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G = C30.D6order 360 = 23·32·5

11st non-split extension by C30 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, A-group

Aliases: C30.11D6, C156(C4×S3), C3⋊D155C4, C326(C4×D5), C3⋊Dic34D5, C6.18(S3×D5), (C3×Dic5)⋊2S3, (C3×C6).22D10, Dic52(C3⋊S3), C32(D30.C2), (C3×C30).10C22, (C32×Dic5)⋊5C2, C52(C4×C3⋊S3), C2.3(D5×C3⋊S3), (C3×C15)⋊18(C2×C4), C10.3(C2×C3⋊S3), (C5×C3⋊Dic3)⋊2C2, (C2×C3⋊D15).2C2, SmallGroup(360,67)

Series: Derived Chief Lower central Upper central

C1C3×C15 — C30.D6
C1C5C15C3×C15C3×C30C32×Dic5 — C30.D6
C3×C15 — C30.D6
C1C2

Generators and relations for C30.D6
 G = < a,b,c | a30=c2=1, b6=a15, bab-1=a19, cac=a-1, cbc=b5 >

Subgroups: 632 in 96 conjugacy classes, 36 normal (14 characteristic)
C1, C2, C2 [×2], C3 [×4], C4 [×2], C22, C5, S3 [×8], C6 [×4], C2×C4, C32, D5 [×2], C10, Dic3 [×4], C12 [×4], D6 [×4], C15 [×4], C3⋊S3 [×2], C3×C6, Dic5, C20, D10, C4×S3 [×4], D15 [×8], C30 [×4], C3⋊Dic3, C3×C12, C2×C3⋊S3, C4×D5, C3×C15, C5×Dic3 [×4], C3×Dic5 [×4], D30 [×4], C4×C3⋊S3, C3⋊D15 [×2], C3×C30, D30.C2 [×4], C32×Dic5, C5×C3⋊Dic3, C2×C3⋊D15, C30.D6
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D5, D6 [×4], C3⋊S3, D10, C4×S3 [×4], C2×C3⋊S3, C4×D5, S3×D5 [×4], C4×C3⋊S3, D30.C2 [×4], D5×C3⋊S3, C30.D6

Smallest permutation representation of C30.D6
On 180 points
Generators in S180
(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)(151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180)
(1 127 78 106 31 169 16 142 63 91 46 154)(2 146 79 95 32 158 17 131 64 110 47 173)(3 135 80 114 33 177 18 150 65 99 48 162)(4 124 81 103 34 166 19 139 66 118 49 151)(5 143 82 92 35 155 20 128 67 107 50 170)(6 132 83 111 36 174 21 147 68 96 51 159)(7 121 84 100 37 163 22 136 69 115 52 178)(8 140 85 119 38 152 23 125 70 104 53 167)(9 129 86 108 39 171 24 144 71 93 54 156)(10 148 87 97 40 160 25 133 72 112 55 175)(11 137 88 116 41 179 26 122 73 101 56 164)(12 126 89 105 42 168 27 141 74 120 57 153)(13 145 90 94 43 157 28 130 75 109 58 172)(14 134 61 113 44 176 29 149 76 98 59 161)(15 123 62 102 45 165 30 138 77 117 60 180)
(1 63)(2 62)(3 61)(4 90)(5 89)(6 88)(7 87)(8 86)(9 85)(10 84)(11 83)(12 82)(13 81)(14 80)(15 79)(16 78)(17 77)(18 76)(19 75)(20 74)(21 73)(22 72)(23 71)(24 70)(25 69)(26 68)(27 67)(28 66)(29 65)(30 64)(32 60)(33 59)(34 58)(35 57)(36 56)(37 55)(38 54)(39 53)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(91 169)(92 168)(93 167)(94 166)(95 165)(96 164)(97 163)(98 162)(99 161)(100 160)(101 159)(102 158)(103 157)(104 156)(105 155)(106 154)(107 153)(108 152)(109 151)(110 180)(111 179)(112 178)(113 177)(114 176)(115 175)(116 174)(117 173)(118 172)(119 171)(120 170)(121 133)(122 132)(123 131)(124 130)(125 129)(126 128)(134 150)(135 149)(136 148)(137 147)(138 146)(139 145)(140 144)(141 143)

