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G = D30.5C8order 480 = 25·3·5

3rd non-split extension by D30 of C8 acting via C8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D30.5C8, C40.53D6, C157M5(2), C24.60D10, Dic15.5C8, C120.56C22, C3⋊C165D5, C6.2(C8×D5), C52C165S3, C53(D6.C8), C8.39(S3×D5), C31(C80⋊C2), C153C8.9C4, C10.11(S3×C8), C20.69(C4×S3), C30.24(C2×C8), (C8×D15).5C2, C12.37(C4×D5), C60.140(C2×C4), (C4×D15).12C4, C2.3(D152C8), C4.17(D30.C2), (C5×C3⋊C16)⋊7C2, (C3×C52C16)⋊10C2, SmallGroup(480,12)

Series: Derived Chief Lower central Upper central

C1C30 — D30.5C8
C1C5C15C30C60C120C3×C52C16 — D30.5C8
C15C30 — D30.5C8
C1C8

Generators and relations for D30.5C8
 G = < a,b,c | a30=b2=1, c8=a15, bab=a-1, cac-1=a19, cbc-1=a3b >

30C2
15C22
15C4
10S3
6D5
15C2×C4
15C8
5Dic3
5D6
3Dic5
3D10
2D15
3C16
5C16
15C2×C8
5C3⋊C8
5C4×S3
3C52C8
3C4×D5
15M5(2)
5C48
5S3×C8
3C80
3C8×D5
5D6.C8
3C80⋊C2

Smallest permutation representation of D30.5C8
On 240 points
Generators in S240
(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)(181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210)(211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)
(1 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)(31 60)(32 59)(33 58)(34 57)(35 56)(36 55)(37 54)(38 53)(39 52)(40 51)(41 50)(42 49)(43 48)(44 47)(45 46)(61 90)(62 89)(63 88)(64 87)(65 86)(66 85)(67 84)(68 83)(69 82)(70 81)(71 80)(72 79)(73 78)(74 77)(75 76)(91 120)(92 119)(93 118)(94 117)(95 116)(96 115)(97 114)(98 113)(99 112)(100 111)(101 110)(102 109)(103 108)(104 107)(105 106)(121 135)(122 134)(123 133)(124 132)(125 131)(126 130)(127 129)(136 150)(137 149)(138 148)(139 147)(140 146)(141 145)(142 144)(151 155)(152 154)(156 180)(157 179)(158 178)(159 177)(160 176)(161 175)(162 174)(163 173)(164 172)(165 171)(166 170)(167 169)(181 205)(182 204)(183 203)(184 202)(185 201)(186 200)(187 199)(188 198)(189 197)(190 196)(191 195)(192 194)(206 210)(207 209)(211 225)(212 224)(213 223)(214 222)(215 221)(216 220)(217 219)(226 240)(227 239)(228 238)(229 237)(230 236)(231 235)(232 234)
(1 226 91 161 46 201 76 121 16 211 106 176 31 186 61 136)(2 215 92 180 47 190 77 140 17 230 107 165 32 205 62 125)(3 234 93 169 48 209 78 129 18 219 108 154 33 194 63 144)(4 223 94 158 49 198 79 148 19 238 109 173 34 183 64 133)(5 212 95 177 50 187 80 137 20 227 110 162 35 202 65 122)(6 231 96 166 51 206 81 126 21 216 111 151 36 191 66 141)(7 220 97 155 52 195 82 145 22 235 112 170 37 210 67 130)(8 239 98 174 53 184 83 134 23 224 113 159 38 199 68 149)(9 228 99 163 54 203 84 123 24 213 114 178 39 188 69 138)(10 217 100 152 55 192 85 142 25 232 115 167 40 207 70 127)(11 236 101 171 56 181 86 131 26 221 116 156 41 196 71 146)(12 225 102 160 57 200 87 150 27 240 117 175 42 185 72 135)(13 214 103 179 58 189 88 139 28 229 118 164 43 204 73 124)(14 233 104 168 59 208 89 128 29 218 119 153 44 193 74 143)(15 222 105 157 60 197 90 147 30 237 120 172 45 182 75 132)

