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G = Dic155C4order 240 = 24·3·5

3rd semidirect product of Dic15 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C30.2Q8, C30.17D4, Dic155C4, C10.2Dic6, C6.2Dic10, C155(C4⋊C4), C6.4(C4×D5), C2.2(C15⋊Q8), C10.11(C4×S3), C30.33(C2×C4), C52(Dic3⋊C4), (C2×C10).10D6, (C2×C6).10D10, C22.9(S3×D5), C32(C10.D4), C2.3(C15⋊D4), C6.13(C5⋊D4), (C2×C30).7C22, (C2×Dic5).2S3, (C2×Dic3).2D5, (C6×Dic5).3C2, C10.13(C3⋊D4), C2.5(D30.C2), (C2×Dic15).6C2, (C10×Dic3).3C2, SmallGroup(240,30)

Series: Derived Chief Lower central Upper central

C1C30 — Dic155C4
C1C5C15C30C2×C30C6×Dic5 — Dic155C4
C15C30 — Dic155C4
C1C22

Generators and relations for Dic155C4
 G = < a,b,c | a30=c4=1, b2=a15, bab-1=a-1, cac-1=a11, cbc-1=a15b >

6C4
10C4
15C4
15C4
3C2×C4
5C2×C4
15C2×C4
2Dic3
5Dic3
5Dic3
10C12
2Dic5
3Dic5
3Dic5
6C20
15C4⋊C4
5C2×Dic3
5C2×C12
3C2×C20
3C2×Dic5
2C5×Dic3
2C3×Dic5
5Dic3⋊C4
3C10.D4

Smallest permutation representation of Dic155C4
Regular action 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)
(1 206 16 191)(2 205 17 190)(3 204 18 189)(4 203 19 188)(5 202 20 187)(6 201 21 186)(7 200 22 185)(8 199 23 184)(9 198 24 183)(10 197 25 182)(11 196 26 181)(12 195 27 210)(13 194 28 209)(14 193 29 208)(15 192 30 207)(31 231 46 216)(32 230 47 215)(33 229 48 214)(34 228 49 213)(35 227 50 212)(36 226 51 211)(37 225 52 240)(38 224 53 239)(39 223 54 238)(40 222 55 237)(41 221 56 236)(42 220 57 235)(43 219 58 234)(44 218 59 233)(45 217 60 232)(61 166 76 151)(62 165 77 180)(63 164 78 179)(64 163 79 178)(65 162 80 177)(66 161 81 176)(67 160 82 175)(68 159 83 174)(69 158 84 173)(70 157 85 172)(71 156 86 171)(72 155 87 170)(73 154 88 169)(74 153 89 168)(75 152 90 167)(91 131 106 146)(92 130 107 145)(93 129 108 144)(94 128 109 143)(95 127 110 142)(96 126 111 141)(97 125 112 140)(98 124 113 139)(99 123 114 138)(100 122 115 137)(101 121 116 136)(102 150 117 135)(103 149 118 134)(104 148 119 133)(105 147 120 132)
(1 91 46 61)(2 102 47 72)(3 113 48 83)(4 94 49 64)(5 105 50 75)(6 116 51 86)(7 97 52 67)(8 108 53 78)(9 119 54 89)(10 100 55 70)(11 111 56 81)(12 92 57 62)(13 103 58 73)(14 114 59 84)(15 95 60 65)(16 106 31 76)(17 117 32 87)(18 98 33 68)(19 109 34 79)(20 120 35 90)(21 101 36 71)(22 112 37 82)(23 93 38 63)(24 104 39 74)(25 115 40 85)(26 96 41 66)(27 107 42 77)(28 118 43 88)(29 99 44 69)(30 110 45 80)(121 211 156 201)(122 222 157 182)(123 233 158 193)(124 214 159 204)(125 225 160 185)(126 236 161 196)(127 217 162 207)(128 228 163 188)(129 239 164 199)(130 220 165 210)(131 231 166 191)(132 212 167 202)(133 223 168 183)(134 234 169 194)(135 215 170 205)(136 226 171 186)(137 237 172 197)(138 218 173 208)(139 229 174 189)(140 240 175 200)(141 221 176 181)(142 232 177 192)(143 213 178 203)(144 224 179 184)(145 235 180 195)(146 216 151 206)(147 227 152 187)(148 238 153 198)(149 219 154 209)(150 230 155 190)

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

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

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

Dic155C4 is a maximal subgroup of
Dic155Q8  Dic151Q8  Dic15⋊Q8  Dic3017C4  Dic5.1Dic6  Dic15.Q8  Dic15.2Q8  Dic3014C4  C605C4⋊C2  Dic3.Dic10  Dic3⋊C4⋊D5  D6⋊Dic5⋊C2  D30.D4  C60.67D4  (C2×C60).C22  (C4×Dic3)⋊D5  C60.46D4  C5⋊(C423S3)  D308Q8  Dic5.7Dic6  Dic3.3Dic10  C10.D4⋊S3  C60.6Q8  Dic15.4Q8  Dic158Q8  (C4×D15)⋊10C4  (C4×Dic5)⋊S3  C20.Dic6  D5×Dic3⋊C4  (D5×Dic3)⋊C4  D6.(C4×D5)  S3×C10.D4  D30⋊Q8  D303Q8  Dic15.D4  D104Dic6  D63Dic10  D64Dic10  D30.2Q8  D30.7D4  C4×C15⋊D4  Dic159D4  D3012D4  C4×C15⋊Q8  C20⋊Dic6  C23.13(S3×D5)  C23.14(S3×D5)  C23.48(S3×D5)  (C2×C30).D4  C30.(C2×D4)  C23.17(S3×D5)  Dic153D4  C1526(C4×D4)  (S3×C10).D4  C1528(C4×D4)  Dic154D4  Dic1518D4  (C2×C30)⋊Q8  (C2×C10)⋊8Dic6  Dic15.48D4
Dic155C4 is a maximal quotient of
C60.14Q8  C30.SD16  C30.20D8  C60.D4  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++++++-+++--++--
imageC1C2C2C2C4S3D4Q8D5D6D10Dic6C4×S3C3⋊D4Dic10C4×D5C5⋊D4S3×D5D30.C2C15⋊D4C15⋊Q8
kernelDic155C4C6×Dic5C10×Dic3C2×Dic15Dic15C2×Dic5C30C30C2×Dic3C2×C10C2×C6C10C10C10C6C6C6C22C2C2C2
# reps111141112122224442222

Matrix representation of Dic155C4 in GL4(𝔽61) generated by

60100
164400
00060
0011
,
01800
17000
00918
00952
,
50000
05000
00600
0011
G:=sub<GL(4,GF(61))| [60,16,0,0,1,44,0,0,0,0,0,1,0,0,60,1],[0,17,0,0,18,0,0,0,0,0,9,9,0,0,18,52],[50,0,0,0,0,50,0,0,0,0,60,1,0,0,0,1] >;

Dic155C4 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}\rtimes_5C_4
% in TeX

G:=Group("Dic15:5C4");
// GroupNames label

G:=SmallGroup(240,30);
// by ID

G=gap.SmallGroup(240,30);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,121,31,490,6917]);
// Polycyclic

G:=Group<a,b,c|a^30=c^4=1,b^2=a^15,b*a*b^-1=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^15*b>;
// generators/relations

Export

Subgroup lattice of Dic155C4 in TeX

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