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G = C30.4Q8order 240 = 24·3·5

1st non-split extension by C30 of Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C30.4Q8, C30.32D4, Dic153C4, C10.4Dic6, C6.4Dic10, C2.1Dic30, C22.4D30, C157(C4⋊C4), C6.8(C4×D5), (C2×C20).2S3, (C2×C60).3C2, C2.4(C4×D15), (C2×C4).1D15, (C2×C12).2D5, C10.15(C4×S3), C30.38(C2×C4), C54(Dic3⋊C4), (C2×C10).22D6, (C2×C6).22D10, C6.14(C5⋊D4), C33(C10.D4), C2.1(C157D4), C10.14(C3⋊D4), (C2×C30).23C22, (C2×Dic15).1C2, SmallGroup(240,73)

Series: Derived Chief Lower central Upper central

C1C30 — C30.4Q8
C1C5C15C30C2×C30C2×Dic15 — C30.4Q8
C15C30 — C30.4Q8
C1C22C2×C4

Generators and relations for C30.4Q8
 G = < a,b,c | a30=b4=1, c2=a15b2, ab=ba, cac-1=a-1, cbc-1=a15b-1 >

2C4
15C4
15C4
30C4
15C2×C4
15C2×C4
2C12
5Dic3
5Dic3
10Dic3
2C20
3Dic5
3Dic5
6Dic5
15C4⋊C4
5C2×Dic3
5C2×Dic3
3C2×Dic5
3C2×Dic5
2C60
2Dic15
5Dic3⋊C4
3C10.D4

Smallest permutation representation of C30.4Q8
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)
(1 116 173 192)(2 117 174 193)(3 118 175 194)(4 119 176 195)(5 120 177 196)(6 91 178 197)(7 92 179 198)(8 93 180 199)(9 94 151 200)(10 95 152 201)(11 96 153 202)(12 97 154 203)(13 98 155 204)(14 99 156 205)(15 100 157 206)(16 101 158 207)(17 102 159 208)(18 103 160 209)(19 104 161 210)(20 105 162 181)(21 106 163 182)(22 107 164 183)(23 108 165 184)(24 109 166 185)(25 110 167 186)(26 111 168 187)(27 112 169 188)(28 113 170 189)(29 114 171 190)(30 115 172 191)(31 229 83 142)(32 230 84 143)(33 231 85 144)(34 232 86 145)(35 233 87 146)(36 234 88 147)(37 235 89 148)(38 236 90 149)(39 237 61 150)(40 238 62 121)(41 239 63 122)(42 240 64 123)(43 211 65 124)(44 212 66 125)(45 213 67 126)(46 214 68 127)(47 215 69 128)(48 216 70 129)(49 217 71 130)(50 218 72 131)(51 219 73 132)(52 220 74 133)(53 221 75 134)(54 222 76 135)(55 223 77 136)(56 224 78 137)(57 225 79 138)(58 226 80 139)(59 227 81 140)(60 228 82 141)
(1 51 158 88)(2 50 159 87)(3 49 160 86)(4 48 161 85)(5 47 162 84)(6 46 163 83)(7 45 164 82)(8 44 165 81)(9 43 166 80)(10 42 167 79)(11 41 168 78)(12 40 169 77)(13 39 170 76)(14 38 171 75)(15 37 172 74)(16 36 173 73)(17 35 174 72)(18 34 175 71)(19 33 176 70)(20 32 177 69)(21 31 178 68)(22 60 179 67)(23 59 180 66)(24 58 151 65)(25 57 152 64)(26 56 153 63)(27 55 154 62)(28 54 155 61)(29 53 156 90)(30 52 157 89)(91 142 182 214)(92 141 183 213)(93 140 184 212)(94 139 185 211)(95 138 186 240)(96 137 187 239)(97 136 188 238)(98 135 189 237)(99 134 190 236)(100 133 191 235)(101 132 192 234)(102 131 193 233)(103 130 194 232)(104 129 195 231)(105 128 196 230)(106 127 197 229)(107 126 198 228)(108 125 199 227)(109 124 200 226)(110 123 201 225)(111 122 202 224)(112 121 203 223)(113 150 204 222)(114 149 205 221)(115 148 206 220)(116 147 207 219)(117 146 208 218)(118 145 209 217)(119 144 210 216)(120 143 181 215)

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

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

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

C30.4Q8 is a maximal subgroup of
Dic55Dic6  Dic35Dic10  Dic151Q8  Dic15⋊Q8  Dic5.2Dic6  Dic15.Q8  Dic15.2Q8  D6⋊C4.D5  Dic3.2Dic10  D10⋊Dic6  D6⋊Dic10  (C2×C12).D10  (C4×Dic3)⋊D5  C5⋊(C423S3)  Dic5.7Dic6  Dic3.3Dic10  D5×Dic3⋊C4  (D5×Dic3)⋊C4  D6.(C4×D5)  S3×C10.D4  (C6×D5).D4  Dic15⋊D4  D61Dic10  D101Dic6  (C2×D12).D5  Dic15.D4  D64Dic10  D6⋊(C4×D5)  C1517(C4×D4)  D6⋊C4⋊D5  D10⋊C4⋊S3  Dic152D4  C4×Dic30  C60.24Q8  C422D15  C423D15  C23.15D30  C222Dic30  C23.8D30  Dic1519D4  D30.28D4  D309D4  Dic1510Q8  C4⋊Dic30  Dic15.3Q8  C4.Dic30  C4⋊C4×D15  D30.29D4  D305Q8  C4⋊C4⋊D15  C60.205D4  C4×C157D4  C23.28D30  C23.22D30  Dic1512D4  Dic154Q8  D307Q8
C30.4Q8 is a maximal quotient of
C60.1Q8  C60.2Q8  C60.26Q8  C60.210D4  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+++++-+++-+-+-
imageC1C2C2C4S3D4Q8D5D6D10Dic6C4×S3C3⋊D4D15Dic10C4×D5C5⋊D4D30Dic30C4×D15C157D4
kernelC30.4Q8C2×Dic15C2×C60Dic15C2×C20C30C30C2×C12C2×C10C2×C6C10C10C10C2×C4C6C6C6C22C2C2C2
# reps121411121222244444888

Matrix representation of C30.4Q8 in GL4(𝔽61) generated by

16000
194300
0001
00601
,
50000
05000
00943
001852
,
04400
43000
005252
00439
G:=sub<GL(4,GF(61))| [1,19,0,0,60,43,0,0,0,0,0,60,0,0,1,1],[50,0,0,0,0,50,0,0,0,0,9,18,0,0,43,52],[0,43,0,0,44,0,0,0,0,0,52,43,0,0,52,9] >;

C30.4Q8 in GAP, Magma, Sage, TeX

C_{30}._4Q_8
% in TeX

G:=Group("C30.4Q8");
// GroupNames label

G:=SmallGroup(240,73);
// by ID

G=gap.SmallGroup(240,73);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,121,31,964,6917]);
// Polycyclic

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

Export

Subgroup lattice of C30.4Q8 in TeX

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