G = C10.452- 1+4 order 320 = 26·5
45th non-split extension by C10 of 2- 1+4 acting via 2- 1+4/C2×Q8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.452- 1+4, (C5×Q8)⋊17D4, Q8⋊8(C5⋊D4), C5⋊6(Q8⋊5D4), (C22×Q8)⋊9D5, C20⋊7D4⋊38C2, (Q8×Dic5)⋊27C2, C20.262(C2×D4), D10⋊3Q8⋊42C2, (C2×Q8).189D10, C20.23D4⋊30C2, (C2×C20).649C23, (C2×C10).309C24, C22⋊3(Q8⋊2D5), (C22×C4).279D10, C10.157(C22×D4), (C2×D20).189C22, C4⋊Dic5.258C22, (Q8×C10).236C22, C22.320(C23×D5), C23.240(C22×D5), D10⋊C4.78C22, (C22×C20).287C22, (C22×C10).427C23, (C2×Dic5).300C23, (C4×Dic5).180C22, (C22×D5).135C23, C23.D5.146C22, C2.45(Q8.10D10), C10.D4.171C22, (Q8×C2×C10)⋊8C2, (C4×C5⋊D4)⋊27C2, C4.70(C2×C5⋊D4), (C2×C10)⋊18(C4○D4), (C2×Q8⋊2D5)⋊18C2, C10.129(C2×C4○D4), C2.36(C2×Q8⋊2D5), (C2×C4×D5).175C22, C2.30(C22×C5⋊D4), (C2×C4).244(C22×D5), (C2×C5⋊D4).147C22, SmallGroup(320,1489)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.452- 1+4
G = < a,b,c,d,e | a10=b4=c2=1, d2=e2=a5b2, bab-1=a-1, ac=ca, ad=da, ae=ea, cbc=b-1, dbd-1=a5b, be=eb, cd=dc, ce=ec, ede-1=a5b2d >
Subgroups: 966 in 290 conjugacy classes, 115 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C4.4D4, C22×Q8, C2×C4○D4, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×Q8, C5×Q8, C22×D5, C22×C10, Q8⋊5D4, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C2×C4×D5, C2×D20, Q8⋊2D5, C2×C5⋊D4, C22×C20, Q8×C10, Q8×C10, Q8×C10, C4×C5⋊D4, C20⋊7D4, Q8×Dic5, D10⋊3Q8, C20.23D4, C2×Q8⋊2D5, Q8×C2×C10, C10.452- 1+4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C22×D4, C2×C4○D4, 2- 1+4, C5⋊D4, C22×D5, Q8⋊5D4, Q8⋊2D5, C2×C5⋊D4, C23×D5, C2×Q8⋊2D5, Q8.10D10, C22×C5⋊D4, C10.452- 1+4
►On 160 points
Generators in S160
(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)
(1 11 28 153)(2 20 29 152)(3 19 30 151)(4 18 21 160)(5 17 22 159)(6 16 23 158)(7 15 24 157)(8 14 25 156)(9 13 26 155)(10 12 27 154)(31 147 43 135)(32 146 44 134)(33 145 45 133)(34 144 46 132)(35 143 47 131)(36 142 48 140)(37 141 49 139)(38 150 50 138)(39 149 41 137)(40 148 42 136)(51 122 63 120)(52 121 64 119)(53 130 65 118)(54 129 66 117)(55 128 67 116)(56 127 68 115)(57 126 69 114)(58 125 70 113)(59 124 61 112)(60 123 62 111)(71 100 83 102)(72 99 84 101)(73 98 85 110)(74 97 86 109)(75 96 87 108)(76 95 88 107)(77 94 89 106)(78 93 90 105)(79 92 81 104)(80 91 82 103)
