?
G = C10.72+ (1+4) order 320 = 26·5
7th non-split extension by C10 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.72+ (1+4), C10.12- (1+4), C20⋊Q8⋊8C2, (C2×C20)⋊3Q8, (C2×C4)⋊3Dic10, C20.67(C2×Q8), C4⋊C4.261D10, C4.Dic10⋊8C2, C10.9(C22×Q8), (C2×C10).44C24, C4.32(C2×Dic10), (C2×C20).135C23, C20.48D4.6C2, (C22×C4).175D10, C2.11(D4⋊6D10), C22.82(C23×D5), C4⋊Dic5.358C22, (C22×C20).74C22, (C2×Dic5).14C23, (C4×Dic5).70C22, C2.11(C22×Dic10), C22.11(C2×Dic10), C10.D4.1C22, C23.223(C22×D5), C23.D5.86C22, (C22×C10).393C23, C2.5(Q8.10D10), C5⋊1(C23.41C23), (C2×Dic10).23C22, C23.21D10.20C2, (C2×C4⋊C4).27D5, (C10×C4⋊C4).20C2, (C2×C10).52(C2×Q8), (C5×C4⋊C4).293C22, (C2×C4).570(C22×D5), SmallGroup(320,1172)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 606 in 206 conjugacy classes, 111 normal (17 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×12], C22, C22 [×2], C22 [×2], C5, C2×C4 [×10], C2×C4 [×10], Q8 [×4], C23, C10 [×3], C10 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×16], C22×C4, C22×C4 [×2], C2×Q8 [×4], Dic5 [×8], C20 [×4], C20 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C4⋊C4, C42⋊C2 [×2], C22⋊Q8 [×4], C42.C2 [×4], C4⋊Q8 [×4], Dic10 [×4], C2×Dic5 [×8], C2×C20 [×10], C2×C20 [×2], C22×C10, C23.41C23, C4×Dic5 [×4], C10.D4 [×8], C4⋊Dic5 [×8], C23.D5 [×4], C5×C4⋊C4 [×4], C2×Dic10 [×4], C22×C20, C22×C20 [×2], C20⋊Q8 [×4], C4.Dic10 [×4], C20.48D4 [×4], C23.21D10 [×2], C10×C4⋊C4, C10.72+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C24, D10 [×7], C22×Q8, 2+ (1+4), 2- (1+4), Dic10 [×4], C22×D5 [×7], C23.41C23, C2×Dic10 [×6], C23×D5, C22×Dic10, D4⋊6D10, Q8.10D10, C10.72+ (1+4)
Generators and relations
G = < a,b,c,d,e | a10=b4=1, c2=e2=a5, d2=b2, bab-1=dad-1=eae-1=a-1, ac=ca, cbc-1=b-1, dbd-1=a5b, be=eb, dcd-1=ece-1=a5c, ede-1=b2d >
►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)
(1 155 25 20)(2 154 26 19)(3 153 27 18)(4 152 28 17)(5 151 29 16)(6 160 30 15)(7 159 21 14)(8 158 22 13)(9 157 23 12)(10 156 24 11)(31 139 46 149)(32 138 47 148)(33 137 48 147)(34 136 49 146)(35 135 50 145)(36 134 41 144)(37 133 42 143)(38 132 43 142)(39 131 44 141)(40 140 45 150)(51 114 66 124)(52 113 67 123)(53 112 68 122)(54 111 69 121)(55 120 70 130)(56 119 61 129)(57 118 62 128)(58 117 63 127)(59 116 64 126)(60 115 65 125)(71 99 87 109)(72 98 88 108)(73 97 89 107)(74 96 90 106)(75 95 81 105)(76 94 82 104)(77 93 83 103)(78 92 84 102)(79 91 85 101)(80 100 86 110)
(1 35 6 40)(2 36 7 31)(3 37 8 32)(4 38 9 33)(5 39 10 34)(11 136 16 131)(12 137 17 132)(13 138 18 133)(14 139 19 134)(15 140 20 135)(21 46 26 41)(22 47 27 42)(23 48 28 43)(24 49 29 44)(25 50 30 45)(51 87 56 82)(52 88 57 83)(53 89 58 84)(54 90 59 85)(55 81 60 86)(61 76 66 71)(62 77 67 72)(63 78 68 73)(64 79 69 74)(65 80 70 75)(91 111 96 116)(92 112 97 117)(93 113 98 118)(94 114 99 119)(95 115 100 120)(101 121 106 126)(102 122 107 127)(103 123 108 128)(104 124 109 129)(105 125 110 130)(141 156 146 151)(142 157 147 152)(143 158 148 153)(144 159 149 154)(145 160 150 155)