G:=sub<Sym(240)| (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)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210)(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)(91,120)(92,119)(93,118)(94,117)(95,116)(96,115)(97,114)(98,113)(99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(127,129)(136,150)(137,149)(138,148)(139,147)(140,146)(141,145)(142,144)(151,155)(152,154)(156,180)(157,179)(158,178)(159,177)(160,176)(161,175)(162,174)(163,173)(164,172)(165,171)(166,170)(167,169)(181,205)(182,204)(183,203)(184,202)(185,201)(186,200)(187,199)(188,198)(189,197)(190,196)(191,195)(192,194)(206,210)(207,209)(211,225)(212,224)(213,223)(214,222)(215,221)(216,220)(217,219)(226,240)(227,239)(228,238)(229,237)(230,236)(231,235)(232,234), (1,226,91,161,46,201,76,121,16,211,106,176,31,186,61,136)(2,215,92,180,47,190,77,140,17,230,107,165,32,205,62,125)(3,234,93,169,48,209,78,129,18,219,108,154,33,194,63,144)(4,223,94,158,49,198,79,148,19,238,109,173,34,183,64,133)(5,212,95,177,50,187,80,137,20,227,110,162,35,202,65,122)(6,231,96,166,51,206,81,126,21,216,111,151,36,191,66,141)(7,220,97,155,52,195,82,145,22,235,112,170,37,210,67,130)(8,239,98,174,53,184,83,134,23,224,113,159,38,199,68,149)(9,228,99,163,54,203,84,123,24,213,114,178,39,188,69,138)(10,217,100,152,55,192,85,142,25,232,115,167,40,207,70,127)(11,236,101,171,56,181,86,131,26,221,116,156,41,196,71,146)(12,225,102,160,57,200,87,150,27,240,117,175,42,185,72,135)(13,214,103,179,58,189,88,139,28,229,118,164,43,204,73,124)(14,233,104,168,59,208,89,128,29,218,119,153,44,193,74,143)(15,222,105,157,60,197,90,147,30,237,120,172,45,182,75,132)>;

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)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210)(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)(91,120)(92,119)(93,118)(94,117)(95,116)(96,115)(97,114)(98,113)(99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(127,129)(136,150)(137,149)(138,148)(139,147)(140,146)(141,145)(142,144)(151,155)(152,154)(156,180)(157,179)(158,178)(159,177)(160,176)(161,175)(162,174)(163,173)(164,172)(165,171)(166,170)(167,169)(181,205)(182,204)(183,203)(184,202)(185,201)(186,200)(187,199)(188,198)(189,197)(190,196)(191,195)(192,194)(206,210)(207,209)(211,225)(212,224)(213,223)(214,222)(215,221)(216,220)(217,219)(226,240)(227,239)(228,238)(229,237)(230,236)(231,235)(232,234), (1,226,91,161,46,201,76,121,16,211,106,176,31,186,61,136)(2,215,92,180,47,190,77,140,17,230,107,165,32,205,62,125)(3,234,93,169,48,209,78,129,18,219,108,154,33,194,63,144)(4,223,94,158,49,198,79,148,19,238,109,173,34,183,64,133)(5,212,95,177,50,187,80,137,20,227,110,162,35,202,65,122)(6,231,96,166,51,206,81,126,21,216,111,151,36,191,66,141)(7,220,97,155,52,195,82,145,22,235,112,170,37,210,67,130)(8,239,98,174,53,184,83,134,23,224,113,159,38,199,68,149)(9,228,99,163,54,203,84,123,24,213,114,178,39,188,69,138)(10,217,100,152,55,192,85,142,25,232,115,167,40,207,70,127)(11,236,101,171,56,181,86,131,26,221,116,156,41,196,71,146)(12,225,102,160,57,200,87,150,27,240,117,175,42,185,72,135)(13,214,103,179,58,189,88,139,28,229,118,164,43,204,73,124)(14,233,104,168,59,208,89,128,29,218,119,153,44,193,74,143)(15,222,105,157,60,197,90,147,30,237,120,172,45,182,75,132) );

