?
G = C10.812- (1+4) order 320 = 26·5
36th non-split extension by C10 of 2- (1+4) acting via 2- (1+4)/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.812- (1+4), C10.602+ (1+4), C20⋊Q8⋊31C2, C4⋊C4.105D10, (C2×D4).97D10, C22⋊C4.26D10, C20.48D4⋊14C2, (C2×C10).195C24, (C2×C20).178C23, (C22×C4).256D10, C2.62(D4⋊6D10), C22.D4.3D5, Dic5.Q8⋊26C2, C20.17D4.10C2, (D4×C10).133C22, C23.D10⋊29C2, C4⋊Dic5.226C22, (C22×C20).86C22, C23.128(C22×D5), C22.216(C23×D5), Dic5.14D4⋊30C2, C23.D5.41C22, (C22×C10).220C23, C5⋊2(C22.57C24), (C4×Dic5).130C22, (C2×Dic10).38C22, (C2×Dic5).100C23, C10.D4.40C22, C23.18D10.3C2, C2.42(D4.10D10), (C22×Dic5).128C22, (C2×C4).59(C22×D5), (C5×C4⋊C4).175C22, (C5×C22⋊C4).50C22, (C5×C22.D4).3C2, SmallGroup(320,1323)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 614 in 196 conjugacy classes, 91 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×13], C22, C22 [×6], C5, C2×C4, C2×C4 [×4], C2×C4 [×10], D4, Q8 [×3], C23 [×2], C10, C10 [×2], C10 [×2], C42 [×3], C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×14], C22×C4, C22×C4, C2×D4, C2×Q8 [×3], Dic5 [×8], C20 [×5], C2×C10, C2×C10 [×6], C22⋊Q8 [×4], C22.D4, C22.D4, C4.4D4, C42.C2 [×2], C42⋊2C2 [×4], C4⋊Q8 [×2], Dic10 [×3], C2×Dic5 [×8], C2×Dic5, C2×C20, C2×C20 [×4], C2×C20, C5×D4, C22×C10 [×2], C22.57C24, C4×Dic5, C4×Dic5 [×2], C10.D4 [×10], C4⋊Dic5 [×4], C23.D5, C23.D5 [×6], C5×C22⋊C4, C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10, C2×Dic10 [×2], C22×Dic5, C22×C20, D4×C10, Dic5.14D4 [×2], C23.D10 [×4], C20⋊Q8 [×2], Dic5.Q8 [×2], C20.48D4 [×2], C23.18D10, C20.17D4, C5×C22.D4, C10.812- (1+4)
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], 2+ (1+4), 2- (1+4) [×2], C22×D5 [×7], C22.57C24, C23×D5, D4⋊6D10, D4.10D10 [×2], C10.812- (1+4)
Generators and relations
G = < a,b,c,d,e | a10=b4=1, c2=a5, d2=e2=a5b2, bab-1=cac-1=eae-1=a-1, ad=da, cbc-1=b-1, bd=db, ebe-1=a5b, dcd-1=a5c, ce=ec, ede-1=a5b2d >
►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)
(1 150 25 138)(2 149 26 137)(3 148 27 136)(4 147 28 135)(5 146 29 134)(6 145 30 133)(7 144 21 132)(8 143 22 131)(9 142 23 140)(10 141 24 139)(11 31 154 44)(12 40 155 43)(13 39 156 42)(14 38 157 41)(15 37 158 50)(16 36 159 49)(17 35 160 48)(18 34 151 47)(19 33 152 46)(20 32 153 45)(51 99 64 101)(52 98 65 110)(53 97 66 109)(54 96 67 108)(55 95 68 107)(56 94 69 106)(57 93 70 105)(58 92 61 104)(59 91 62 103)(60 100 63 102)(71 126 85 114)(72 125 86 113)(73 124 87 112)(74 123 88 111)(75 122 89 120)(76 121 90 119)(77 130 81 118)(78 129 82 117)(79 128 83 116)(80 127 84 115)
(1 118 6 113)(2 117 7 112)(3 116 8 111)(4 115 9 120)(5 114 10 119)(11 64 16 69)(12 63 17 68)(13 62 18 67)(14 61 19 66)(15 70 20 65)(21 124 26 129)(22 123 27 128)(23 122 28 127)(24 121 29 126)(25 130 30 125)(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)(51 159 56 154)(52 158 57 153)(53 157 58 152)(54 156 59 151)(55 155 60 160)(71 139 76 134)(72 138 77 133)(73 137 78 132)(74 136 79 131)(75 135 80 140)(81 145 86 150)(82 144 87 149)(83 143 88 148)(84 142 89 147)(85 141 90 146)
