Copied to
clipboard

G = C32×C15order 135 = 33·5

Abelian group of type [3,3,15]

direct product, abelian, monomial, 3-elementary

Aliases: C32×C15, SmallGroup(135,5)

Series: Derived Chief Lower central Upper central

C1 — C32×C15
C1C5C15C3×C15 — C32×C15
C1 — C32×C15
C1 — C32×C15

Generators and relations for C32×C15
 G = < a,b,c | a3=b3=c15=1, ab=ba, ac=ca, bc=cb >

Subgroups: 56, all normal (4 characteristic)
C1, C3 [×13], C5, C32 [×13], C15 [×13], C33, C3×C15 [×13], C32×C15
Quotients: C1, C3 [×13], C5, C32 [×13], C15 [×13], C33, C3×C15 [×13], C32×C15

Smallest permutation representation of C32×C15
Regular action on 135 points
Generators in S135
(1 91 46)(2 92 47)(3 93 48)(4 94 49)(5 95 50)(6 96 51)(7 97 52)(8 98 53)(9 99 54)(10 100 55)(11 101 56)(12 102 57)(13 103 58)(14 104 59)(15 105 60)(16 106 68)(17 107 69)(18 108 70)(19 109 71)(20 110 72)(21 111 73)(22 112 74)(23 113 75)(24 114 61)(25 115 62)(26 116 63)(27 117 64)(28 118 65)(29 119 66)(30 120 67)(31 123 87)(32 124 88)(33 125 89)(34 126 90)(35 127 76)(36 128 77)(37 129 78)(38 130 79)(39 131 80)(40 132 81)(41 133 82)(42 134 83)(43 135 84)(44 121 85)(45 122 86)
(1 42 16)(2 43 17)(3 44 18)(4 45 19)(5 31 20)(6 32 21)(7 33 22)(8 34 23)(9 35 24)(10 36 25)(11 37 26)(12 38 27)(13 39 28)(14 40 29)(15 41 30)(46 83 68)(47 84 69)(48 85 70)(49 86 71)(50 87 72)(51 88 73)(52 89 74)(53 90 75)(54 76 61)(55 77 62)(56 78 63)(57 79 64)(58 80 65)(59 81 66)(60 82 67)(91 134 106)(92 135 107)(93 121 108)(94 122 109)(95 123 110)(96 124 111)(97 125 112)(98 126 113)(99 127 114)(100 128 115)(101 129 116)(102 130 117)(103 131 118)(104 132 119)(105 133 120)
(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)

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

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

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

C32×C15 is a maximal subgroup of   C33⋊D5

135 conjugacy classes

class 1 3A···3Z5A5B5C5D15A···15CZ
order13···3555515···15
size11···111111···1

135 irreducible representations

dim1111
type+
imageC1C3C5C15
kernelC32×C15C3×C15C33C32
# reps1264104

Matrix representation of C32×C15 in GL3(𝔽31) generated by

500
0250
005
,
500
0250
0025
,
2500
070
0028
G:=sub<GL(3,GF(31))| [5,0,0,0,25,0,0,0,5],[5,0,0,0,25,0,0,0,25],[25,0,0,0,7,0,0,0,28] >;

C32×C15 in GAP, Magma, Sage, TeX

C_3^2\times C_{15}
% in TeX

G:=Group("C3^2xC15");
// GroupNames label

G:=SmallGroup(135,5);
// by ID

G=gap.SmallGroup(135,5);
# by ID

G:=PCGroup([4,-3,-3,-3,-5]);
// Polycyclic

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

׿
×
𝔽