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## G = C30.7M4(2)  order 480 = 25·3·5

### 2nd non-split extension by C30 of M4(2) acting via M4(2)/C4=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C30.7M4(2)
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — C6×Dic5 — C2×C15⋊C8 — C30.7M4(2)
 Lower central C15 — C30 — C30.7M4(2)
 Upper central C1 — C22 — C2×C4

Generators and relations for C30.7M4(2)
G = < a,b,c | a30=b8=c2=1, bab-1=a23, cac=a19, cbc=a15b5 >

Subgroups: 428 in 100 conjugacy classes, 41 normal (31 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, C23, D5, C10, C12, C2×C6, C2×C6, C15, C2×C8, C22×C4, Dic5, C20, D10, D10, C2×C10, C3⋊C8, C2×C12, C2×C12, C22×C6, C3×D5, C30, C22⋊C8, C5⋊C8, C4×D5, C2×Dic5, C2×C20, C22×D5, C2×C3⋊C8, C22×C12, C3×Dic5, C60, C6×D5, C6×D5, C2×C30, C2×C5⋊C8, C2×C4×D5, C12.55D4, C15⋊C8, D5×C12, C6×Dic5, C2×C60, D5×C2×C6, D10⋊C8, C2×C15⋊C8, D5×C2×C12, C30.7M4(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, C2×F5, C2×C3⋊C8, C4.Dic3, C6.D4, C3⋊F5, D5⋊C8, C4.F5, C22⋊F5, C12.55D4, C2×C3⋊F5, D10⋊C8, C60.C4, C12.F5, D10.D6, C30.7M4(2)

Smallest permutation representation of C30.7M4(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 64 219 52 187 175 113 138)(2 81 238 45 188 162 102 131)(3 68 227 38 189 179 91 124)(4 85 216 31 190 166 110 147)(5 72 235 54 191 153 99 140)(6 89 224 47 192 170 118 133)(7 76 213 40 193 157 107 126)(8 63 232 33 194 174 96 149)(9 80 221 56 195 161 115 142)(10 67 240 49 196 178 104 135)(11 84 229 42 197 165 93 128)(12 71 218 35 198 152 112 121)(13 88 237 58 199 169 101 144)(14 75 226 51 200 156 120 137)(15 62 215 44 201 173 109 130)(16 79 234 37 202 160 98 123)(17 66 223 60 203 177 117 146)(18 83 212 53 204 164 106 139)(19 70 231 46 205 151 95 132)(20 87 220 39 206 168 114 125)(21 74 239 32 207 155 103 148)(22 61 228 55 208 172 92 141)(23 78 217 48 209 159 111 134)(24 65 236 41 210 176 100 127)(25 82 225 34 181 163 119 150)(26 69 214 57 182 180 108 143)(27 86 233 50 183 167 97 136)(28 73 222 43 184 154 116 129)(29 90 211 36 185 171 105 122)(30 77 230 59 186 158 94 145)
(1 187)(2 206)(3 195)(4 184)(5 203)(6 192)(7 181)(8 200)(9 189)(10 208)(11 197)(12 186)(13 205)(14 194)(15 183)(16 202)(17 191)(18 210)(19 199)(20 188)(21 207)(22 196)(23 185)(24 204)(25 193)(26 182)(27 201)(28 190)(29 209)(30 198)(31 58)(32 47)(33 36)(34 55)(35 44)(37 52)(38 41)(39 60)(40 49)(42 57)(43 46)(45 54)(48 51)(50 59)(53 56)(61 82)(62 71)(63 90)(64 79)(65 68)(66 87)(67 76)(69 84)(70 73)(72 81)(74 89)(75 78)(77 86)(80 83)(85 88)(91 221)(92 240)(93 229)(94 218)(95 237)(96 226)(97 215)(98 234)(99 223)(100 212)(101 231)(102 220)(103 239)(104 228)(105 217)(106 236)(107 225)(108 214)(109 233)(110 222)(111 211)(112 230)(113 219)(114 238)(115 227)(116 216)(117 235)(118 224)(119 213)(120 232)(121 130)(122 149)(123 138)(124 127)(125 146)(126 135)(128 143)(129 132)(131 140)(133 148)(134 137)(136 145)(139 142)(141 150)(144 147)(151 154)(152 173)(153 162)(155 170)(156 159)(157 178)(158 167)(160 175)(161 164)(163 172)(165 180)(166 169)(168 177)(171 174)(176 179)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 5 6A 6B 6C 6D 6E 6F 6G 8A ··· 8H 10A 10B 10C 12A 12B 12C 12D 12E 12F 12G 12H 15A 15B 20A 20B 20C 20D 30A ··· 30F 60A ··· 60H order 1 2 2 2 2 2 3 4 4 4 4 4 4 5 6 6 6 6 6 6 6 8 ··· 8 10 10 10 12 12 12 12 12 12 12 12 15 15 20 20 20 20 30 ··· 30 60 ··· 60 size 1 1 1 1 10 10 2 2 2 5 5 5 5 4 2 2 2 10 10 10 10 30 ··· 30 4 4 4 2 2 2 2 10 10 10 10 4 4 4 4 4 4 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 D6 Dic3 Dic3 M4(2) C3⋊D4 C3⋊C8 C4.Dic3 F5 C2×F5 C3⋊F5 D5⋊C8 C4.F5 C22⋊F5 C2×C3⋊F5 C60.C4 C12.F5 D10.D6 kernel C30.7M4(2) C2×C15⋊C8 D5×C2×C12 C2×C60 D5×C2×C6 C6×D5 C2×C4×D5 C3×Dic5 C2×Dic5 C2×C20 C22×D5 C30 Dic5 D10 C10 C2×C12 C2×C6 C2×C4 C6 C6 C6 C22 C2 C2 C2 # reps 1 2 1 2 2 8 1 2 1 1 1 2 4 4 4 1 1 2 2 2 2 2 4 4 4

Matrix representation of C30.7M4(2) in GL6(𝔽241)

 240 0 0 0 0 0 0 240 0 0 0 0 0 0 229 115 229 0 0 0 0 229 115 229 0 0 12 12 0 127 0 0 114 126 126 114
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 151 0 181 181 0 0 181 181 0 151 0 0 60 211 60 0 0 0 90 30 30 90
,
 1 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 1 1 1 1 0 0 0 0 0 240 0 0 0 0 240 0

G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,229,0,12,114,0,0,115,229,12,126,0,0,229,115,0,126,0,0,0,229,127,114],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,151,181,60,90,0,0,0,181,211,30,0,0,181,0,60,30,0,0,181,151,0,90],[1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,1,0,0,0,0,0,1,0,0,0,0,0,1,0,240,0,0,0,1,240,0] >;

C30.7M4(2) in GAP, Magma, Sage, TeX

C_{30}._7M_4(2)
% in TeX

G:=Group("C30.7M4(2)");
// GroupNames label

G:=SmallGroup(480,308);
// by ID

G=gap.SmallGroup(480,308);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,120,100,2693,14118,4724]);
// Polycyclic

G:=Group<a,b,c|a^30=b^8=c^2=1,b*a*b^-1=a^23,c*a*c=a^19,c*b*c=a^15*b^5>;
// generators/relations

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