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×C20).178C23, (C2×C10).195C24, (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, C22.216(C23×D5), C23.128(C22×D5), Dic5.14D4⋊30C2, C23.D5.41C22, (C22×C10).220C23, C5⋊2(C22.57C24), (C2×Dic10).38C22, (C2×Dic5).100C23, (C4×Dic5).130C22, 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
Generators and relations for C10.812- 1+4
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 >
Subgroups: 614 in 196 conjugacy classes, 91 normal (27 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, C42, C22⋊C4, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, Dic5, C20, C2×C10, C2×C10, C22⋊Q8, C22.D4, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C2×C20, C5×D4, C22×C10, C22.57C24, C4×Dic5, C4×Dic5, C10.D4, C4⋊Dic5, C23.D5, C23.D5, C5×C22⋊C4, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×Dic10, C22×Dic5, C22×C20, D4×C10, Dic5.14D4, C23.D10, C20⋊Q8, Dic5.Q8, C20.48D4, C23.18D10, C20.17D4, C5×C22.D4, C10.812- 1+4
Quotients: C1, C2, C22, C23, D5, C24, D10, 2+ 1+4, 2- 1+4, C22×D5, C22.57C24, C23×D5, D4⋊6D10, D4.10D10, C10.812- 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)
(1 150 25 140)(2 149 26 139)(3 148 27 138)(4 147 28 137)(5 146 29 136)(6 145 30 135)(7 144 21 134)(8 143 22 133)(9 142 23 132)(10 141 24 131)(11 39 156 44)(12 38 157 43)(13 37 158 42)(14 36 159 41)(15 35 160 50)(16 34 151 49)(17 33 152 48)(18 32 153 47)(19 31 154 46)(20 40 155 45)(51 99 66 109)(52 98 67 108)(53 97 68 107)(54 96 69 106)(55 95 70 105)(56 94 61 104)(57 93 62 103)(58 92 63 102)(59 91 64 101)(60 100 65 110)(71 124 87 114)(72 123 88 113)(73 122 89 112)(74 121 90 111)(75 130 81 120)(76 129 82 119)(77 128 83 118)(78 127 84 117)(79 126 85 116)(80 125 86 115)
(1 120 6 115)(2 119 7 114)(3 118 8 113)(4 117 9 112)(5 116 10 111)(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 35)(2 46 21 36)(3 47 22 37)(4 48 23 38)(5 49 24 39)(6 50 25 40)(7 41 26 31)(8 42 27 32)(9 43 28 33)(10 44 29 34)(11 136 151 141)(12 137 152 142)(13 138 153 143)(14 139 154 144)(15 140 155 145)(16 131 156 146)(17 132 157 147)(18 133 158 148)(19 134 159 149)(20 135 160 150)(51 82 61 71)(52 83 62 72)(53 84 63 73)(54 85 64 74)(55 86 65 75)(56 87 66 76)(57 88 67 77)(58 89 68 78)(59 90 69 79)(60 81 70 80)(91 111 106 126)(92 112 107 127)(93 113 108 128)(94 114 109 129)(95 115 110 130)(96 116 101 121)(97 117 102 122)(98 118 103 123)(99 119 104 124)(100 120 105 125)
(1 155 30 15)(2 154 21 14)(3 153 22 13)(4 152 23 12)(5 151 24 11)(6 160 25 20)(7 159 26 19)(8 158 27 18)(9 157 28 17)(10 156 29 16)(31 149 41 134)(32 148 42 133)(33 147 43 132)(34 146 44 131)(35 145 45 140)(36 144 46 139)(37 143 47 138)(38 142 48 137)(39 141 49 136)(40 150 50 135)(51 124 61 119)(52 123 62 118)(53 122 63 117)(54 121 64 116)(55 130 65 115)(56 129 66 114)(57 128 67 113)(58 127 68 112)(59 126 69 111)(60 125 70 120)(71 99 82 104)(72 98 83 103)(73 97 84 102)(74 96 85 101)(75 95 86 110)(76 94 87 109)(77 93 88 108)(78 92 89 107)(79 91 90 106)(80 100 81 105)