(1 95 25 105)(2 94 26 104)(3 93 27 103)(4 92 28 102)(5 91 29 101)(6 100 30 110)(7 99 21 109)(8 98 22 108)(9 97 23 107)(10 96 24 106)(11 79 156 85)(12 78 157 84)(13 77 158 83)(14 76 159 82)(15 75 160 81)(16 74 151 90)(17 73 152 89)(18 72 153 88)(19 71 154 87)(20 80 155 86)(31 114 46 124)(32 113 47 123)(33 112 48 122)(34 111 49 121)(35 120 50 130)(36 119 41 129)(37 118 42 128)(38 117 43 127)(39 116 44 126)(40 115 45 125)(51 134 66 144)(52 133 67 143)(53 132 68 142)(54 131 69 141)(55 140 70 150)(56 139 61 149)(57 138 62 148)(58 137 63 147)(59 136 64 146)(60 135 65 145)
(1 115 6 120)(2 114 7 119)(3 113 8 118)(4 112 9 117)(5 111 10 116)(11 59 16 54)(12 58 17 53)(13 57 18 52)(14 56 19 51)(15 55 20 60)(21 129 26 124)(22 128 27 123)(23 127 28 122)(24 126 29 121)(25 125 30 130)(31 99 36 94)(32 98 37 93)(33 97 38 92)(34 96 39 91)(35 95 40 100)(41 104 46 109)(42 103 47 108)(43 102 48 107)(44 101 49 106)(45 110 50 105)(61 154 66 159)(62 153 67 158)(63 152 68 157)(64 151 69 156)(65 160 70 155)(71 144 76 149)(72 143 77 148)(73 142 78 147)(74 141 79 146)(75 150 80 145)(81 140 86 135)(82 139 87 134)(83 138 88 133)(84 137 89 132)(85 136 90 131)
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), (1,155,25,20)(2,154,26,19)(3,153,27,18)(4,152,28,17)(5,151,29,16)(6,160,30,15)(7,159,21,14)(8,158,22,13)(9,157,23,12)(10,156,24,11)(31,139,46,149)(32,138,47,148)(33,137,48,147)(34,136,49,146)(35,135,50,145)(36,134,41,144)(37,133,42,143)(38,132,43,142)(39,131,44,141)(40,140,45,150)(51,114,66,124)(52,113,67,123)(53,112,68,122)(54,111,69,121)(55,120,70,130)(56,119,61,129)(57,118,62,128)(58,117,63,127)(59,116,64,126)(60,115,65,125)(71,99,87,109)(72,98,88,108)(73,97,89,107)(74,96,90,106)(75,95,81,105)(76,94,82,104)(77,93,83,103)(78,92,84,102)(79,91,85,101)(80,100,86,110), (1,35,6,40)(2,36,7,31)(3,37,8,32)(4,38,9,33)(5,39,10,34)(11,136,16,131)(12,137,17,132)(13,138,18,133)(14,139,19,134)(15,140,20,135)(21,46,26,41)(22,47,27,42)(23,48,28,43)(24,49,29,44)(25,50,30,45)(51,87,56,82)(52,88,57,83)(53,89,58,84)(54,90,59,85)(55,81,60,86)(61,76,66,71)(62,77,67,72)(63,78,68,73)(64,79,69,74)(65,80,70,75)(91,111,96,116)(92,112,97,117)(93,113,98,118)(94,114,99,119)(95,115,100,120)(101,121,106,126)(102,122,107,127)(103,123,108,128)(104,124,109,129)(105,125,110,130)(141,156,146,151)(142,157,147,152)(143,158,148,153)(144,159,149,154)(145,160,150,155), (1,95,25,105)(2,94,26,104)(3,93,27,103)(4,92,28,102)(5,91,29,101)(6,100,30,110)(7,99,21,109)(8,98,22,108