(1 45 30 37)(2 46 21 38)(3 47 22 39)(4 48 23 40)(5 49 24 31)(6 50 25 32)(7 41 26 33)(8 42 27 34)(9 43 28 35)(10 44 29 36)(11 134 159 141)(12 135 160 142)(13 136 151 143)(14 137 152 144)(15 138 153 145)(16 139 154 146)(17 140 155 147)(18 131 156 148)(19 132 157 149)(20 133 158 150)(51 90 69 71)(52 81 70 72)(53 82 61 73)(54 83 62 74)(55 84 63 75)(56 85 64 76)(57 86 65 77)(58 87 66 78)(59 88 67 79)(60 89 68 80)(91 111 108 128)(92 112 109 129)(93 113 110 130)(94 114 101 121)(95 115 102 122)(96 116 103 123)(97 117 104 124)(98 118 105 125)(99 119 106 126)(100 120 107 127)
(1 153 30 15)(2 152 21 14)(3 151 22 13)(4 160 23 12)(5 159 24 11)(6 158 25 20)(7 157 26 19)(8 156 27 18)(9 155 28 17)(10 154 29 16)(31 141 49 134)(32 150 50 133)(33 149 41 132)(34 148 42 131)(35 147 43 140)(36 146 44 139)(37 145 45 138)(38 144 46 137)(39 143 47 136)(40 142 48 135)(51 126 69 119)(52 125 70 118)(53 124 61 117)(54 123 62 116)(55 122 63 115)(56 121 64 114)(57 130 65 113)(58 129 66 112)(59 128 67 111)(60 127 68 120)(71 99 90 106)(72 98 81 105)(73 97 82 104)(74 96 83 103)(75 95 84 102)(76 94 85 101)(77 93 86 110)(78 92 87 109)(79 91 88 108)(80 100 89 107)
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), (1,150,25,138)(2,149,26,137)(3,148,27,136)(4,147,28,135)(5,146,29,134)(6,145,30,133)(7,144,21,132)(8,143,22,131)(9,142,23,140)(10,141,24,139)(11,31,154,44)(12,40,155,43)(13,39,156,42)(14,38,157,41)(15,37,158,50)(16,36,159,49)(17,35,160,48)(18,34,151,47)(19,33,152,46)(20,32,153,45)(51,99,64,101)(52,98,65,110)(53,97,66,109)(54,96,67,108)(55,95,68,107)(56,94,69,106)(57,93,70,105)(58,92,61,104)(59,91,62,103)(60,100,63,102)(71,126,85,114)(72,125,86,113)(73,124,87,112)(74,123,88,111)(75,122,89,120)(76,121,90,119)(77,130,81,118)(78,129,82,117)(79,128,83,116)(80,127,84,115), (1,118,6,113)(2,117,7,112)(3,116,8,111)(4,115,9,120)(5,114,10,119)(11,64,16,69)(12,63,17,68)(13,62,18,67)(14,61,19,66)(15,70,20,65)(21,124,26,129)(22,123,27,128)(23,122,28,127)(24,121,29,126)(25,130,30,125)(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)(51,159,56,154)(52,158,57,153)(53,157,58,152)(54,156,59,151)(55,155,60,160)(71,139,76,134)(72,138,77,133)(73,137,78,132)(74,136,79,131)(75,135,80,140)(81,145,86,150)(82,144,87,149)(83,143,88,148)(84,142,89,147)(85,141,90,146), (1,45,30,37)(2,46,21,38)(3,47,22,39)(4,48,23,40)(5,49,24,31)(6,50,25,32)(7,41,26,33)(8,42,27,34)(9,43,28,35)(10,44,29,36)(11,134,159,141)(12,135,160,142)(13,136,151,143)(14,137,152,144)(15,138,153,145)(16,139,154,146)(17,140,155,147)(18,131,156,148)(19,132,157,149)(20,133,158,150)(51,90,69,71)(52,81,70,72)(53,82,61,73)(54,83,62,74)(55,84,63,75)(56,85,64,76)(57,86,65,77)(58,87,66,78)(59,88,67,79)(60,89,68,80)(91,111,108,128)(92,112,109,