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), (1,150,25,140)(2,149,26,139)(3,148,27,138)(4,147,28,137)(5,146,29,136)(6,145,30,135)(7,144,21,134)(8,143,22,133)(9,142,23,132)(10,141,24,131)(11,39,156,44)(12,38,157,43)(13,37,158,42)(14,36,159,41)(15,35,160,50)(16,34,151,49)(17,33,152,48)(18,32,153,47)(19,31,154,46)(20,40,155,45)(51,99,66,109)(52,98,67,108)(53,97,68,107)(54,96,69,106)(55,95,70,105)(56,94,61,104)(57,93,62,103)(58,92,63,102)(59,91,64,101)(60,100,65,110)(71,124,87,114)(72,123,88,113)(73,122,89,112)(74,121,90,111)(75,130,81,120)(76,129,82,119)(77,128,83,118)(78,127,84,117)(79,126,85,116)(80,125,86,115), (1,120,6,115)(2,119,7,114)(3,118,8,113)(4,117,9,112)(5,116,10,111)(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), (1,45,30,35)(2,46,21,36)(3,47,22,37)(4,48,23,38)(5,49,24,39)(6,50,25,40)(7,41,26,31)(8,42,27,32)(9,43,28,33)(10,44,29,34)(11,136,151,141)(12,137,152,142)(13,138,153,143)(14,139,154,144)(15,140,155,145)(16,131,156,146)(17,132,157,147)(18,133,158,148)(19,134,159,149)(20,135,160,150)(51,82,61,71)(52,83,62,72)(53,84,63,73)(54,85,64,74)(55,86,65,75)(56,87,66,76)(57,88,67,77)(58,89,68,78)(59,90,69,79)(60,81,70,80)(91,111,106,126)(92,112,107,127)(93,113,108,128)(94,114,109,129)(95,115,110,130)(96,116,101,121)(97,117,102,122)(98,118,103,123)(99,119,104,124)(100,120,105,125), (1,155,30,15)(2,154,21,14)(3,153,22,13)(4,152,23,12)(5,151,24,11)(6,160,25,20)(7,159,26,19)(8,158,27,18)(9,157,28,17)(10,156,29,16)(31,149,41,134)(32,148,42,133)(33,147,43,132)(34,146,44,131)(35,145,45,140)(36,144,46,139)(37,143,47,138)(38,142,48,137)(39,141,49,136)(40,150,50,135)(51,124,61,119)(52,123,62,118)(53,122,63,117)(54,121,64,116)(55,130,65,115)(56,129,66,114)(57,128,67,113)(58,127,68,112)(59,126,69,111)(60,125,70,120)(71,99,82,104)(72,98,83,103)(73,97,84,102)(74,96,85,101)(75,95,86,110)(76,94,87,109)(77,93,88,108)(78,92,89,107)(79,91,90,106)(80,100,81,105)>;
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), (1,150,25,140)(2,149,26,139)(3,148,27,138)(4,147,28,137)(5,146,29,136)(6,145,30,135)(7,144,21,134)(8,143,22,133)(9,142,23,132)(10,141,24,131)(11,39,156,44)(12,38,157,43)(13,37,158,42)(14,36,159,41)(15,35,160,50)(16,34,151,49)(17,33,152,48)(18,32,153,47)(19,31,154,46)(20,40,155,45)(51,99,66,109)(52,98,67,108)(53,97,68,107)(54,96,69,106)(55,95,70,105)(56,94,61,104)(57,93,62,103)(58,92,63,102)(59,91,64,101)(60,100,65,110)(71,124,87,114)(72,123,88,113)(73,122,89,112)(74,121,90,111)(75,130,81,120)(76,129,82,119)(77,128,83,118)(78,127,84,117)(79,126,85,116)(80,125,86,115), (1,120,6,115)(2,119,7,114)(3,118,8,113)(4,117,9,112)(5,116,10,111)(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), (1,45,30,35)(2,46,21,36)(3,47,22,37)(4,48,23,38)(5,49,24,39)(6,50,25,40)(7,41,26,31)(8,42,27,32)(9,43,28,33)(10,44,29,34)(11,136,151,141)(12,137,152,142)(13,138,153,143)(14,139,154,144)(15,140,155,145)(16,131,156,146)(17,132,157,147)(18,133,158,148)(19,134,159,149)(20,135,160,150)(51,82,61,71)(52,83,62,72)(53,84,63,73)(54,85,64,74)(55,86,65,75)(56,87,66,76)(57,88,67,77)(58,89,68,78)(59