G:=sub<Sym(180)| (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)(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180), (1,127,78,106,31,169,16,142,63,91,46,154)(2,146,79,95,32,158,17,131,64,110,47,173)(3,135,80,114,33,177,18,150,65,99,48,162)(4,124,81,103,34,166,19,139,66,118,49,151)(5,143,82,92,35,155,20,128,67,107,50,170)(6,132,83,111,36,174,21,147,68,96,51,159)(7,121,84,100,37,163,22,136,69,115,52,178)(8,140,85,119,38,152,23,125,70,104,53,167)(9,129,86,108,39,171,24,144,71,93,54,156)(10,148,87,97,40,160,25,133,72,112,55,175)(11,137,88,116,41,179,26,122,73,101,56,164)(12,126,89,105,42,168,27,141,74,120,57,153)(13,145,90,94,43,157,28,130,75,109,58,172)(14,134,61,113,44,176,29,149,76,98,59,161)(15,123,62,102,45,165,30,138,77,117,60,180), (1,63)(2,62)(3,61)(4,90)(5,89)(6,88)(7,87)(8,86)(9,85)(10,84)(11,83)(12,82)(13,81)(14,80)(15,79)(16,78)(17,77)(18,76)(19,75)(20,74)(21,73)(22,72)(23,71)(24,70)(25,69)(26,68)(27,67)(28,66)(29,65)(30,64)(32,60)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(91,169)(92,168)(93,167)(94,166)(95,165)(96,164)(97,163)(98,162)(99,161)(100,160)(101,159)(102,158)(103,157)(104,156)(105,155)(106,154)(107,153)(108,152)(109,151)(110,180)(111,179)(112,178)(113,177)(114,176)(115,175)(116,174)(117,173)(118,172)(119,171)(120,170)(121,133)(122,132)(123,131)(124,130)(125,129)(126,128)(134,150)(135,149)(136,148)(137,147)(138,146)(139,145)(140,144)(141,143)>;

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)(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180), (1,127,78,106,31,169,16,142,63,91,46,154)(2,146,79,95,32,158,17,131,64,110,47,173)(3,135,80,114,33,177,18,150,65,99,48,162)(4,124,81,103,34,166,19,139,66,118,49,151)(5,143,82,92,35,155,20,128,67,107,50,170)(6,132,83,111,36,174,21,147,68,96,51,159)(7,121,84,100,37,163,22,136,69,115,52,178)(8,140,85,119,38,152,23,125,70,104,53,167)(9,129,86,108,39,171,24,144,71,93,54,156)(10,148,87,97,40,160,25,133,72,112,55,175)(11,137,88,116,41,179,26,122,73,101,56,164)(12,126,89,105,42,168,27,141,74,120,57,153)(13,145,90,94,43,157,28,130,75,109,58,172)(14,134,61,113,44,176,29,149,76,98,59,161)(15,123,62,102,45,165,30,138,77,117,60,180), (1,63)(2,62)(3,61)(4,90)(5,89)(6,88)(7,87)(8,86)(9,85)(10,84)(11,83)(12,82)(13,81)(14,80)(15,79)(16,78)(17,77)(18,76)(19,75)(20,74)(21,73)(22,72)(23,71)(24,70)(25,69)(26,68)(27,67)(28,66)(29,65)(30,64)(32,60)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(91,169)(92,168)(93,167)(94,166)(95,165)(96,164)(97,163)(98,162)(99,161)(100,160)(101,159)(102,158)(103,157)(104,156)(105,155)(106,154)(107,153)(108,152)(109,151)(110,180)(111,179)(112,178)(113,177)(114,176)(115,175)(116,174)(117,173)(118,172)(119,171)(120,170)(121,133)(122,132)(123,131)(124,130)(125,129)(126,128)(134,150)(135,149)(136,148)(137,147)(138,146)(139,145)(140,144)(141,143) );

