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G = C605C4order 240 = 24·3·5

1st semidirect product of C60 and C4 acting via C4/C2=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C605C4, C4⋊Dic15, C2.1D60, C6.4D20, C30.5Q8, C203Dic3, C121Dic5, C30.22D4, C10.4D12, C2.2Dic30, C6.5Dic10, C10.5Dic6, C22.5D30, C158(C4⋊C4), (C2×C20).3S3, (C2×C60).5C2, C53(C4⋊Dic3), C32(C4⋊Dic5), (C2×C4).3D15, (C2×C12).3D5, C30.52(C2×C4), (C2×C6).23D10, (C2×C10).23D6, C6.9(C2×Dic5), C2.4(C2×Dic15), (C2×C30).24C22, (C2×Dic15).2C2, C10.16(C2×Dic3), SmallGroup(240,74)

Series: Derived Chief Lower central Upper central

C1C30 — C605C4
C1C5C15C30C2×C30C2×Dic15 — C605C4
C15C30 — C605C4
C1C22C2×C4

Generators and relations for C605C4
 G = < a,b | a60=b4=1, bab-1=a-1 >

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

Smallest permutation representation of C605C4
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)
(1 179 110 222)(2 178 111 221)(3 177 112 220)(4 176 113 219)(5 175 114 218)(6 174 115 217)(7 173 116 216)(8 172 117 215)(9 171 118 214)(10 170 119 213)(11 169 120 212)(12 168 61 211)(13 167 62 210)(14 166 63 209)(15 165 64 208)(16 164 65 207)(17 163 66 206)(18 162 67 205)(19 161 68 204)(20 160 69 203)(21 159 70 202)(22 158 71 201)(23 157 72 200)(24 156 73 199)(25 155 74 198)(26 154 75 197)(27 153 76 196)(28 152 77 195)(29 151 78 194)(30 150 79 193)(31 149 80 192)(32 148 81 191)(33 147 82 190)(34 146 83 189)(35 145 84 188)(36 144 85 187)(37 143 86 186)(38 142 87 185)(39 141 88 184)(40 140 89 183)(41 139 90 182)(42 138 91 181)(43 137 92 240)(44 136 93 239)(45 135 94 238)(46 134 95 237)(47 133 96 236)(48 132 97 235)(49 131 98 234)(50 130 99 233)(51 129 100 232)(52 128 101 231)(53 127 102 230)(54 126 103 229)(55 125 104 228)(56 124 105 227)(57 123 106 226)(58 122 107 225)(59 121 108 224)(60 180 109 223)

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

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

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

C605C4 is a maximal subgroup of
C6.D40  C10.D24  C6.Dic20  C10.Dic12  C60.7Q8  C60.Q8  C60.8Q8  C60.5Q8  C60.1Q8  C60.2Q8  Dic308C4  C12010C4  C1209C4  D608C4  D4⋊Dic15  Q82Dic15  Dic3×Dic10  Dic5×Dic6  Dic5.1Dic6  (S3×C20)⋊5C4  C605C4⋊C2  Dic3.Dic10  (C4×D5)⋊Dic3  C60.67D4  (C2×C60).C22  C60.46D4  C60.6Q8  C20.Dic6  D5×C4⋊Dic3  D10.17D12  Dic5×D12  D62Dic10  D102Dic6  Dic3×D20  S3×C4⋊Dic5  C60⋊D4  C604D4  D6.9D20  C60⋊Q8  C204Dic6  C4×Dic30  C608Q8  C60.24Q8  C4×D60  C222Dic30  C23.8D30  D30.28D4  C22.D60  C4⋊Dic30  Dic15.3Q8  C4.Dic30  C4⋊C4×D15  C4⋊C47D15  D306Q8  C4⋊C4⋊D15  C60.205D4  C23.26D30  C6029D4  D4×Dic15  C602D4  Q8×Dic15  D307Q8
C605C4 is a maximal quotient of
C605C8  C12010C4  C1209C4  C4.18D60  C30.29C42

66 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F5A5B6A6B6C10A···10F12A12B12C12D15A15B15C15D20A···20H30A···30L60A···60P
order122234444445566610···10121212121515151520···2030···3060···60
size111122230303030222222···2222222222···22···22···2

66 irreducible representations

dim111122222222222222222
type+++++-+-+-+-++-+-+-+
imageC1C2C2C4S3D4Q8D5Dic3D6Dic5D10Dic6D12D15Dic10D20Dic15D30Dic30D60
kernelC605C4C2×Dic15C2×C60C60C2×C20C30C30C2×C12C20C2×C10C12C2×C6C10C10C2×C4C6C6C4C22C2C2
# reps121411122142224448488

Matrix representation of C605C4 in GL3(𝔽61) generated by

100
05528
03339
,
5000
0458
05216
G:=sub<GL(3,GF(61))| [1,0,0,0,55,33,0,28,39],[50,0,0,0,45,52,0,8,16] >;

C605C4 in GAP, Magma, Sage, TeX

C_{60}\rtimes_5C_4
% in TeX

G:=Group("C60:5C4");
// GroupNames label

G:=SmallGroup(240,74);
// by ID

G=gap.SmallGroup(240,74);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,121,55,964,6917]);
// Polycyclic

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

Export

Subgroup lattice of C605C4 in TeX

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