(1 38)(2 39)(3 40)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 138)(12 139)(13 140)(14 131)(15 132)(16 133)(17 134)(18 135)(19 136)(20 137)(21 43)(22 44)(23 45)(24 46)(25 47)(26 48)(27 49)(28 50)(29 41)(30 42)(51 88)(52 89)(53 90)(54 81)(55 82)(56 83)(57 84)(58 85)(59 86)(60 87)(61 74)(62 75)(63 76)(64 77)(65 78)(66 79)(67 80)(68 71)(69 72)(70 73)(91 128)(92 129)(93 130)(94 121)(95 122)(96 123)(97 124)(98 125)(99 126)(100 127)(101 114)(102 115)(103 116)(104 117)(105 118)(106 119)(107 120)(108 111)(109 112)(110 113)(141 154)(142 155)(143 156)(144 157)(145 158)(146 159)(147 160)(148 151)(149 152)(150 153)
(1 85 23 78)(2 86 24 79)(3 87 25 80)(4 88 26 71)(5 89 27 72)(6 90 28 73)(7 81 29 74)(8 82 30 75)(9 83 21 76)(10 84 22 77)(11 105 158 98)(12 106 159 99)(13 107 160 100)(14 108 151 91)(15 109 152 92)(16 110 153 93)(17 101 154 94)(18 102 155 95)(19 103 156 96)(20 104 157 97)(31 51 48 68)(32 52 49 69)(33 53 50 70)(34 54 41 61)(35 55 42 62)(36 56 43 63)(37 57 44 64)(38 58 45 65)(39 59 46 66)(40 60 47 67)(111 148 128 131)(112 149 129 132)(113 150 130 133)(114 141 121 134)(115 142 122 135)(116 143 123 136)(117 144 124 137)(118 145 125 138)(119 146 126 139)(120 147 127 140)
(1 98 23 105)(2 99 24 106)(3 100 25 107)(4 91 26 108)(5 92 27 109)(6 93 28 110)(7 94 29 101)(8 95 30 102)(9 96 21 103)(10 97 22 104)(11 85 158 78)(12 86 159 79)(13 87 160 80)(14 88 151 71)(15 89 152 72)(16 90 153 73)(17 81 154 74)(18 82 155 75)(19 83 156 76)(20 84 157 77)(31 128 48 111)(32 129 49 112)(33 130 50 113)(34 121 41 114)(35 122 42 115)(36 123 43 116)(37 124 44 117)(38 125 45 118)(39 126 46 119)(40 127 47 120)(51 148 68 131)(52 149 69 132)(53 150 70 133)(54 141 61 134)(55 142 62 135)(56 143 63 136)(57 144 64 137)(58 145 65 138)(59 146 66 139)(60 147 67 140)
G:=sub<Sym(160)| (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), (1,11,28,153)(2,20,29,152)(3,19,30,151)(4,18,21,160)(5,17,22,159)(6,16,23,158)(7,15,24,157)(8,14,25,156)(9,13,26,155)(10,12,27,154)(31,147,43,135)(32,146,44,134)(33,145,45,133)(34,144,46,132)(35,143,47,131)(36,142,48,140)(37,141,49,139)(38,150,50,138)(39,149,41,137)(40,148,42,136)(51,122,63,120)(52,121,64,119)(53,130,65,118)(54,129,66,117)(55,128,67,116)(56,127,68,115)(57,126,69,114)(58,125,70,113)(59,124,61,112)(60,123,62,111)(71,100,83,102)(72,99,84,101)(73,98,85,110)(74,97,86,109)(75,96,87,108)(76,95,88,107)(77,94,89,106)(78,93,90,105)(79,92,81,104)(80,91,82,103), (1,38)(2,39)(3,40)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,138)(12,139)(13,140)(14,131)(15,132)(16,133)(17,134)(18,135)(19,136)(20,137)(21,43)(22,44)(23,45)(24,46)(25,47)(26,48)(27,49)(28,50)(29,41)(30,42)(51,88)(52,89)(53,90)(54,81)(55,82)(56,83)(57,84)(58,85)(59,86)(60,87)(61,74)(62,75)(63,76)(64,77)(65,78)(66,79)(67,80)(68,71)(69,72)(70,73)(91,128)(92,129)(93,130)(94,121)(95,122)(96,123)(97,124)(98,125)(99,126)(100,127)(101,114)(102,115)(103,116)(104,117)(105,118)(106