129)(93,113,110,130)(94,114,101,121)(95,115,102,122)(96,116,103,123)(97,117,104,124)(98,118,105,125)(99,119,106,126)(100,120,107,127), (1,153,30,15)(2,152,21,14)(3,151,22,13)(4,160,23,12)(5,159,24,11)(6,158,25,20)(7,157,26,19)(8,156,27,18)(9,155,28,17)(10,154,29,16)(31,141,49,134)(32,150,50,133)(33,149,41,132)(34,148,42,131)(35,147,43,140)(36,146,44,139)(37,145,45,138)(38,144,46,137)(39,143,47,136)(40,142,48,135)(51,126,69,119)(52,125,70,118)(53,124,61,117)(54,123,62,116)(55,122,63,115)(56,121,64,114)(57,130,65,113)(58,129,66,112)(59,128,67,111)(60,127,68,120)(71,99,90,106)(72,98,81,105)(73,97,82,104)(74,96,83,103)(75,95,84,102)(76,94,85,101)(77,93,86,110)(78,92,87,109)(79,91,88,108)(80,100,89,107)>;
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), (1,150,25,138)(2,149,26,137)(3,148,27,136)(4,147,28,135)(5,146,29,134)(6,145,30,133)(7,144,21,132)(8,143,22,131)(9,142,23,140)(10,141,24,139)(11,31,154,44)(12,40,155,43)(13,39,156,42)(14,38,157,41)(15,37,158,50)(16,36,159,49)(17,35,160,48)(18,34,151,47)(19,33,152,46)(20,32,153,45)(51,99,64,101)(52,98,65,110)(53,97,66,109)(54,96,67,108)(55,95,68,107)(56,94,69,106)(57,93,70,105)(58,92,61,104)(59,91,62,103)(60,100,63,102)(71,126,85,114)(72,125,86,113)(73,124,87,112)(74,123,88,111)(75,122,89,120)(76,121,90,119)(77,130,81,118)(78,129,82,117)(79,128,83,116)(80,127,84,115), (1,118,6,113)(2,117,7,112)(3,116,8,111)(4,115,9,120)(5,114,10,119)(11,64,16,69)(12,63,17,68)(13,62,18,67)(14,61,19,66)(15,70,20,65)(21,124,26,129)(22,123,27,128)(23,122,28,127)(24,121,29,126)(25,130,30,125)(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)(51,159,56,154)(52,158,57,153)(53,157,58,152)(54,156,59,151)(55,155,60,160)(71,139,76,134)(72,138,77,133)(73,137,78,132)(74,136,79,131)(75,135,80,140)(81,145,86,150)(82,144,87,149)(83,143,88,148)(84,142,89,147)(85,141,90,146), (1,45,30,37)(2,46,21,38)(3,47,22,39)(4,48,23,40)(5,49,24,31)(6,50,25,32)(7,41,26,33)(8,42,27,34)(9,43,28,35)(10,44,29,36)(11,134,159,141)(12,135,160,142)(13,136,151,143)(14,137,152,144)(15,138,153,145)(16,139,154,146)(17,140,155,147)(18,131,156,148)(19,132,157,149)(20,133,158,150)(51,90,69,71)(52,81,70,72)(53,82,61,73)(54,83,62,74)(55,84,63,75)(56,85,64,76)(57,86,65,77)(58,87,66,78)(59,88,67,79)(60,89,68,80)(91,111,108,128)(92,112,109,129)(93,113,110,130)(94,114,101,121)(95,115,102,122)(96,116,103,123)(97,117,104,124)(98,118,105,125)(99,119,106,126)(100,120,107,127), (1,153,30,15)(2,152,21,14)(3,151,22,13)(4,160,23,12)(5,159,24,11)(6,158,25,20)(7,157,26,19)(8,156,27,18)(9,155,28,17)(10,154,29,16)(31,141,49,134)(32,150,50,133)(33,149,41,132)(34,148,42,131)(35,147,43,140)(36,146,44,139)(37,145,45,138)(38,144,46,137)(39,143,47,136)(40,142,48,135)(51,126,69,119)(52,125,70,118)(53,124,61,117)(54,123,62,116)(55,122,63,115)(56,121,64,114)(57,130,65,113)(58,129,66,112)(59,128,67,111)(60,127,68,120)(71,99,90,106)(72,98,81,105)(73,97,82,104)(74,96,83,103)(75,95,84,102)(76,94,85,101)(77,93,86,110)(78,92,87,109)(79,91,88,108)(80,100,89,107) );
