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G = D304C8order 480 = 25·3·5

2nd semidirect product of D30 and C8 acting via C8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D304C8, C60.95D4, C20.55D12, C12.55D20, C30.16M4(2), C53(D6⋊C8), C6.4(C8×D5), C10.13(S3×C8), C157(C22⋊C8), C30.30(C2×C8), C31(D101C8), (C2×C20).323D6, C6.2(C8⋊D5), C10.7(C8⋊S3), C10.16(D6⋊C4), (C2×C12).327D10, C20.60(C3⋊D4), C4.27(C5⋊D12), C4.27(C3⋊D20), C12.63(C5⋊D4), C2.1(D304C4), C2.4(D152C8), C6.1(D10⋊C4), C30.40(C22⋊C4), (C2×C60).225C22, (C2×Dic15).19C4, (C22×D15).11C4, C2.2(D30.5C4), C22.10(D30.C2), (C2×C3⋊C8)⋊10D5, (C10×C3⋊C8)⋊10C2, (C6×C52C8)⋊10C2, (C2×C52C8)⋊10S3, (C2×C4×D15).19C2, (C2×C6).18(C4×D5), (C2×C10).41(C4×S3), (C2×C30).79(C2×C4), (C2×C4).228(S3×D5), SmallGroup(480,33)

Series: Derived Chief Lower central Upper central

C1C30 — D304C8
C1C5C15C30C60C2×C60C6×C52C8 — D304C8
C15C30 — D304C8
C1C2×C4

Generators and relations for D304C8
 G = < a,b,c | a30=b2=c8=1, bab=a-1, cac-1=a11, cbc-1=a25b >

Subgroups: 572 in 100 conjugacy classes, 42 normal (40 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4, C22, C22 [×4], C5, S3 [×2], C6 [×3], C8 [×2], C2×C4, C2×C4 [×3], C23, D5 [×2], C10 [×3], Dic3, C12 [×2], D6 [×4], C2×C6, C15, C2×C8 [×2], C22×C4, Dic5, C20 [×2], D10 [×4], C2×C10, C3⋊C8, C24, C4×S3 [×2], C2×Dic3, C2×C12, C22×S3, D15 [×2], C30 [×3], C22⋊C8, C52C8, C40, C4×D5 [×2], C2×Dic5, C2×C20, C22×D5, C2×C3⋊C8, C2×C24, S3×C2×C4, Dic15, C60 [×2], D30 [×2], D30 [×2], C2×C30, C2×C52C8, C2×C40, C2×C4×D5, D6⋊C8, C5×C3⋊C8, C3×C52C8, C4×D15 [×2], C2×Dic15, C2×C60, C22×D15, D101C8, C6×C52C8, C10×C3⋊C8, C2×C4×D15, D304C8
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C8 [×2], C2×C4, D4 [×2], D5, D6, C22⋊C4, C2×C8, M4(2), D10, C4×S3, D12, C3⋊D4, C22⋊C8, C4×D5, D20, C5⋊D4, S3×C8, C8⋊S3, D6⋊C4, S3×D5, C8×D5, C8⋊D5, D10⋊C4, D6⋊C8, D30.C2, C3⋊D20, C5⋊D12, D101C8, D152C8, D30.5C4, D304C4, D304C8

Smallest permutation representation of D304C8
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 49)(32 48)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(50 60)(51 59)(52 58)(53 57)(54 56)(61 87)(62 86)(63 85)(64 84)(65 83)(66 82)(67 81)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)(88 90)(91 116)(92 115)(93 114)(94 113)(95 112)(96 111)(97 110)(98 109)(99 108)(100 107)(101 106)(102 105)(103 104)(117 120)(118 119)(121 136)(122 135)(123 134)(124 133)(125 132)(126 131)(127 130)(128 129)(137 150)(138 149)(139 148)(140 147)(141 146)(142 145)(143 144)(151 177)(152 176)(153 175)(154 174)(155 173)(156 172)(157 171)(158 170)(159 169)(160 168)(161 167)(162 166)(163 165)(178 180)(181 196)(182 195)(183 194)(184 193)(185 192)(186 191)(187 190)(188 189)(197 210)(198 209)(199 208)(200 207)(201 206)(202 205)(203 204)(211 231)(212 230)(213 229)(214 228)(215 227)(216 226)(217 225)(218 224)(219 223)(220 222)(232 240)(233 239)(234 238)(235 237)
(1 214 189 33 144 157 119 82)(2 225 190 44 145 168 120 63)(3 236 191 55 146 179 91 74)(4 217 192 36 147 160 92 85)(5 228 193 47 148 171 93 66)(6 239 194 58 149 152 94 77)(7 220 195 39 150 163 95 88)(8 231 196 50 121 174 96 69)(9 212 197 31 122 155 97 80)(10 223 198 42 123 166 98 61)(11 234 199 53 124 177 99 72)(12 215 200 34 125 158 100 83)(13 226 201 45 126 169 101 64)(14 237 202 56 127 180 102 75)(15 218 203 37 128 161 103 86)(16 229 204 48 129 172 104 67)(17 240 205 59 130 153 105 78)(18 221 206 40 131 164 106 89)(19 232 207 51 132 175 107 70)(20 213 208 32 133 156 108 81)(21 224 209 43 134 167 109 62)(22 235 210 54 135 178 110 73)(23 216 181 35 136 159 111 84)(24 227 182 46 137 170 112 65)(25 238 183 57 138 151 113 76)(26 219 184 38 139 162 114 87)(27 230 185 49 140 173 115 68)(28 211 186 60 141 154 116 79)(29 222 187 41 142 165 117 90)(30 233 188 52 143 176 118 71)

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

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

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

84 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F5A5B6A6B6C8A8B8C8D8E8F8G8H10A···10F12A12B12C12D15A15B20A···20H24A···24H30A···30F40A···40P60A···60H
order1222223444444556668888888810···1012121212151520···2024···2430···3040···4060···60
size11113030211113030222226666101010102···22222442···210···104···46···64···4

84 irreducible representations

dim11111112222222222222222444444
type+++++++++++++++
imageC1C2C2C2C4C4C8S3D4D5D6M4(2)D10D12C3⋊D4C4×S3D20C5⋊D4C4×D5S3×C8C8⋊S3C8×D5C8⋊D5S3×D5C3⋊D20C5⋊D12D30.C2D152C8D30.5C4
kernelD304C8C6×C52C8C10×C3⋊C8C2×C4×D15C2×Dic15C22×D15D30C2×C52C8C60C2×C3⋊C8C2×C20C30C2×C12C20C20C2×C10C12C12C2×C6C10C10C6C6C2×C4C4C4C22C2C2
# reps11112281221222224444488222244

Matrix representation of D304C8 in GL5(𝔽241)

10000
052100
0240000
00001
000240240
,
2400000
052100
018918900
00001
00010
,
300000
08719800
04315400
000640
000177177

G:=sub<GL(5,GF(241))| [1,0,0,0,0,0,52,240,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,1,240],[240,0,0,0,0,0,52,189,0,0,0,1,189,0,0,0,0,0,0,1,0,0,0,1,0],[30,0,0,0,0,0,87,43,0,0,0,198,154,0,0,0,0,0,64,177,0,0,0,0,177] >;

D304C8 in GAP, Magma, Sage, TeX

D_{30}\rtimes_4C_8
% in TeX

G:=Group("D30:4C8");
// GroupNames label

G:=SmallGroup(480,33);
// by ID

G=gap.SmallGroup(480,33);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,92,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^30=b^2=c^8=1,b*a*b=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^25*b>;
// generators/relations

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