)(9,97,23,107)(10,96,24,106)(11,79,156,85)(12,78,157,84)(13,77,158,83)(14,76,159,82)(15,75,160,81)(16,74,151,90)(17,73,152,89)(18,72,153,88)(19,71,154,87)(20,80,155,86)(31,114,46,124)(32,113,47,123)(33,112,48,122)(34,111,49,121)(35,120,50,130)(36,119,41,129)(37,118,42,128)(38,117,43,127)(39,116,44,126)(40,115,45,125)(51,134,66,144)(52,133,67,143)(53,132,68,142)(54,131,69,141)(55,140,70,150)(56,139,61,149)(57,138,62,148)(58,137,63,147)(59,136,64,146)(60,135,65,145), (1,115,6,120)(2,114,7,119)(3,113,8,118)(4,112,9,117)(5,111,10,116)(11,59,16,54)(12,58,17,53)(13,57,18,52)(14,56,19,51)(15,55,20,60)(21,129,26,124)(22,128,27,123)(23,127,28,122)(24,126,29,121)(25,125,30,130)(31,99,36,94)(32,98,37,93)(33,97,38,92)(34,96,39,91)(35,95,40,100)(41,104,46,109)(42,103,47,108)(43,102,48,107)(44,101,49,106)(45,110,50,105)(61,154,66,159)(62,153,67,158)(63,152,68,157)(64,151,69,156)(65,160,70,155)(71,144,76,149)(72,143,77,148)(73,142,78,147)(74,141,79,146)(75,150,80,145)(81,140,86,135)(82,139,87,134)(83,138,88,133)(84,137,89,132)(85,136,90,131)>;
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), (1,155,25,20)(2,154,26,19)(3,153,27,18)(4,152,28,17)(5,151,29,16)(6,160,30,15)(7,159,21,14)(8,158,22,13)(9,157,23,12)(10,156,24,11)(31,139,46,149)(32,138,47,148)(33,137,48,147)(34,136,49,146)(35,135,50,145)(36,134,41,144)(37,133,42,143)(38,132,43,142)(39,131,44,141)(40,140,45,150)(51,114,66,124)(52,113,67,123)(53,112,68,122)(54,111,69,121)(55,120,70,130)(56,119,61,129)(57,118,62,128)(58,117,63,127)(59,116,64,126)(60,115,65,125)(71,99,87,109)(72,98,88,108)(73,97,89,107)(74,96,90,106)(75,95,81,105)(76,94,82,104)(77,93,83,103)(78,92,84,102)(79,91,85,101)(80,100,86,110), (1,35,6,40)(2,36,7,31)(3,37,8,32)(4,38,9,33)(5,39,10,34)(11,136,16,131)(12,137,17,132)(13,138,18,133)(14,139,19,134)(15,140,20,135)(21,46,26,41)(22,47,27,42)(23,48,28,43)(24,49,29,44)(25,50,30,45)(51,87,56,82)(52,88,57,83)(53,89,58,84)(54,90,59,85)(55,81,60,86)(61,76,66,71)(62,77,67,72)(63,78,68,73)(64,79,69,74)(65,80,70,75)(91,111,96,116)(92,112,97,117)(93,113,98,118)(94,114,99,119)(95,115,100,120)(101,121,106,126)(102,122,107,127)(103,123,108,128)(104,124,109,129)(105,125,110,130)(141,156,146,151)(142,157,147,152)(143,158,148,153)(144,159,149,154)(145,160,150,155), (1,95,25,105)(2,94,26,104)(3,93,27,103)(4,92,28,102)(5,91,29,101)(6,100,30,110)(7,99,21,109)(8,98,22,108)(9,97,23,107)(10,96,24,106)(11,79,156,85)(12,78,157,84)(13,77,158,83)(14,76,159,82)(15,75,160,81)(16,74,151,90)(17,73,152,89)(18,72,153,88)(19,71,154,87)(20,80,155,86)(31,114,46,124)(32,113,47,123)(33,112,48,122)(34,111,49,121)(35,120,50,130)(36,119,41,129)(37,118,42,128)(38,117,43,127)(39,116,44,126)(40,115,45,125)(51,134,66,144)(52,133,67,143)(53,132,68,142)(54,131,