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)], [(1,150,25,138),(2,149,26,137),(3,148,27,136),(4,147,28,135),(5,146,29,134),(6,145,30,133),(7,144,21,132),(8,143,22,131),(9,142,23,140),(10,141,24,139),(11,31,154,44),(12,40,155,43),(13,39,156,42),(14,38,157,41),(15,37,158,50),(16,36,159,49),(17,35,160,48),(18,34,151,47),(19,33,152,46),(20,32,153,45),(51,99,64,101),(52,98,65,110),(53,97,66,109),(54,96,67,108),(55,95,68,107),(56,94,69,106),(57,93,70,105),(58,92,61,104),(59,91,62,103),(60,100,63,102),(71,126,85,114),(72,125,86,113),(73,124,87,112),(74,123,88,111),(75,122,89,120),(76,121,90,119),(77,130,81,118),(78,129,82,117),(79,128,83,116),(80,127,84,115)], [(1,118,6,113),(2,117,7,112),(3,116,8,111),(4,115,9,120),(5,114,10,119),(11,64,16,69),(12,63,17,68),(13,62,18,67),(14,61,19,66),(15,70,20,65),(21,124,26,129),(22,123,27,128),(23,122,28,127),(24,121,29,126),(25,130,30,125),(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),(51,159,56,154),(52,158,57,153),(53,157,58,152),(54,156,59,151),(55,155,60,160),(71,139,76,134),(72,138,77,133),(73,137,78,132),(74,136,79,131),(75,135,80,140),(81,145,86,150),(82,144,87,149),(83,143,88,148),(84,142,89,147),(85,141,90,146)], [(1,45,30,37),(2,46,21,38),(3,47,22,39),(4,48,23,40),(5,49,24,31),(6,50,25,32),(7,41,26,33),(8,42,27,34),(9,43,28,35),(10,44,29,36),(11,134,159,141),(12,135,160,142),(13,136,151,143),(14,137,152,144),(15,138,153,145),(16,139,154,146),(17,140,155,147),(18,131,156,148),(19,132,157,149),(20,133,158,150),(51,90,69,71),(52,81,70,72),(53,82,61,73),(54,83,62,74),(55,84,63,75),(56,85,64,76),(57,86,65,77),(58,87,66,78),(59,88,67,79),(60,89,68,80),(91,111,108,128),(92,112,109,129),(93,113,110,130),(94,114,101,121),(95,115,102,122),(96,116,103,123),(97,117,104,124),(98,118,105,125),(99,119,106,126),(100,120,107,127)], [(1,153,30,15),(2,152,21,14),(3,151,22,13),(4,160,23,12),(5,159,24,11),(6,158,25,20),(7,157,26,19),(8,156,27,18),(9,155,28,17),(10,154,29,16),(31,141,49,134),(32,150,50,133),(33,149,41,132),(34,148,42,131),(35,147,43,140),(36,146,44,139),(37,145,45,138),(38,144,46,137),(39,143,47,136),(40,142,48,135),(51,126,69,119),(52,125,70,118),(53,124,61,117),(54,123,62,116),(55,122,63,115),(56,121,64,114),(57,130,65,113),(58,129,66,112),(59,128,67,111),(60,127,68,120),(71,99,90,106),(72,98,81,105),(73,97,82,104),(74,96,83,103),(75,95,84,102),(76,94,85,101),(77,93,86,110),(78,92,87,109),(79,91,88,108),(80,100,89,107)])
Matrix representation ►G ⊆ GL8(𝔽41)
| 35 | 35 | 0 | 0 | 0 | 0 | 0 | 0 |
| 6 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 39 | 27 | 0 | 0 | 0 | 0 | 0 | 0 |
| 15 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 39 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 0 | 15 | 37 |
| 0 | 0 | 0 | 0 | 15 | 0 | 26 | 26 |