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),(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)], [(1,127,78,106,31,169,16,142,63,91,46,154),(2,146,79,95,32,158,17,131,64,110,47,173),(3,135,80,114,33,177,18,150,65,99,48,162),(4,124,81,103,34,166,19,139,66,118,49,151),(5,143,82,92,35,155,20,128,67,107,50,170),(6,132,83,111,36,174,21,147,68,96,51,159),(7,121,84,100,37,163,22,136,69,115,52,178),(8,140,85,119,38,152,23,125,70,104,53,167),(9,129,86,108,39,171,24,144,71,93,54,156),(10,148,87,97,40,160,25,133,72,112,55,175),(11,137,88,116,41,179,26,122,73,101,56,164),(12,126,89,105,42,168,27,141,74,120,57,153),(13,145,90,94,43,157,28,130,75,109,58,172),(14,134,61,113,44,176,29,149,76,98,59,161),(15,123,62,102,45,165,30,138,77,117,60,180)], [(1,63),(2,62),(3,61),(4,90),(5,89),(6,88),(7,87),(8,86),(9,85),(10,84),(11,83),(12,82),(13,81),(14,80),(15,79),(16,78),(17,77),(18,76),(19,75),(20,74),(21,73),(22,72),(23,71),(24,70),(25,69),(26,68),(27,67),(28,66),(29,65),(30,64),(32,60),(33,59),(34,58),(35,57),(36,56),(37,55),(38,54),(39,53),(40,52),(41,51),(42,50),(43,49),(44,48),(45,47),(91,169),(92,168),(93,167),(94,166),(95,165),(96,164),(97,163),(98,162),(99,161),(100,160),(101,159),(102,158),(103,157),(104,156),(105,155),(106,154),(107,153),(108,152),(109,151),(110,180),(111,179),(112,178),(113,177),(114,176),(115,175),(116,174),(117,173),(118,172),(119,171),(120,170),(121,133),(122,132),(123,131),(124,130),(125,129),(126,128),(134,150),(135,149),(136,148),(137,147),(138,146),(139,145),(140,144),(141,143)])

48 conjugacy classes

class 1 2A2B2C3A3B3C3D4A4B4C4D5A5B6A6B6C6D10A10B12A···12H15A···15H20A20B20C20D30A···30H
order122233334444556666101012···1215···152020202030···30
size114545222255992222222210···104···4181818184···4

48 irreducible representations

dim1111122222244
type++++++++++
imageC1C2C2C2C4S3D5D6D10C4×S3C4×D5S3×D5D30.C2
kernelC30.D6C32×Dic5C5×C3⋊Dic3C2×C3⋊D15C3⋊D15C3×Dic5C3⋊Dic3C30C3×C6C15C32C6C3
# reps1111442428488

Matrix representation of C30.D6 in GL6(𝔽61)

60600000
100000
0060100
0060000
0000160
00004517
,
010000
60600000
001000
000100
0000411
0000457
,
010000
100000
000100
001000
0000171
00001744

G:=sub<GL(6,GF(61))| [60,1,0,0,0,0,60,0,0,0,0,0,0,0,60,60,0,0,0,0,1,0,0,0,0,0,0,0,1,45,0,0,0,0,60,17],[0,60,0,0,0,0,1,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,4,0,0,0,0,11,57],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,17,17,0,0,0,0,1,44] >;

C30.D6 in GAP, Magma, Sage, TeX

C_{30}.D_6
% in TeX

G:=Group("C30.D6");
// GroupNames label

G:=SmallGroup(360,67);
// by ID

G=gap.SmallGroup(360,67);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-5,24,31,201,730,10373]);
// Polycyclic

G:=Group<a,b,c|a^30=c^2=1,b^6=a^15,b*a*b^-1=a^19,c*a*c=a^-1,c*b*c=b^5>;
// generators/relations

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