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G = C6.Dic10order 240 = 24·3·5

3rd non-split extension by C6 of Dic10 acting via Dic10/Dic5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C30.3Q8, Dic3⋊Dic5, C30.18D4, C6.12D20, C6.3Dic10, C10.3Dic6, C156(C4⋊C4), C31(C4⋊Dic5), C2.3(C15⋊Q8), C10.21(C4×S3), C30.34(C2×C4), C53(Dic3⋊C4), (C5×Dic3)⋊4C4, (C2×C6).11D10, (C2×C10).11D6, C2.5(S3×Dic5), C6.5(C2×Dic5), C2.3(C3⋊D20), C10.6(C3⋊D4), (C2×C30).8C22, (C2×Dic5).3S3, (C6×Dic5).4C2, (C2×Dic3).3D5, C22.10(S3×D5), (C10×Dic3).4C2, (C2×Dic15).7C2, SmallGroup(240,31)

Series: Derived Chief Lower central Upper central

C1C30 — C6.Dic10
C1C5C15C30C2×C30C6×Dic5 — C6.Dic10
C15C30 — C6.Dic10
C1C22

Generators and relations for C6.Dic10
 G = < a,b,c | a6=b20=1, c2=a3b10, bab-1=a-1, ac=ca, cbc-1=b-1 >

3C4
3C4
10C4
30C4
3C2×C4
5C2×C4
15C2×C4
10C12
10Dic3
2Dic5
3C20
3C20
6Dic5
15C4⋊C4
5C2×Dic3
5C2×C12
3C2×C20
3C2×Dic5
2C3×Dic5
2Dic15
5Dic3⋊C4
3C4⋊Dic5

Smallest permutation representation of C6.Dic10
Regular action on 240 points
Generators in S240
(1 79 96 169 60 118)(2 119 41 170 97 80)(3 61 98 171 42 120)(4 101 43 172 99 62)(5 63 100 173 44 102)(6 103 45 174 81 64)(7 65 82 175 46 104)(8 105 47 176 83 66)(9 67 84 177 48 106)(10 107 49 178 85 68)(11 69 86 179 50 108)(12 109 51 180 87 70)(13 71 88 161 52 110)(14 111 53 162 89 72)(15 73 90 163 54 112)(16 113 55 164 91 74)(17 75 92 165 56 114)(18 115 57 166 93 76)(19 77 94 167 58 116)(20 117 59 168 95 78)(21 127 199 160 240 203)(22 204 221 141 200 128)(23 129 181 142 222 205)(24 206 223 143 182 130)(25 131 183 144 224 207)(26 208 225 145 184 132)(27 133 185 146 226 209)(28 210 227 147 186 134)(29 135 187 148 228 211)(30 212 229 149 188 136)(31 137 189 150 230 213)(32 214 231 151 190 138)(33 139 191 152 232 215)(34 216 233 153 192 140)(35 121 193 154 234 217)(36 218 235 155 194 122)(37 123 195 156 236 219)(38 220 237 157 196 124)(39 125 197 158 238 201)(40 202 239 159 198 126)
(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)
(1 225 179 122)(2 224 180 121)(3 223 161 140)(4 222 162 139)(5 221 163 138)(6 240 164 137)(7 239 165 136)(8 238 166 135)(9 237 167 134)(10 236 168 133)(11 235 169 132)(12 234 170 131)(13 233 171 130)(14 232 172 129)(15 231 173 128)(16 230 174 127)(17 229 175 126)(18 228 176 125)(19 227 177 124)(20 226 178 123)(21 74 150 45)(22 73 151 44)(23 72 152 43)(24 71 153 42)(25 70 154 41)(26 69 155 60)(27 68 156 59)(28 67 157 58)(29 66 158 57)(30 65 159 56)(31 64 160 55)(32 63 141 54)(33 62 142 53)(34 61 143 52)(35 80 144 51)(36 79 145 50)(37 78 146 49)(38 77 147 48)(39 76 148 47)(40 75 149 46)(81 199 113 213)(82 198 114 212)(83 197 115 211)(84 196 116 210)(85 195 117 209)(86 194 118 208)(87 193 119 207)(88 192 120 206)(89 191 101 205)(90 190 102 204)(91 189 103 203)(92 188 104 202)(93 187 105 201)(94 186 106 220)(95 185 107 219)(96 184 108 218)(97 183 109 217)(98 182 110 216)(99 181 111 215)(100 200 112 214)

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

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

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

C6.Dic10 is a maximal subgroup of
Dic35Dic10  Dic151Q8  Dic3⋊Dic10  Dic5×Dic6  Dic5.2Dic6  Dic15.Q8  C4⋊Dic3⋊D5  Dic3.Dic10  Dic157Q8  Dic3.2Dic10  D6⋊Dic10  D30.35D4  C60.68D4  (C2×C12).D10  (C4×Dic15)⋊C2  D6⋊Dic5.C2  (S3×C20)⋊7C4  C5⋊(C423S3)  Dic5.7Dic6  Dic3.3Dic10  Dic15.4Q8  C12.Dic10  (C4×Dic5)⋊S3  C20.Dic6  C60.48D4  D3010Q8  D5×Dic3⋊C4  D10.19(C4×S3)  Dic34D20  D30.C2⋊C4  D30.Q8  D61Dic10  D101Dic6  D102Dic6  D303Q8  S3×C4⋊Dic5  D6.D20  D304Q8  D63Dic10  C4×C3⋊D20  D6⋊C4⋊D5  D6⋊D20  D6.9D20  C4×C15⋊Q8  C204Dic6  C23.26(S3×D5)  C23.13(S3×D5)  C23.14(S3×D5)  C6.(C2×D20)  C6.D4⋊D5  C23.17(S3×D5)  (C6×D5)⋊D4  Dic5×C3⋊D4  D307D4  Dic1517D4  D30.16D4  (C2×C6)⋊D20  (C2×C30)⋊Q8  (C2×C10)⋊8Dic6  Dic15.48D4
C6.Dic10 is a maximal quotient of
C60.15Q8  C60.Q8  C60.5Q8  C12.59D20  C30.24C42

42 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F5A5B6A6B6C10A···10F12A12B12C12D15A15B20A···20H30A···30F
order122234444445566610···1012121212151520···2030···30
size111126610103030222222···210101010446···64···4

42 irreducible representations

dim111112222222222224444
type++++++-++-+--++-+-
imageC1C2C2C2C4S3D4Q8D5D6Dic5D10Dic6C4×S3C3⋊D4Dic10D20S3×D5S3×Dic5C3⋊D20C15⋊Q8
kernelC6.Dic10C6×Dic5C10×Dic3C2×Dic15C5×Dic3C2×Dic5C30C30C2×Dic3C2×C10Dic3C2×C6C10C10C10C6C6C22C2C2C2
# reps111141112142222442222

Matrix representation of C6.Dic10 in GL4(𝔽61) generated by

1000
0100
0001
00601
,
252900
592700
004112
005320
,
343000
572700
005218
00439
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,0,60,0,0,1,1],[25,59,0,0,29,27,0,0,0,0,41,53,0,0,12,20],[34,57,0,0,30,27,0,0,0,0,52,43,0,0,18,9] >;

C6.Dic10 in GAP, Magma, Sage, TeX

C_6.{\rm Dic}_{10}
% in TeX

G:=Group("C6.Dic10");
// GroupNames label

G:=SmallGroup(240,31);
// by ID

G=gap.SmallGroup(240,31);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,121,55,490,6917]);
// Polycyclic

G:=Group<a,b,c|a^6=b^20=1,c^2=a^3*b^10,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C6.Dic10 in TeX

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