69,141)(55,140,70,150)(56,139,61,149)(57,138,62,148)(58,137,63,147)(59,136,64,146)(60,135,65,145), (1,115,6,120)(2,114,7,119)(3,113,8,118)(4,112,9,117)(5,111,10,116)(11,59,16,54)(12,58,17,53)(13,57,18,52)(14,56,19,51)(15,55,20,60)(21,129,26,124)(22,128,27,123)(23,127,28,122)(24,126,29,121)(25,125,30,130)(31,99,36,94)(32,98,37,93)(33,97,38,92)(34,96,39,91)(35,95,40,100)(41,104,46,109)(42,103,47,108)(43,102,48,107)(44,101,49,106)(45,110,50,105)(61,154,66,159)(62,153,67,158)(63,152,68,157)(64,151,69,156)(65,160,70,155)(71,144,76,149)(72,143,77,148)(73,142,78,147)(74,141,79,146)(75,150,80,145)(81,140,86,135)(82,139,87,134)(83,138,88,133)(84,137,89,132)(85,136,90,131) );
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)], [(1,155,25,20),(2,154,26,19),(3,153,27,18),(4,152,28,17),(5,151,29,16),(6,160,30,15),(7,159,21,14),(8,158,22,13),(9,157,23,12),(10,156,24,11),(31,139,46,149),(32,138,47,148),(33,137,48,147),(34,136,49,146),(35,135,50,145),(36,134,41,144),(37,133,42,143),(38,132,43,142),(39,131,44,141),(40,140,45,150),(51,114,66,124),(52,113,67,123),(53,112,68,122),(54,111,69,121),(55,120,70,130),(56,119,61,129),(57,118,62,128),(58,117,63,127),(59,116,64,126),(60,115,65,125),(71,99,87,109),(72,98,88,108),(73,97,89,107),(74,96,90,106),(75,95,81,105),(76,94,82,104),(77,93,83,103),(78,92,84,102),(79,91,85,101),(80,100,86,110)], [(1,35,6,40),(2,36,7,31),(3,37,8,32),(4,38,9,33),(5,39,10,34),(11,136,16,131),(12,137,17,132),(13,138,18,133),(14,139,19,134),(15,140,20,135),(21,46,26,41),(22,47,27,42),(23,48,28,43),(24,49,29,44),(25,50,30,45),(51,87,56,82),(52,88,57,83),(53,89,58,84),(54,90,59,85),(55,81,60,86),(61,76,66,71),(62,77,67,72),(63,78,68,73),(64,79,69,74),(65,80,70,75),(91,111,96,116),(92,112,97,117),(93,113,98,118),(94,114,99,119),(95,115,100,120),(101,121,106,126),(102,122,107,127),(103,123,108,128),(104,124,109,129),(105,125,110,130),(141,156,146,151),(142,157,147,152),(143,158,148,153),(144,159,149,154),(145,160,150,155)], [(1,95,25,105),(2,94,26,104),(3,93,27,103),(4,92,28,102),(5,91,29,101),(6,100,30,110),(7,99,21,109),(8,98,22,108),(9,97,23,107),(10,96,24,106),(11,79,156,85),(12,78,157,84),(13,77,158,83),(14,76,159,82),(15,75,160,81),(16,74,151,90),(17,73,152,89),(18,72,153,88),(19,71,154,87),(20,80,155,86),(31,114,46,124),(32,113,47,123),(33,112,48,122),(34,111,49,121),(35,120,50,130),(36,119,41,129),(37,118,42,128),(38,117,43,127),(39,116,44,126),(40,115,45,125),(51,134,66,144),(52,133,67,143),(53,132,68,142),(54,131,69,141),(55,140,70,150),(56,139,61,149),(57,138,62,148),(58,137,63,147),(59,136,64,146),(60,135,65,145)], [(1,115,6,120),(2,114,7,119),(3,113,8,118),(4,112,9,117),(5,111,10,116),(11,59,16,54),(12,58,17,53),(13,57,18,52),(14,56,19,51),(15,55,20,60),(21,