?
G = C10.442- (1+4) order 320 = 26·5
44th non-split extension by C10 of 2- (1+4) acting via 2- (1+4)/C2×Q8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.442- (1+4), (C22×Q8)⋊8D5, (C2×C20).216D4, C20.261(C2×D4), D10⋊3Q8⋊41C2, (C2×Q8).188D10, Dic5⋊Q8⋊30C2, C20.23D4⋊29C2, (C2×C10).308C24, (C2×C20).648C23, C10.156(C22×D4), (C22×C4).278D10, (C2×D20).287C22, C4⋊Dic5.390C22, (Q8×C10).235C22, C22.319(C23×D5), C23.239(C22×D5), D10⋊C4.77C22, C23.21D10⋊34C2, C23.23D10⋊29C2, (C22×C20).440C22, (C22×C10).426C23, C5⋊6(C23.38C23), (C2×Dic5).159C23, (C4×Dic5).179C22, C10.D4.90C22, (C22×D5).134C23, C23.D5.132C22, C2.44(Q8.10D10), (C2×Dic10).316C22, (Q8×C2×C10)⋊7C2, C4.99(C2×C5⋊D4), (C2×C4○D20).25C2, (C2×C10).590(C2×D4), (C2×C4).94(C5⋊D4), (C2×C4×D5).174C22, C2.29(C22×C5⋊D4), C22.37(C2×C5⋊D4), (C2×C4).634(C22×D5), (C2×C5⋊D4).146C22, SmallGroup(320,1488)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 846 in 270 conjugacy classes, 111 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×2], C22 [×8], C5, C2×C4 [×2], C2×C4 [×8], C2×C4 [×14], D4 [×6], Q8 [×10], C23, C23 [×2], D5 [×2], C10, C10 [×2], C10 [×2], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×10], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×3], C2×Q8 [×4], C2×Q8 [×5], C4○D4 [×4], Dic5 [×6], C20 [×4], C20 [×4], D10 [×6], C2×C10, C2×C10 [×2], C2×C10 [×2], C42⋊C2, C22⋊Q8 [×4], C22.D4 [×4], C4.4D4 [×2], C4⋊Q8 [×2], C22×Q8, C2×C4○D4, Dic10 [×2], C4×D5 [×4], D20 [×2], C2×Dic5 [×6], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×8], C2×C20 [×4], C5×Q8 [×8], C22×D5 [×2], C22×C10, C23.38C23, C4×Dic5 [×2], C10.D4 [×8], C4⋊Dic5 [×2], D10⋊C4 [×8], C23.D5 [×2], C2×Dic10, C2×C4×D5 [×2], C2×D20, C4○D20 [×4], C2×C5⋊D4 [×2], C22×C20, C22×C20 [×2], Q8×C10 [×4], Q8×C10 [×4], C23.21D10, C23.23D10 [×4], Dic5⋊Q8 [×2], D10⋊3Q8 [×4], C20.23D4 [×2], C2×C4○D20, Q8×C2×C10, C10.442- (1+4)
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, 2- (1+4) [×2], C5⋊D4 [×4], C22×D5 [×7], C23.38C23, C2×C5⋊D4 [×6], C23×D5, Q8.10D10 [×2], C22×C5⋊D4, C10.442- (1+4)
Generators and relations
G = < a,b,c,d,e | a10=b4=1, c2=a5, d2=e2=a5b2, bab-1=cac-1=a-1, ad=da, ae=ea, cbc-1=a5b-1, dbd-1=ebe-1=a5b, dcd-1=ece-1=a5c, 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 31 25 50)(2 40 26 49)(3 39 27 48)(4 38 28 47)(5 37 29 46)(6 36 30 45)(7 35 21 44)(8 34 22 43)(9 33 23 42)(10 32 24 41)(11 131 156 150)(12 140 157 149)(13 139 158 148)(14 138 159 147)(15 137 160 146)(16 136 151 145)(17 135 152 144)(18 134 153 143)(19 133 154 142)(20 132 155 141)(51 90 70 71)(52 89 61 80)(53 88 62 79)(54 87 63 78)(55 86 64 77)(56 85 65 76)(57 84 66 75)(58 83 67 74)(59 82 68 73)(60 81 69 72)(91 130 110 111)(92 129 101 120)(93 128 102 119)(94 127 103 118)(95 126 104 117)(96 125 105 116)(97 124 106 115)(98 123 107 114)(99 122 108 113)(100 121 109 112)
(1 71 6 76)(2 80 7 75)(3 79 8 74)(4 78 9 73)(5 77 10 72)(11 91 16 96)(12 100 17 95)(13 99 18 94)(14 98 19 93)(15 97 20 92)(21 84 26 89)(22 83 27 88)(23 82 28 87)(24 81 29 86)(25 90 30 85)(31 65 36 70)(32 64 37 69)(33 63 38 68)(34 62 39 67)(35 61 40 66)(41 55 46 60)(42 54 47 59)(43 53 48 58)(44 52 49 57)(45 51 50 56)(101 160 106 155)(102 159 107 154)(103 158 108 153)(104 157 109 152)(105 156 110 151)(111 131 116 136)(112 140 117 135)(113 139 118 134)(114 138 119 133)(115 137 120 132)(121 149 126 144)(122 148 127 143)(123 147 128 142)(124 146 129 141)(125 145 130 150)