| 0 | 0 | 0 | 0 | 15 | 39 | 28 | 39 |
| 0 | 0 | 0 | 0 | 0 | 13 | 28 | 39 |
| 27 | 39 | 3 | 33 | 0 | 0 | 0 | 0 |
| 37 | 14 | 39 | 5 | 0 | 0 | 0 | 0 |
| 31 | 25 | 23 | 14 | 0 | 0 | 0 | 0 |
| 37 | 35 | 9 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 26 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 | 2 | 13 |
| 0 | 0 | 0 | 0 | 15 | 28 | 28 | 39 |
| 40 | 0 | 3 | 20 | 0 | 0 | 0 | 0 |
| 0 | 40 | 1 | 20 | 0 | 0 | 0 | 0 |
| 40 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |
| 39 | 6 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 1 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 1 | 1 | 0 |
| 2 | 14 | 21 | 8 | 0 | 0 | 0 | 0 |
| 26 | 39 | 6 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 39 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 0 | 37 | 15 |
| 0 | 0 | 0 | 0 | 15 | 0 | 26 | 26 |
| 0 | 0 | 0 | 0 | 15 | 28 | 39 | 28 |
| 0 | 0 | 0 | 0 | 0 | 2 | 39 | 28 |
G:=sub<GL(8,GF(41))| [35,6,0,0,0,0,0,0,35,40,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,35,34,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[39,15,0,0,0,0,0,0,27,2,0,0,0,0,0,0,0,0,28,3,0,0,0,0,0,0,39,13,0,0,0,0,0,0,0,0,15,15,15,0,0,0,0,0,0,0,39,13,0,0,0,0,15,26,28,28,0,0,0,0,37,26,39,39],[27,37,31,37,0,0,0,0,39,14,25,35,0,0,0,0,3,39,23,9,0,0,0,0,33,5,14,18,0,0,0,0,0,0,0,0,15,15,0,15,0,0,0,0,15,26,28,28,0,0,0,0,0,0,2,28,0,0,0,0,0,0,13,39],[40,0,40,39,0,0,0,0,0,40,1,6,0,0,0,0,3,1,1,0,0,0,0,0,20,20,0,1,0,0,0,0,0,0,0,0,40,0,40,40,0,0,0,0,2,1,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[2,26,0,0,0,0,0,0,14,39,0,0,0,0,0,0,21,6,28,3,0,0,0,0,8,12,39,13,0,0,0,0,0,0,0,0,15,15,15,0,0,0,0,0,0,0,28,2,0,0,0,0,37,26,39,39,0,0,0,0,15,26,28,28] >;
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4E | 4F | ··· | 4M | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 20A | ··· | 20H | 20I | ··· | 20N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | ··· | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | D10 | D10 | D10 | D10 | 2+ (1+4) | 2- (1+4) | D4⋊6D10 | D4.10D10 |
| kernel | C10.812- (1+4) | Dic5.14D4 | C23.D10 | C20⋊Q8 | Dic5.Q8 | C20.48D4 | C23.18D10 | C20.17D4 | C5×C22.D4 | C22.D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C10 | C10 | C2 | C2 |
| # reps | 1 | 2 | 4 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 6 | 4 | 2 | 2 | 1 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_{10}._{81}2_-^{(1+4)} % in TeX
G:=Group("C10.81ES-(2,2)"); // GroupNames label
G:=SmallGroup(320,1323);
// by ID
G=gap.SmallGroup(320,1323);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,758,219,184,1571,570,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=1,c^2=a^5,d^2=e^2=a^5*b^2,b*a*b^-1=c*a*c^-1=e*a*e^-1=a^-1,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,e*b*e^-1=a^5*b,d*c*d^-1=a^5*c,c*e=e*c,e*d*e^-1=a^5*b^2*d>;
// generators/relations
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