129,26,124),(22,128,27,123),(23,127,28,122),(24,126,29,121),(25,125,30,130),(31,99,36,94),(32,98,37,93),(33,97,38,92),(34,96,39,91),(35,95,40,100),(41,104,46,109),(42,103,47,108),(43,102,48,107),(44,101,49,106),(45,110,50,105),(61,154,66,159),(62,153,67,158),(63,152,68,157),(64,151,69,156),(65,160,70,155),(71,144,76,149),(72,143,77,148),(73,142,78,147),(74,141,79,146),(75,150,80,145),(81,140,86,135),(82,139,87,134),(83,138,88,133),(84,137,89,132),(85,136,90,131)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 25 | 0 | 0 | 0 | 0 | 0 |
| 36 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 39 | 0 | 31 | 0 |
| 0 | 0 | 23 | 0 | 0 | 31 |
| 33 | 5 | 0 | 0 | 0 | 0 |
| 28 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 6 | 1 |
| 0 | 0 | 0 | 0 | 34 | 19 |
| 0 | 0 | 4 | 21 | 5 | 35 |
| 0 | 0 | 36 | 38 | 4 | 28 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 37 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 39 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 40 |
| 0 | 0 | 0 | 26 | 1 | 0 |
| 10 | 4 | 0 | 0 | 0 | 0 |
| 26 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 2 | 0 |
| 0 | 0 | 15 | 0 | 32 | 40 |
| 0 | 0 | 31 | 0 | 29 | 0 |
| 0 | 0 | 24 | 40 | 15 | 0 |
| 33 | 5 | 0 | 0 | 0 | 0 |
| 28 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 | 0 | 39 |
| 0 | 0 | 29 | 0 | 40 | 9 |
| 0 | 0 | 17 | 1 | 0 | 12 |
| 0 | 0 | 31 | 0 | 0 | 26 |
G:=sub<GL(6,GF(41))| [25,36,0,0,0,0,0,23,0,0,0,0,0,0,4,0,39,23,0,0,0,4,0,0,0,0,0,0,31,0,0,0,0,0,0,31],[33,28,0,0,0,0,5,8,0,0,0,0,0,0,8,0,4,36,0,0,0,0,21,38,0,0,6,34,5,4,0,0,1,19,35,28],[9,37,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,39,9,12,26,0,0,0,0,0,1,0,0,0,0,40,0],[10,26,0,0,0,0,4,31,0,0,0,0,0,0,12,15,31,24,0,0,0,0,0,40,0,0,2,32,29,15,0,0,0,40,0,0],[33,28,0,0,0,0,5,8,0,0,0,0,0,0,15,29,17,31,0,0,0,0,1,0,0,0,0,40,0,0,0,0,39,9,12,26] >;
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | D10 | D10 | Dic10 | 2+ (1+4) | 2- (1+4) | D4⋊6D10 | Q8.10D10 |
| kernel | C10.72+ (1+4) | C20⋊Q8 | C4.Dic10 | C20.48D4 | C23.21D10 | C10×C4⋊C4 | C2×C20 | C2×C4⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C10 | C10 | C2 | C2 |
| # reps | 1 | 4 | 4 | 4 | 2 | 1 | 4 | 2 | 8 | 6 | 16 | 1 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_{10}._72_+^{(1+4)} % in TeX
G:=Group("C10.7ES+(2,2)"); // GroupNames label
G:=SmallGroup(320,1172);
// by ID
G=gap.SmallGroup(320,1172);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,232,758,675,570,80,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=1,c^2=e^2=a^5,d^2=b^2,b*a*b^-1=d*a*d^-1=e*a*e^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1,d*b*d^-1=a^5*b,b*e=e*b,d*c*d^-1=e*c*e^-1=a^5*c,e*d*e^-1=b^2*d>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