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G = D5×C33order 330 = 2·3·5·11

Direct product of C33 and D5

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: D5×C33, C5⋊C66, C553C6, C1656C2, C152C22, SmallGroup(330,6)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C33
C1C5C55C165 — D5×C33
C5 — D5×C33
C1C33

Generators and relations for D5×C33
 G = < a,b,c | a33=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

5C2
5C6
5C22
5C66

Smallest permutation representation of D5×C33
On 165 points
Generators in S165
(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)(133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165)
(1 127 164 96 51)(2 128 165 97 52)(3 129 133 98 53)(4 130 134 99 54)(5 131 135 67 55)(6 132 136 68 56)(7 100 137 69 57)(8 101 138 70 58)(9 102 139 71 59)(10 103 140 72 60)(11 104 141 73 61)(12 105 142 74 62)(13 106 143 75 63)(14 107 144 76 64)(15 108 145 77 65)(16 109 146 78 66)(17 110 147 79 34)(18 111 148 80 35)(19 112 149 81 36)(20 113 150 82 37)(21 114 151 83 38)(22 115 152 84 39)(23 116 153 85 40)(24 117 154 86 41)(25 118 155 87 42)(26 119 156 88 43)(27 120 157 89 44)(28 121 158 90 45)(29 122 159 91 46)(30 123 160 92 47)(31 124 161 93 48)(32 125 162 94 49)(33 126 163 95 50)
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 61)(12 62)(13 63)(14 64)(15 65)(16 66)(17 34)(18 35)(19 36)(20 37)(21 38)(22 39)(23 40)(24 41)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 49)(33 50)(67 131)(68 132)(69 100)(70 101)(71 102)(72 103)(73 104)(74 105)(75 106)(76 107)(77 108)(78 109)(79 110)(80 111)(81 112)(82 113)(83 114)(84 115)(85 116)(86 117)(87 118)(88 119)(89 120)(90 121)(91 122)(92 123)(93 124)(94 125)(95 126)(96 127)(97 128)(98 129)(99 130)

G:=sub<Sym(165)| (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)(133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165), (1,127,164,96,51)(2,128,165,97,52)(3,129,133,98,53)(4,130,134,99,54)(5,131,135,67,55)(6,132,136,68,56)(7,100,137,69,57)(8,101,138,70,58)(9,102,139,71,59)(10,103,140,72,60)(11,104,141,73,61)(12,105,142,74,62)(13,106,143,75,63)(14,107,144,76,64)(15,108,145,77,65)(16,109,146,78,66)(17,110,147,79,34)(18,111,148,80,35)(19,112,149,81,36)(20,113,150,82,37)(21,114,151,83,38)(22,115,152,84,39)(23,116,153,85,40)(24,117,154,86,41)(25,118,155,87,42)(26,119,156,88,43)(27,120,157,89,44)(28,121,158,90,45)(29,122,159,91,46)(30,123,160,92,47)(31,124,161,93,48)(32,125,162,94,49)(33,126,163,95,50), (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,61)(12,62)(13,63)(14,64)(15,65)(16,66)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,41)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,49)(33,50)(67,131)(68,132)(69,100)(70,101)(71,102)(72,103)(73,104)(74,105)(75,106)(76,107)(77,108)(78,109)(79,110)(80,111)(81,112)(82,113)(83,114)(84,115)(85,116)(86,117)(87,118)(88,119)(89,120)(90,121)(91,122)(92,123)(93,124)(94,125)(95,126)(96,127)(97,128)(98,129)(99,130)>;

