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G = D609C4order 480 = 25·3·5

3rd semidirect product of D60 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D609C4, C60.1D4, C4.9D60, C30.38D8, C12.9D20, C20.9D12, C30.33SD16, C4⋊C41D15, C4.1(C4×D15), C20.47(C4×S3), C60.77(C2×C4), (C2×D60).6C2, C12.15(C4×D5), (C2×C4).34D30, (C2×C20).66D6, C53(C6.D8), C6.8(Q8⋊D5), C32(D206C4), C2.2(D4⋊D15), C6.16(D4⋊D5), (C2×C30).137D4, (C2×C12).67D10, C1511(D4⋊C4), C10.16(D4⋊S3), C10.29(D6⋊C4), (C2×C60).52C22, C2.5(D303C4), C30.71(C22⋊C4), C10.8(Q82S3), C2.2(Q82D15), C6.14(D10⋊C4), C22.14(C157D4), (C5×C4⋊C4)⋊1S3, (C3×C4⋊C4)⋊1D5, (C15×C4⋊C4)⋊1C2, (C2×C153C8)⋊1C2, (C2×C6).69(C5⋊D4), (C2×C10).69(C3⋊D4), SmallGroup(480,169)

Series: Derived Chief Lower central Upper central

C1C60 — D609C4
C1C5C15C30C2×C30C2×C60C2×D60 — D609C4
C15C30C60 — D609C4
C1C22C2×C4C4⋊C4

Generators and relations for D609C4
 G = < a,b,c | a60=b2=c4=1, bab=a-1, cac-1=a31, cbc-1=a15b >

Subgroups: 804 in 100 conjugacy classes, 41 normal (39 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4, C22, C22 [×4], C5, S3 [×2], C6 [×3], C8, C2×C4, C2×C4, D4 [×3], C23, D5 [×2], C10 [×3], C12 [×2], C12, D6 [×4], C2×C6, C15, C4⋊C4, C2×C8, C2×D4, C20 [×2], C20, D10 [×4], C2×C10, C3⋊C8, D12 [×3], C2×C12, C2×C12, C22×S3, D15 [×2], C30 [×3], D4⋊C4, C52C8, D20 [×3], C2×C20, C2×C20, C22×D5, C2×C3⋊C8, C3×C4⋊C4, C2×D12, C60 [×2], C60, D30 [×4], C2×C30, C2×C52C8, C5×C4⋊C4, C2×D20, C6.D8, C153C8, D60 [×2], D60, C2×C60, C2×C60, C22×D15, D206C4, C2×C153C8, C15×C4⋊C4, C2×D60, D609C4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, D6, C22⋊C4, D8, SD16, D10, C4×S3, D12, C3⋊D4, D15, D4⋊C4, C4×D5, D20, C5⋊D4, D6⋊C4, D4⋊S3, Q82S3, D30, D10⋊C4, D4⋊D5, Q8⋊D5, C6.D8, C4×D15, D60, C157D4, D206C4, D303C4, D4⋊D15, Q82D15, D609C4