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),(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210),(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19),(13,18),(14,17),(15,16),(31,60),(32,59),(33,58),(34,57),(35,56),(36,55),(37,54),(38,53),(39,52),(40,51),(41,50),(42,49),(43,48),(44,47),(45,46),(61,90),(62,89),(63,88),(64,87),(65,86),(66,85),(67,84),(68,83),(69,82),(70,81),(71,80),(72,79),(73,78),(74,77),(75,76),(91,120),(92,119),(93,118),(94,117),(95,116),(96,115),(97,114),(98,113),(99,112),(100,111),(101,110),(102,109),(103,108),(104,107),(105,106),(121,135),(122,134),(123,133),(124,132),(125,131),(126,130),(127,129),(136,150),(137,149),(138,148),(139,147),(140,146),(141,145),(142,144),(151,155),(152,154),(156,180),(157,179),(158,178),(159,177),(160,176),(161,175),(162,174),(163,173),(164,172),(165,171),(166,170),(167,169),(181,205),(182,204),(183,203),(184,202),(185,201),(186,200),(187,199),(188,198),(189,197),(190,196),(191,195),(192,194),(206,210),(207,209),(211,225),(212,224),(213,223),(214,222),(215,221),(216,220),(217,219),(226,240),(227,239),(228,238),(229,237),(230,236),(231,235),(232,234)], [(1,226,91,161,46,201,76,121,16,211,106,176,31,186,61,136),(2,215,92,180,47,190,77,140,17,230,107,165,32,205,62,125),(3,234,93,169,48,209,78,129,18,219,108,154,33,194,63,144),(4,223,94,158,49,198,79,148,19,238,109,173,34,183,64,133),(5,212,95,177,50,187,80,137,20,227,110,162,35,202,65,122),(6,231,96,166,51,206,81,126,21,216,111,151,36,191,66,141),(7,220,97,155,52,195,82,145,22,235,112,170,37,210,67,130),(8,239,98,174,53,184,83,134,23,224,113,159,38,199,68,149),(9,228,99,163,54,203,84,123,24,213,114,178,39,188,69,138),(10,217,100,152,55,192,85,142,25,232,115,167,40,207,70,127),(11,236,101,171,56,181,86,131,26,221,116,156,41,196,71,146),(12,225,102,160,57,200,87,150,27,240,117,175,42,185,72,135),(13,214,103,179,58,189,88,139,28,229,118,164,43,204,73,124),(14,233,104,168,59,208,89,128,29,218,119,153,44,193,74,143),(15,222,105,157,60,197,90,147,30,237,120,172,45,182,75,132)])

84 conjugacy classes

class 1 2A2B 3 4A4B4C5A5B 6 8A8B8C8D8E8F10A10B12A12B15A15B16A16B16C16D16E16F16G16H20A20B20C20D24A24B24C24D30A30B40A···40H48A···48H60A60B60C60D80A···80P120A···120H
order122344455688888810101212151516161616161616162020202024242424303040···4048···486060606080···80120···120
size1130211302221111303022224466661010101022222222442···210···1044446···64···4

84 irreducible representations

dim11111111222222222224444
type++++++++++
imageC1C2C2C2C4C4C8C8S3D5D6D10C4×S3M5(2)C4×D5S3×C8C8×D5D6.C8C80⋊C2S3×D5D30.C2D152C8D30.5C8
kernelD30.5C8C5×C3⋊C16C3×C52C16C8×D15C153C8C4×D15Dic15D30C52C16C3⋊C16C40C24C20C15C12C10C6C5C3C8C4C2C1
# reps111122441212244488162248

Matrix representation of D30.5C8 in GL4(𝔽241) generated by

190100
240000
002401
002400
,
190100
515100
0001
0010
,
23310900
183800
001770
000177
G:=sub<GL(4,GF(241))| [190,240,0,0,1,0,0,0,0,0,240,240,0,0,1,0],[190,51,0,0,1,51,0,0,0,0,0,1,0,0,1,0],[233,183,0,0,109,8,0,0,0,0,177,0,0,0,0,177] >;

D30.5C8 in GAP, Magma, Sage, TeX

D_{30}._5C_8
% in TeX

G:=Group("D30.5C8");
// GroupNames label

G:=SmallGroup(480,12);
// by ID

G=gap.SmallGroup(480,12);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,253,36,58,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^30=b^2=1,c^8=a^15,b*a*b=a^-1,c*a*c^-1=a^19,c*b*c^-1=a^3*b>;
// generators/relations

Export

Subgroup lattice of D30.5C8 in TeX

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