,90,69,79)(60,81,70,80)(91,111,106,126)(92,112,107,127)(93,113,108,128)(94,114,109,129)(95,115,110,130)(96,116,101,121)(97,117,102,122)(98,118,103,123)(99,119,104,124)(100,120,105,125), (1,155,30,15)(2,154,21,14)(3,153,22,13)(4,152,23,12)(5,151,24,11)(6,160,25,20)(7,159,26,19)(8,158,27,18)(9,157,28,17)(10,156,29,16)(31,149,41,134)(32,148,42,133)(33,147,43,132)(34,146,44,131)(35,145,45,140)(36,144,46,139)(37,143,47,138)(38,142,48,137)(39,141,49,136)(40,150,50,135)(51,124,61,119)(52,123,62,118)(53,122,63,117)(54,121,64,116)(55,130,65,115)(56,129,66,114)(57,128,67,113)(58,127,68,112)(59,126,69,111)(60,125,70,120)(71,99,82,104)(72,98,83,103)(73,97,84,102)(74,96,85,101)(75,95,86,110)(76,94,87,109)(77,93,88,108)(78,92,89,107)(79,91,90,106)(80,100,81,105) );
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)], [(1,150,25,140),(2,149,26,139),(3,148,27,138),(4,147,28,137),(5,146,29,136),(6,145,30,135),(7,144,21,134),(8,143,22,133),(9,142,23,132),(10,141,24,131),(11,39,156,44),(12,38,157,43),(13,37,158,42),(14,36,159,41),(15,35,160,50),(16,34,151,49),(17,33,152,48),(18,32,153,47),(19,31,154,46),(20,40,155,45),(51,99,66,109),(52,98,67,108),(53,97,68,107),(54,96,69,106),(55,95,70,105),(56,94,61,104),(57,93,62,103),(58,92,63,102),(59,91,64,101),(60,100,65,110),(71,124,87,114),(72,123,88,113),(73,122,89,112),(74,121,90,111),(75,130,81,120),(76,129,82,119),(77,128,83,118),(78,127,84,117),(79,126,85,116),(80,125,86,115)], [(1,120,6,115),(2,119,7,114),(3,118,8,113),(4,117,9,112),(5,116,10,111),(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)], [(1,45,30,35),(2,46,21,36),(3,47,22,37),(4,48,23,38),(5,49,24,39),(6,50,25,40),(7,41,26,31),(8,42,27,32),(9,43,28,33),(10,44,29,34),(11,136,151,141),(12,137,152,142),(13,138,153,143),(14,139,154,144),(15,140,155,145),(16,131,156,146),(17,132,157,147),(18,133,158,148),(19,134,159,149),(20,135,160,150),(51,82,61,71),(52,83,62,72),(53,84,63,73),(54,85,64,74),(55,86,65,75),(56,87,66,76),(57,88,67,77),(58,89,68,78),(59,90,69,79),(60,81,70,80),(91,111,106,126),(92,112,107,127),(93,113,108,128),(94,114,109,129),(95,115,110,130),(96,116,101,121),(97,117,102,122),(98,118,103,123),(99,119,104,124),(100,120,105,125)], [(1,155,30,15),(2,154,21,14),(3,153,22,13),(4,152,23,12),(5,151,24,11),(6,160,25,20),(7,159,26,19),(8,158,27,18),(9,157,28,17),(10,156,29,16),(31,149,41,134),(32,148,42,133),(33,147,43,132),(34,146,44,131),(35,145,45,140),(36,144,46,139),(37,143,47,138),(38,142,48,137),(39,141,49,136),(40,150,50,135),(51,124,61,119),(52,123,62,118),(53,122,63,117),(54,121,64,116),(55,130,65,115),(56,129,66,114),(57,128,67,113),(58,127,68,112),(59,126,69,111),(60,125,70,120),(71,99,82,104),(72,98,83,103),(73,97,84,102),(74,96,85,101),(75,95,86,110),(76,94,87,109),(77,93,88,108),(78,92,89,107),(79,91,90,106),(80,100,81,105)]])
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 |
Matrix representation of C10.812- 1+4 ►in 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] >;
C10.812- 1+4 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