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G = C30.Q8order 240 = 24·3·5

1st non-split extension by C30 of Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C30.1Q8, C30.16D4, C10.12D12, C6.1Dic10, C10.1Dic6, Dic51Dic3, C154(C4⋊C4), C52(C4⋊Dic3), C6.14(C4×D5), (C2×C6).9D10, (C2×C10).9D6, C2.1(C15⋊Q8), C30.32(C2×C4), (C3×Dic5)⋊2C4, C6.6(C5⋊D4), C2.5(D5×Dic3), C22.8(S3×D5), C2.3(C5⋊D12), C31(C10.D4), (C2×C30).6C22, (C6×Dic5).2C2, (C2×Dic3).1D5, (C2×Dic5).1S3, (C2×Dic15).5C2, (C10×Dic3).2C2, C10.12(C2×Dic3), SmallGroup(240,29)

Series: Derived Chief Lower central Upper central

C1C30 — C30.Q8
C1C5C15C30C2×C30C6×Dic5 — C30.Q8
C15C30 — C30.Q8
C1C22

Generators and relations for C30.Q8
 G = < a,b,c | a30=b4=1, c2=a15b2, bab-1=a11, cac-1=a19, cbc-1=a15b-1 >

5C4
5C4
6C4
30C4
3C2×C4
5C2×C4
15C2×C4
2Dic3
5C12
5C12
10Dic3
6Dic5
6C20
15C4⋊C4
5C2×C12
5C2×Dic3
3C2×Dic5
3C2×C20
2C5×Dic3
2Dic15
5C4⋊Dic3
3C10.D4

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

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

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

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

C30.Q8 is a maximal subgroup of
Dic55Dic6  Dic3⋊Dic10  Dic15⋊Q8  Dic3×Dic10  Dic156Q8  Dic5.1Dic6  Dic5.2Dic6  Dic15.2Q8  D6⋊C4.D5  C4⋊Dic5⋊S3  Dic3.2Dic10  D10⋊Dic6  D30.34D4  (D5×C12)⋊C4  (C4×Dic3)⋊D5  C60.45D4  (C4×Dic15)⋊C2  D6⋊Dic5.C2  Dic5⋊Dic6  Dic5.7Dic6  Dic3.3Dic10  C10.D4⋊S3  C60.6Q8  Dic15.4Q8  D309Q8  C12.Dic10  S3×C10.D4  (S3×Dic5)⋊C4  D30.23(C2×C4)  D30.Q8  Dic54D12  D61Dic10  D30⋊Q8  D5×C4⋊Dic3  D10.16D12  D10.17D12  D62Dic10  D302Q8  D101Dic6  D104Dic6  C4×C5⋊D12  D10⋊D12  D10⋊C4⋊S3  C4×C15⋊Q8  C60⋊Q8  (C6×Dic5)⋊7C4  C23.13(S3×D5)  C23.14(S3×D5)  D306D4  C10.(C2×D12)  (C2×C10).D12  Dic3×C5⋊D4  (S3×C10).D4  Dic1516D4  D30.16D4  (S3×C10)⋊D4  (C2×C10)⋊4D12  (C2×C30)⋊Q8  (C2×C10)⋊8Dic6  Dic15.48D4
C30.Q8 is a maximal quotient of
C60.13Q8  C60.7Q8  C60.8Q8  C60.105D4  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++++++-+-++-+-+-+-
imageC1C2C2C2C4S3D4Q8D5Dic3D6D10Dic6D12Dic10C4×D5C5⋊D4S3×D5D5×Dic3C5⋊D12C15⋊Q8
kernelC30.Q8C6×Dic5C10×Dic3C2×Dic15C3×Dic5C2×Dic5C30C30C2×Dic3Dic5C2×C10C2×C6C10C10C6C6C6C22C2C2C2
# reps111141112212224442222

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

06000
11800
0011
00600
,
11000
01100
001725
00844
,
33600
222800
003815
004623
G:=sub<GL(4,GF(61))| [0,1,0,0,60,18,0,0,0,0,1,60,0,0,1,0],[11,0,0,0,0,11,0,0,0,0,17,8,0,0,25,44],[33,22,0,0,6,28,0,0,0,0,38,46,0,0,15,23] >;

C30.Q8 in GAP, Magma, Sage, TeX

C_{30}.Q_8
% in TeX

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

G:=SmallGroup(240,29);
// by ID

G=gap.SmallGroup(240,29);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,121,31,490,6917]);
// Polycyclic

G:=Group<a,b,c|a^30=b^4=1,c^2=a^15*b^2,b*a*b^-1=a^11,c*a*c^-1=a^19,c*b*c^-1=a^15*b^-1>;
// generators/relations

Export

Subgroup lattice of C30.Q8 in TeX

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