,119)(107,120)(108,111)(109,112)(110,113)(141,154)(142,155)(143,156)(144,157)(145,158)(146,159)(147,160)(148,151)(149,152)(150,153), (1,85,23,78)(2,86,24,79)(3,87,25,80)(4,88,26,71)(5,89,27,72)(6,90,28,73)(7,81,29,74)(8,82,30,75)(9,83,21,76)(10,84,22,77)(11,105,158,98)(12,106,159,99)(13,107,160,100)(14,108,151,91)(15,109,152,92)(16,110,153,93)(17,101,154,94)(18,102,155,95)(19,103,156,96)(20,104,157,97)(31,51,48,68)(32,52,49,69)(33,53,50,70)(34,54,41,61)(35,55,42,62)(36,56,43,63)(37,57,44,64)(38,58,45,65)(39,59,46,66)(40,60,47,67)(111,148,128,131)(112,149,129,132)(113,150,130,133)(114,141,121,134)(115,142,122,135)(116,143,123,136)(117,144,124,137)(118,145,125,138)(119,146,126,139)(120,147,127,140), (1,98,23,105)(2,99,24,106)(3,100,25,107)(4,91,26,108)(5,92,27,109)(6,93,28,110)(7,94,29,101)(8,95,30,102)(9,96,21,103)(10,97,22,104)(11,85,158,78)(12,86,159,79)(13,87,160,80)(14,88,151,71)(15,89,152,72)(16,90,153,73)(17,81,154,74)(18,82,155,75)(19,83,156,76)(20,84,157,77)(31,128,48,111)(32,129,49,112)(33,130,50,113)(34,121,41,114)(35,122,42,115)(36,123,43,116)(37,124,44,117)(38,125,45,118)(39,126,46,119)(40,127,47,120)(51,148,68,131)(52,149,69,132)(53,150,70,133)(54,141,61,134)(55,142,62,135)(56,143,63,136)(57,144,64,137)(58,145,65,138)(59,146,66,139)(60,147,67,140)>;
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), (1,11,28,153)(2,20,29,152)(3,19,30,151)(4,18,21,160)(5,17,22,159)(6,16,23,158)(7,15,24,157)(8,14,25,156)(9,13,26,155)(10,12,27,154)(31,147,43,135)(32,146,44,134)(33,145,45,133)(34,144,46,132)(35,143,47,131)(36,142,48,140)(37,141,49,139)(38,150,50,138)(39,149,41,137)(40,148,42,136)(51,122,63,120)(52,121,64,119)(53,130,65,118)(54,129,66,117)(55,128,67,116)(56,127,68,115)(57,126,69,114)(58,125,70,113)(59,124,61,112)(60,123,62,111)(71,100,83,102)(72,99,84,101)(73,98,85,110)(74,97,86,109)(75,96,87,108)(76,95,88,107)(77,94,89,106)(78,93,90,105)(79,92,81,104)(80,91,82,103), (1,38)(2,39)(3,40)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,138)(12,139)(13,140)(14,131)(15,132)(16,133)(17,134)(18,135)(19,136)(20,137)(21,43)(22,44)(23,45)(24,46)(25,47)(26,48)(27,49)(28,50)(29,41)(30,42)(51,88)(52,89)(53,90)(54,81)(55,82)(56,83)(57,84)(58,85)(59,86)(60,87)(61,74)(62,75)(63,76)(64,77)(65,78)(66,79)(67,80)(68,71)(69,72)(70,73)(91,128)(92,129)(93,130)(94,121)(95,122)(96,123)(97,124)(98,125)(99,126)(100,127)(101,114)(102,115)(103,116)(104,117)(105,118)(106,119)(107,120)(108,111)(109,112)(110,113)(141,154)(142,155)(143,156)(144,157)(145,158)(146,159)(147,160)(148,151)(149,152)(150,153), (1,85,23,78)(2,86,24,79)(3,87,25,80)(4,88,26,71)(5,89,27,72)(6,90