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)(133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165), (1,127,164,96,51)(2,128,165,97,52)(3,129,133,98,53)(4,130,134,99,54)(5,131,135,67,55)(6,132,136,68,56)(7,100,137,69,57)(8,101,138,70,58)(9,102,139,71,59)(10,103,140,72,60)(11,104,141,73,61)(12,105,142,74,62)(13,106,143,75,63)(14,107,144,76,64)(15,108,145,77,65)(16,109,146,78,66)(17,110,147,79,34)(18,111,148,80,35)(19,112,149,81,36)(20,113,150,82,37)(21,114,151,83,38)(22,115,152,84,39)(23,116,153,85,40)(24,117,154,86,41)(25,118,155,87,42)(26,119,156,88,43)(27,120,157,89,44)(28,121,158,90,45)(29,122,159,91,46)(30,123,160,92,47)(31,124,161,93,48)(32,125,162,94,49)(33,126,163,95,50), (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,61)(12,62)(13,63)(14,64)(15,65)(16,66)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,41)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,49)(33,50)(67,131)(68,132)(69,100)(70,101)(71,102)(72,103)(73,104)(74,105)(75,106)(76,107)(77,108)(78,109)(79,110)(80,111)(81,112)(82,113)(83,114)(84,115)(85,116)(86,117)(87,118)(88,119)(89,120)(90,121)(91,122)(92,123)(93,124)(94,125)(95,126)(96,127)(97,128)(98,129)(99,130) );

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),(133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165)], [(1,127,164,96,51),(2,128,165,97,52),(3,129,133,98,53),(4,130,134,99,54),(5,131,135,67,55),(6,132,136,68,56),(7,100,137,69,57),(8,101,138,70,58),(9,102,139,71,59),(10,103,140,72,60),(11,104,141,73,61),(12,105,142,74,62),(13,106,143,75,63),(14,107,144,76,64),(15,108,145,77,65),(16,109,146,78,66),(17,110,147,79,34),(18,111,148,80,35),(19,112,149,81,36),(20,113,150,82,37),(21,114,151,83,38),(22,115,152,84,39),(23,116,153,85,40),(24,117,154,86,41),(25,118,155,87,42),(26,119,156,88,43),(27,120,157,89,44),(28,121,158,90,45),(29,122,159,91,46),(30,123,160,92,47),(31,124,161,93,48),(32,125,162,94,49),(33,126,163,95,50)], [(1,51),(2,52),(3,53),(4,54),(5,55),(6,56),(7,57),(8,58),(9,59),(10,60),(11,61),(12,62),(13,63),(14,64),(15,65),(16,66),(17,34),(18,35),(19,36),(20,37),(21,38),(22,39),(23,40),(24,41),(25,42),(26,43),(27,44),(28,45),(29,46),(30,47),(31,48),(32,49),(33,50),(67,131),(68,132),(69,100),(70,101),(71,102),(72,103),(73,104),(74,105),(75,106),(76,107),(77,108),(78,109),(79,110),(80,111),(81,112),(82,113),(83,114),(84,115),(85,116),(86,117),(87,118),(88,119),(89,120),(90,121),(91,122),(92,123),(93,124),(94,125),(95,126),(96,127),(97,128),(98,129),(99,130)])

132 conjugacy classes

class 1  2 3A3B5A5B6A6B11A···11J15A15B15C15D22A···22J33A···33T55A···55T66A···66T165A···165AN
order1233556611···111515151522···2233···3355···5566···66165···165
size151122551···122225···51···12···25···52···2

132 irreducible representations

dim111111112222
type+++
imageC1C2C3C6C11C22C33C66D5C3×D5D5×C11D5×C33
kernelD5×C33C165D5×C11C55C3×D5C15D5C5C33C11C3C1
# reps112210102020242040

Matrix representation of D5×C33 in GL2(𝔽331) generated by

2190
0219
,
1161
3300
,
01
10
G:=sub<GL(2,GF(331))| [219,0,0,219],[116,330,1,0],[0,1,1,0] >;

D5×C33 in GAP, Magma, Sage, TeX

D_5\times C_{33}
% in TeX

G:=Group("D5xC33");
// GroupNames label

G:=SmallGroup(330,6);
// by ID

G=gap.SmallGroup(330,6);
# by ID

G:=PCGroup([4,-2,-3,-11,-5,4227]);
// Polycyclic

G:=Group<a,b,c|a^33=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of D5×C33 in TeX

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