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## G = C5×C9○He3order 405 = 34·5

### Direct product of C5 and C9○He3

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

Aliases: C5×C9○He3, C45.C32, He3.2C15, C15.3C33, 3- 1+23C15, C9.(C3×C15), (C3×C45)⋊5C3, (C3×C9)⋊5C15, (C5×He3).2C3, C32.6(C3×C15), (C3×C15).5C32, C3.3(C32×C15), (C5×3- 1+2)⋊3C3, SmallGroup(405,14)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C5×C9○He3
 Chief series C1 — C3 — C15 — C3×C15 — C5×He3 — C5×C9○He3
 Lower central C1 — C3 — C5×C9○He3
 Upper central C1 — C45 — C5×C9○He3

Generators and relations for C5×C9○He3
G = < a,b,c,d,e | a5=b9=c3=e3=1, d1=b6, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=b3c, de=ed >

Subgroups: 82 in 66 conjugacy classes, 58 normal (10 characteristic)
C1, C3, C3, C5, C9, C9, C32, C15, C15, C3×C9, He3, 3- 1+2, C45, C45, C3×C15, C9○He3, C3×C45, C5×He3, C5×3- 1+2, C5×C9○He3
Quotients: C1, C3, C5, C32, C15, C33, C3×C15, C9○He3, C32×C15, C5×C9○He3

Smallest permutation representation of C5×C9○He3
On 135 points
Generators in S135
(1 65 43 26 48)(2 66 44 27 49)(3 67 45 19 50)(4 68 37 20 51)(5 69 38 21 52)(6 70 39 22 53)(7 71 40 23 54)(8 72 41 24 46)(9 64 42 25 47)(10 122 36 128 104)(11 123 28 129 105)(12 124 29 130 106)(13 125 30 131 107)(14 126 31 132 108)(15 118 32 133 100)(16 119 33 134 101)(17 120 34 135 102)(18 121 35 127 103)(55 93 73 110 88)(56 94 74 111 89)(57 95 75 112 90)(58 96 76 113 82)(59 97 77 114 83)(60 98 78 115 84)(61 99 79 116 85)(62 91 80 117 86)(63 92 81 109 87)
(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 73 124)(2 74 125)(3 75 126)(4 76 118)(5 77 119)(6 78 120)(7 79 121)(8 80 122)(9 81 123)(10 46 91)(11 47 92)(12 48 93)(13 49 94)(14 50 95)(15 51 96)(16 52 97)(17 53 98)(18 54 99)(19 57 108)(20 58 100)(21 59 101)(22 60 102)(23 61 103)(24 62 104)(25 63 105)(26 55 106)(27 56 107)(28 64 109)(29 65 110)(30 66 111)(31 67 112)(32 68 113)(33 69 114)(34 70 115)(35 71 116)(36 72 117)(37 82 133)(38 83 134)(39 84 135)(40 85 127)(41 86 128)(42 87 129)(43 88 130)(44 89 131)(45 90 132)
(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)
(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)

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

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

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

165 conjugacy classes

 class 1 3A 3B 3C ··· 3J 5A 5B 5C 5D 9A ··· 9F 9G ··· 9V 15A ··· 15H 15I ··· 15AN 45A ··· 45X 45Y ··· 45CJ 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 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3 1 ··· 1 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 C9○He3 C5×C9○He3 kernel C5×C9○He3 C3×C45 C5×He3 C5×3- 1+2 C9○He3 C3×C9 He3 3- 1+2 C5 C1 # reps 1 8 2 16 4 32 8 64 6 24

Matrix representation of C5×C9○He3 in GL3(𝔽181) generated by

 125 0 0 0 125 0 0 0 125
,
 65 0 0 0 65 0 0 0 65
,
 55 106 92 131 126 57 0 132 0
,
 132 0 0 0 132 0 0 0 132
,
 1 55 21 0 132 0 0 0 48
G:=sub<GL(3,GF(181))| [125,0,0,0,125,0,0,0,125],[65,0,0,0,65,0,0,0,65],[55,131,0,106,126,132,92,57,0],[132,0,0,0,132,0,0,0,132],[1,0,0,55,132,0,21,0,48] >;

C5×C9○He3 in GAP, Magma, Sage, TeX

C_5\times C_9\circ {\rm He}_3
% in TeX

G:=Group("C5xC9oHe3");
// GroupNames label

G:=SmallGroup(405,14);
// by ID

G=gap.SmallGroup(405,14);
# by ID

G:=PCGroup([5,-3,-3,-3,-5,-3,1381,237]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^9=c^3=e^3=1,d^1=b^6,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^3*c,d*e=e*d>;
// generators/relations

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