62 59)(57 63 60)(73 79 76)(74 80 77)(75 81 78)(82 88 85)(83 89 86)(84 90 87)(91 97 94)(92 98 95)(93 99 96)(100 103 106)(101 104 107)(102 105 108)(109 115 112)(110 116 113)(111 117 114)(118 121 124)(119 122 125)(120 123 126)(127 130 133)(128 131 134)(129 132 135)
(1 123 76 7 120 73 4 126 79)(2 118 77 8 124 74 5 121 80)(3 122 78 9 119 75 6 125 81)(10 99 47 16 96 53 13 93 50)(11 97 54 17 94 51 14 91 48)(12 95 52 18 92 49 15 98 46)(19 106 63 25 103 60 22 100 57)(20 101 55 26 107 61 23 104 58)(21 105 56 27 102 62 24 108 59)(28 109 65 34 115 71 31 112 68)(29 116 72 35 113 69 32 110 66)(30 114 70 36 111 67 33 117 64)(37 127 88 43 133 85 40 130 82)(38 131 89 44 128 86 41 134 83)(39 135 90 45 132 87 42 129 84)

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

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

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

85 conjugacy classes

 class 1 3A 3B 3C 3D 5A 5B 5C 5D 9A ··· 9F 9G ··· 9L 15A ··· 15H 15I ··· 15P 45A ··· 45X 45Y ··· 45AV order 1 3 3 3 3 5 5 5 5 9 ··· 9 9 ··· 9 15 ··· 15 15 ··· 15 45 ··· 45 45 ··· 45 size 1 1 1 3 3 1 1 1 1 3 ··· 3 9 ··· 9 1 ··· 1 3 ··· 3 3 ··· 3 9 ··· 9

85 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 3 type + image C1 C3 C3 C5 C15 C15 He3 C3.He3 C5×He3 C5×C3.He3 kernel C5×C3.He3 C3×C45 C5×3- 1+2 C3.He3 C3×C9 3- 1+2 C15 C5 C3 C1 # reps 1 2 6 4 8 24 2 6 8 24

Matrix representation of C5×C3.He3 in GL3(𝔽181) generated by

 135 0 0 0 135 0 0 0 135
,
 132 0 0 0 132 0 0 0 132
,
 43 0 0 0 43 0 43 0 65
,
 1 0 0 1 132 0 133 0 48
,
 1 131 0 0 180 1 1 180 0
G:=sub<GL(3,GF(181))| [135,0,0,0,135,0,0,0,135],[132,0,0,0,132,0,0,0,132],[43,0,43,0,43,0,0,0,65],[1,1,133,0,132,0,0,0,48],[1,0,1,131,180,180,0,1,0] >;

C5×C3.He3 in GAP, Magma, Sage, TeX

C_5\times C_3.{\rm He}_3
% in TeX

G:=Group("C5xC3.He3");
// GroupNames label

G:=SmallGroup(405,10);
// by ID

G=gap.SmallGroup(405,10);
# by ID

G:=PCGroup([5,-3,-3,-5,-3,-3,675,481,906,3603]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^3=d^3=1,c^3=b^-1,e^3=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b*c*d^-1,e*d*e^-1=b^-1*d>;
// generators/relations

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