,28,73)(7,81,29,74)(8,82,30,75)(9,83,21,76)(10,84,22,77)(11,105,158,98)(12,106,159,99)(13,107,160,100)(14,108,151,91)(15,109,152,92)(16,110,153,93)(17,101,154,94)(18,102,155,95)(19,103,156,96)(20,104,157,97)(31,51,48,68)(32,52,49,69)(33,53,50,70)(34,54,41,61)(35,55,42,62)(36,56,43,63)(37,57,44,64)(38,58,45,65)(39,59,46,66)(40,60,47,67)(111,148,128,131)(112,149,129,132)(113,150,130,133)(114,141,121,134)(115,142,122,135)(116,143,123,136)(117,144,124,137)(118,145,125,138)(119,146,126,139)(120,147,127,140), (1,98,23,105)(2,99,24,106)(3,100,25,107)(4,91,26,108)(5,92,27,109)(6,93,28,110)(7,94,29,101)(8,95,30,102)(9,96,21,103)(10,97,22,104)(11,85,158,78)(12,86,159,79)(13,87,160,80)(14,88,151,71)(15,89,152,72)(16,90,153,73)(17,81,154,74)(18,82,155,75)(19,83,156,76)(20,84,157,77)(31,128,48,111)(32,129,49,112)(33,130,50,113)(34,121,41,114)(35,122,42,115)(36,123,43,116)(37,124,44,117)(38,125,45,118)(39,126,46,119)(40,127,47,120)(51,148,68,131)(52,149,69,132)(53,150,70,133)(54,141,61,134)(55,142,62,135)(56,143,63,136)(57,144,64,137)(58,145,65,138)(59,146,66,139)(60,147,67,140) );
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)], [(1,11,28,153),(2,20,29,152),(3,19,30,151),(4,18,21,160),(5,17,22,159),(6,16,23,158),(7,15,24,157),(8,14,25,156),(9,13,26,155),(10,12,27,154),(31,147,43,135),(32,146,44,134),(33,145,45,133),(34,144,46,132),(35,143,47,131),(36,142,48,140),(37,141,49,139),(38,150,50,138),(39,149,41,137),(40,148,42,136),(51,122,63,120),(52,121,64,119),(53,130,65,118),(54,129,66,117),(55,128,67,116),(56,127,68,115),(57,126,69,114),(58,125,70,113),(59,124,61,112),(60,123,62,111),(71,100,83,102),(72,99,84,101),(73,98,85,110),(74,97,86,109),(75,96,87,108),(76,95,88,107),(77,94,89,106),(78,93,90,105),(79,92,81,104),(80,91,82,103)], [(1,38),(2,39),(3,40),(4,31),(5,32),(6,33),(7,34),(8,35),(9,36),(10,37),(11,138),(12,139),(13,140),(14,131),(15,132),(16,133),(17,134),(18,135),(19,136),(20,137),(21,43),(22,44),(23,45),(24,46),(25,47),(26,48),(27,49),(28,50),(29,41),(30,42),(51,88),(52,89),(53,90),(54,81),(55,82),(56,83),(57,84),(58,85),(59,86),(60,87),(61,74),(62,75),(63,76),(64,77),(65,78),(66,79),(67,80),(68,71),(69,72),(70,73),(91,128),(92,129),(93,130),(94,121),(95,122),(96,123),(97,124),(98,125),(99,126),(100,127),(101,114),(102,115),(103,116),(104,117),(105,118),(106,119),(107,120),(108,111),(109,112),(110,113),(141,154),(142,155),(143,156),(144,157),(145,158),(146,159),(147,160),(148,151),(149,152),(150,153)], [(1,85,23,78),(2,86,24,79),(3,87,25,80),(4,88,26,71),(5,89,27,72),(6,90,28,73),(7,81,29,74),(8,82,30,75),(9,83,21,76),(10,84,22,77),(11,105,158,