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G = C30.22M4(2)  order 480 = 25·3·5

3rd non-split extension by C30 of M4(2) acting via M4(2)/C22=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C30.22M4(2)
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — C6×Dic5 — C2×C15⋊C8 — C30.22M4(2)
 Lower central C15 — C30 — C30.22M4(2)
 Upper central C1 — C22 — C23

Generators and relations for C30.22M4(2)
G = < a,b,c | a30=b8=c2=1, bab-1=a17, ac=ca, cbc=a15b5 >

Subgroups: 364 in 100 conjugacy classes, 45 normal (31 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C8, C2×C4, C23, C10, C10, C12, C2×C6, C2×C6, C2×C6, C15, C2×C8, C22×C4, Dic5, Dic5, C2×C10, C2×C10, C2×C10, C3⋊C8, C2×C12, C22×C6, C30, C30, C22⋊C8, C5⋊C8, C2×Dic5, C2×Dic5, C22×C10, C2×C3⋊C8, C22×C12, C3×Dic5, C3×Dic5, C2×C30, C2×C30, C2×C30, C2×C5⋊C8, C22×Dic5, C12.55D4, C15⋊C8, C6×Dic5, C6×Dic5, C22×C30, C23.2F5, C2×C15⋊C8, C2×C6×Dic5, C30.22M4(2)
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, Dic3, D6, C22⋊C4, C2×C8, M4(2), F5, C3⋊C8, C2×Dic3, C3⋊D4, C22⋊C8, C5⋊C8, C2×F5, C2×C3⋊C8, C4.Dic3, C6.D4, C3⋊F5, C2×C5⋊C8, C22.F5, C22⋊F5, C12.55D4, C15⋊C8, C2×C3⋊F5, C23.2F5, C2×C15⋊C8, C158M4(2), D10.D6, C30.22M4(2)

Smallest permutation representation of C30.22M4(2)
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 38 133 86 183 103 222 164)(2 31 122 73 184 96 211 151)(3 54 141 90 185 119 230 168)(4 47 130 77 186 112 219 155)(5 40 149 64 187 105 238 172)(6 33 138 81 188 98 227 159)(7 56 127 68 189 91 216 176)(8 49 146 85 190 114 235 163)(9 42 135 72 191 107 224 180)(10 35 124 89 192 100 213 167)(11 58 143 76 193 93 232 154)(12 51 132 63 194 116 221 171)(13 44 121 80 195 109 240 158)(14 37 140 67 196 102 229 175)(15 60 129 84 197 95 218 162)(16 53 148 71 198 118 237 179)(17 46 137 88 199 111 226 166)(18 39 126 75 200 104 215 153)(19 32 145 62 201 97 234 170)(20 55 134 79 202 120 223 157)(21 48 123 66 203 113 212 174)(22 41 142 83 204 106 231 161)(23 34 131 70 205 99 220 178)(24 57 150 87 206 92 239 165)(25 50 139 74 207 115 228 152)(26 43 128 61 208 108 217 169)(27 36 147 78 209 101 236 156)(28 59 136 65 210 94 225 173)(29 52 125 82 181 117 214 160)(30 45 144 69 182 110 233 177)
(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 91)(42 92)(43 93)(44 94)(45 95)(46 96)(47 97)(48 98)(49 99)(50 100)(51 101)(52 102)(53 103)(54 104)(55 105)(56 106)(57 107)(58 108)(59 109)(60 110)(61 154)(62 155)(63 156)(64 157)(65 158)(66 159)(67 160)(68 161)(69 162)(70 163)(71 164)(72 165)(73 166)(74 167)(75 168)(76 169)(77 170)(78 171)(79 172)(80 173)(81 174)(82 175)(83 176)(84 177)(85 178)(86 179)(87 180)(88 151)(89 152)(90 153)```

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 5 6A ··· 6G 8A ··· 8H 10A ··· 10G 12A ··· 12H 15A 15B 30A ··· 30N order 1 2 2 2 2 2 3 4 4 4 4 4 4 5 6 ··· 6 8 ··· 8 10 ··· 10 12 ··· 12 15 15 30 ··· 30 size 1 1 1 1 2 2 2 5 5 5 5 10 10 4 2 ··· 2 30 ··· 30 4 ··· 4 10 ··· 10 4 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 type + + + + + - + - + - + - + image C1 C2 C2 C4 C4 C8 S3 D4 Dic3 D6 Dic3 M4(2) C3⋊D4 C3⋊C8 C4.Dic3 F5 C5⋊C8 C2×F5 C3⋊F5 C22.F5 C22⋊F5 C15⋊C8 C2×C3⋊F5 C15⋊8M4(2) D10.D6 kernel C30.22M4(2) C2×C15⋊C8 C2×C6×Dic5 C6×Dic5 C22×C30 C2×C30 C22×Dic5 C3×Dic5 C2×Dic5 C2×Dic5 C22×C10 C30 Dic5 C2×C10 C10 C22×C6 C2×C6 C2×C6 C23 C6 C6 C22 C22 C2 C2 # reps 1 2 1 2 2 8 1 2 1 1 1 2 4 4 4 1 2 1 2 2 2 4 2 4 4

Matrix representation of C30.22M4(2) in GL6(𝔽241)

 225 0 0 0 0 0 62 15 0 0 0 0 0 0 225 16 0 0 0 0 116 109 0 0 0 0 26 57 0 42 0 0 72 26 184 57
,
 103 172 0 0 0 0 69 138 0 0 0 0 0 0 140 150 69 0 0 0 2 60 0 69 0 0 154 171 101 91 0 0 209 18 239 181
,
 1 0 0 0 0 0 237 240 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 209 69 240 0 0 0 213 124 0 240

`G:=sub<GL(6,GF(241))| [225,62,0,0,0,0,0,15,0,0,0,0,0,0,225,116,26,72,0,0,16,109,57,26,0,0,0,0,0,184,0,0,0,0,42,57],[103,69,0,0,0,0,172,138,0,0,0,0,0,0,140,2,154,209,0,0,150,60,171,18,0,0,69,0,101,239,0,0,0,69,91,181],[1,237,0,0,0,0,0,240,0,0,0,0,0,0,1,0,209,213,0,0,0,1,69,124,0,0,0,0,240,0,0,0,0,0,0,240] >;`

C30.22M4(2) in GAP, Magma, Sage, TeX

`C_{30}._{22}M_4(2)`
`% in TeX`

`G:=Group("C30.22M4(2)");`
`// GroupNames label`

`G:=SmallGroup(480,317);`
`// by ID`

`G=gap.SmallGroup(480,317);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,100,2693,14118,4724]);`
`// Polycyclic`

`G:=Group<a,b,c|a^30=b^8=c^2=1,b*a*b^-1=a^17,a*c=c*a,c*b*c=a^15*b^5>;`
`// generators/relations`

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