Smallest permutation representation of D609C4
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 60)(2 59)(3 58)(4 57)(5 56)(6 55)(7 54)(8 53)(9 52)(10 51)(11 50)(12 49)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 42)(20 41)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(61 108)(62 107)(63 106)(64 105)(65 104)(66 103)(67 102)(68 101)(69 100)(70 99)(71 98)(72 97)(73 96)(74 95)(75 94)(76 93)(77 92)(78 91)(79 90)(80 89)(81 88)(82 87)(83 86)(84 85)(109 120)(110 119)(111 118)(112 117)(113 116)(114 115)(121 175)(122 174)(123 173)(124 172)(125 171)(126 170)(127 169)(128 168)(129 167)(130 166)(131 165)(132 164)(133 163)(134 162)(135 161)(136 160)(137 159)(138 158)(139 157)(140 156)(141 155)(142 154)(143 153)(144 152)(145 151)(146 150)(147 149)(176 180)(177 179)(181 221)(182 220)(183 219)(184 218)(185 217)(186 216)(187 215)(188 214)(189 213)(190 212)(191 211)(192 210)(193 209)(194 208)(195 207)(196 206)(197 205)(198 204)(199 203)(200 202)(222 240)(223 239)(224 238)(225 237)(226 236)(227 235)(228 234)(229 233)(230 232)
(1 239 115 156)(2 210 116 127)(3 181 117 158)(4 212 118 129)(5 183 119 160)(6 214 120 131)(7 185 61 162)(8 216 62 133)(9 187 63 164)(10 218 64 135)(11 189 65 166)(12 220 66 137)(13 191 67 168)(14 222 68 139)(15 193 69 170)(16 224 70 141)(17 195 71 172)(18 226 72 143)(19 197 73 174)(20 228 74 145)(21 199 75 176)(22 230 76 147)(23 201 77 178)(24 232 78 149)(25 203 79 180)(26 234 80 151)(27 205 81 122)(28 236 82 153)(29 207 83 124)(30 238 84 155)(31 209 85 126)(32 240 86 157)(33 211 87 128)(34 182 88 159)(35 213 89 130)(36 184 90 161)(37 215 91 132)(38 186 92 163)(39 217 93 134)(40 188 94 165)(41 219 95 136)(42 190 96 167)(43 221 97 138)(44 192 98 169)(45 223 99 140)(46 194 100 171)(47 225 101 142)(48 196 102 173)(49 227 103 144)(50 198 104 175)(51 229 105 146)(52 200 106 177)(53 231 107 148)(54 202 108 179)(55 233 109 150)(56 204 110 121)(57 235 111 152)(58 206 112 123)(59 237 113 154)(60 208 114 125)

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

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

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

84 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D5A5B6A6B6C8A8B8C8D10A···10F12A···12F15A15B15C15D20A···20L30A···30L60A···60X
order1222223444455666888810···1012···121515151520···2030···3060···60
size111160602224422222303030302···24···422224···42···24···4

84 irreducible representations

dim111112222222222222222222444444
type++++++++++++++++++++++
imageC1C2C2C2C4S3D4D4D5D6D8SD16D10C4×S3D12C3⋊D4D15C4×D5D20C5⋊D4D30C4×D15D60C157D4D4⋊S3Q82S3D4⋊D5Q8⋊D5D4⋊D15Q82D15
kernelD609C4C2×C153C8C15×C4⋊C4C2×D60D60C5×C4⋊C4C60C2×C30C3×C4⋊C4C2×C20C30C30C2×C12C20C20C2×C10C4⋊C4C12C12C2×C6C2×C4C4C4C22C10C10C6C6C2C2
# reps111141112122222244444888112244

Matrix representation of D609C4 in GL6(𝔽241)

010000
24010000
0005200
0019019000
000012
0000240240
,
24010000
010000
00933000
001714800
000010
0000240240
,
1711400000
101700000
0019715600
00884400
00000219
00002300

G:=sub<GL(6,GF(241))| [0,240,0,0,0,0,1,1,0,0,0,0,0,0,0,190,0,0,0,0,52,190,0,0,0,0,0,0,1,240,0,0,0,0,2,240],[240,0,0,0,0,0,1,1,0,0,0,0,0,0,93,17,0,0,0,0,30,148,0,0,0,0,0,0,1,240,0,0,0,0,0,240],[171,101,0,0,0,0,140,70,0,0,0,0,0,0,197,88,0,0,0,0,156,44,0,0,0,0,0,0,0,230,0,0,0,0,219,0] >;

D609C4 in GAP, Magma, Sage, TeX

D_{60}\rtimes_9C_4
% in TeX

G:=Group("D60:9C4");
// GroupNames label

G:=SmallGroup(480,169);
// by ID

G=gap.SmallGroup(480,169);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,36,675,346,80,2693,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^31,c*b*c^-1=a^15*b>;
// generators/relations

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