98),(12,106,159,99),(13,107,160,100),(14,108,151,91),(15,109,152,92),(16,110,153,93),(17,101,154,94),(18,102,155,95),(19,103,156,96),(20,104,157,97),(31,51,48,68),(32,52,49,69),(33,53,50,70),(34,54,41,61),(35,55,42,62),(36,56,43,63),(37,57,44,64),(38,58,45,65),(39,59,46,66),(40,60,47,67),(111,148,128,131),(112,149,129,132),(113,150,130,133),(114,141,121,134),(115,142,122,135),(116,143,123,136),(117,144,124,137),(118,145,125,138),(119,146,126,139),(120,147,127,140)], [(1,98,23,105),(2,99,24,106),(3,100,25,107),(4,91,26,108),(5,92,27,109),(6,93,28,110),(7,94,29,101),(8,95,30,102),(9,96,21,103),(10,97,22,104),(11,85,158,78),(12,86,159,79),(13,87,160,80),(14,88,151,71),(15,89,152,72),(16,90,153,73),(17,81,154,74),(18,82,155,75),(19,83,156,76),(20,84,157,77),(31,128,48,111),(32,129,49,112),(33,130,50,113),(34,121,41,114),(35,122,42,115),(36,123,43,116),(37,124,44,117),(38,125,45,118),(39,126,46,119),(40,127,47,120),(51,148,68,131),(52,149,69,132),(53,150,70,133),(54,141,61,134),(55,142,62,135),(56,143,63,136),(57,144,64,137),(58,145,65,138),(59,146,66,139),(60,147,67,140)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | 2- 1+4 | Q8⋊2D5 | Q8.10D10 |
| kernel | C10.452- 1+4 | C4×C5⋊D4 | C20⋊7D4 | Q8×Dic5 | D10⋊3Q8 | C20.23D4 | C2×Q8⋊2D5 | Q8×C2×C10 | C5×Q8 | C22×Q8 | C2×C10 | C22×C4 | C2×Q8 | Q8 | C10 | C22 | C2 |
| # reps | 1 | 3 | 3 | 1 | 3 | 3 | 1 | 1 | 4 | 2 | 4 | 6 | 8 | 16 | 1 | 4 | 4 |
Matrix representation of C10.452- 1+4 ►in GL4(𝔽41) generated by
| 34 | 34 | 0 | 0 |
| 7 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 17 | 40 | 0 | 0 |
| 3 | 24 | 0 | 0 |
| 0 | 0 | 14 | 31 |
| 0 | 0 | 40 | 27 |
| 17 | 40 | 0 | 0 |
| 1 | 24 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 24 | 1 | 0 | 0 |
| 40 | 17 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 24 | 9 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 38 | 8 |
| 0 | 0 | 9 | 3 |
G:=sub<GL(4,GF(41))| [34,7,0,0,34,1,0,0,0,0,40,0,0,0,0,40],[17,3,0,0,40,24,0,0,0,0,14,40,0,0,31,27],[17,1,0,0,40,24,0,0,0,0,1,0,0,0,0,1],[24,40,0,0,1,17,0,0,0,0,32,24,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,38,9,0,0,8,3] >;
C10.452- 1+4 in GAP, Magma, Sage, TeX
C_{10}._{45}2_-^{1+4} % in TeX
G:=Group("C10.45ES-(2,2)"); // GroupNames label
G:=SmallGroup(320,1489);
// by ID
G=gap.SmallGroup(320,1489);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,387,184,675,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=c^2=1,d^2=e^2=a^5*b^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,d*b*d^-1=a^5*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=a^5*b^2*d>;
// generators/relations