(1 151 30 11)(2 152 21 12)(3 153 22 13)(4 154 23 14)(5 155 24 15)(6 156 25 16)(7 157 26 17)(8 158 27 18)(9 159 28 19)(10 160 29 20)(31 150 45 136)(32 141 46 137)(33 142 47 138)(34 143 48 139)(35 144 49 140)(36 145 50 131)(37 146 41 132)(38 147 42 133)(39 148 43 134)(40 149 44 135)(51 116 65 130)(52 117 66 121)(53 118 67 122)(54 119 68 123)(55 120 69 124)(56 111 70 125)(57 112 61 126)(58 113 62 127)(59 114 63 128)(60 115 64 129)(71 110 85 96)(72 101 86 97)(73 102 87 98)(74 103 88 99)(75 104 89 100)(76 105 90 91)(77 106 81 92)(78 107 82 93)(79 108 83 94)(80 109 84 95)
(1 45 30 31)(2 46 21 32)(3 47 22 33)(4 48 23 34)(5 49 24 35)(6 50 25 36)(7 41 26 37)(8 42 27 38)(9 43 28 39)(10 44 29 40)(11 136 151 150)(12 137 152 141)(13 138 153 142)(14 139 154 143)(15 140 155 144)(16 131 156 145)(17 132 157 146)(18 133 158 147)(19 134 159 148)(20 135 160 149)(51 90 65 76)(52 81 66 77)(53 82 67 78)(54 83 68 79)(55 84 69 80)(56 85 70 71)(57 86 61 72)(58 87 62 73)(59 88 63 74)(60 89 64 75)(91 116 105 130)(92 117 106 121)(93 118 107 122)(94 119 108 123)(95 120 109 124)(96 111 110 125)(97 112 101 126)(98 113 102 127)(99 114 103 128)(100 115 104 129)
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), (1,31,25,50)(2,40,26,49)(3,39,27,48)(4,38,28,47)(5,37,29,46)(6,36,30,45)(7,35,21,44)(8,34,22,43)(9,33,23,42)(10,32,24,41)(11,131,156,150)(12,140,157,149)(13,139,158,148)(14,138,159,147)(15,137,160,146)(16,136,151,145)(17,135,152,144)(18,134,153,143)(19,133,154,142)(20,132,155,141)(51,90,70,71)(52,89,61,80)(53,88,62,79)(54,87,63,78)(55,86,64,77)(56,85,65,76)(57,84,66,75)(58,83,67,74)(59,82,68,73)(60,81,69,72)(91,130,110,111)(92,129,101,120)(93,128,102,119)(94,127,103,118)(95,126,104,117)(96,125,105,116)(97,124,106,115)(98,123,107,114)(99,122,108,113)(100,121,109,112), (1,71,6,76)(2,80,7,75)(3,79,8,74)(4,78,9,73)(5,77,10,72)(11,91,16,96)(12,100,17,95)(13,99,18,94)(14,98,19,93)(15,97,20,92)(21,84,26,89)(22,83,27,88)(23,82,28,87)(24,81,29,86)(25,90,30,85)(31,65,36,70)(32,64,37,69)(33,63,38,68)(34,62,39,67)(35,61,40,66)(41,55,46,60)(42,54,47,59)(43,53,48,58)(44,52,49,57)(45,51,50,56)(101,160,106,155)(102,159,107,154)(103,158,108,153)(104,157,109,152)(105,156,110,151)(111,131,116,136)(112,140,117,135)(113,139,118,134)(114,138,119,133)(115,137,120,132)(121,149,126,144)(122,148,127,143)(123,147,128,142)(124,146,129,141)(125,145,130,150), (1,151,30,11)(2,152,21,12)(3,153,22,13)(4,154,23,14)(5,155,24,15)(6,156,25,16)(7,157,26,17)(8,158,27,18)(9,159,28,19)(10,160,29,20)(31,150,45,136)(32,141,46,137)(33,142,47,138)(34,143,48,139)(35,144,49,140)(36,145,50,131)(37,146,41,132)(38,147,42,133)(39,148,43,134)(40,149,44,135)(51,116,65,130)(52,117,66,121)(53,118,67,122)(54,119,68,123)(55,120,69,124)(56,111,70,125)(57,112,61,126)(58,113,62,127)(59,114,63,128)(60,115,64,129)(71,110,85,96)(72,101,86,97)(73,102,87,98)(74,103,88,99)(75,104,89,100)(76,105,90,91)(77,106,81,92)(78,107,82,93)(79,108,83,94)(80,109,84,95), (1,45,30,31)(2,46,21,32)(3,47,22,33)(4,48,23,34)(5,49,24,35)(6,50,25,36)(7,41,26,37)(8,42,27,38)(9,43,28,39)(10,44,29,40)(11,136,151,150)(12,137,152,141)(13,138,153,142)(14,139,154,143)(15,140,155,144)(16,131,156,145)(17,132,157,146)(18,133,158,147)(19,134,159,148)(20,135,160,149)(51,90,65,76)(52,81,66,77)(53,82,67,78)(54,83,68,79)(55,84,69,80)(56,85,70,71)(57,86,61,72)(58,87,62,73)(59,88,63,74)(60,89,64,75)(91,116,105,130)(92,117,106,121)(93,118,107,122)(94,119,108,123)(95,120,109,124)(96,111,110,125)(97,112,101,126)(98,113,102,127)(99,114,103,128)(100,115,104,129)>;
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), (1,31,25,50)(2,40,26,49)(3,39,27,48)(4,38,28,47)(5,37,29,46)(6,36,30,45)(7,35,21,44)(8,34,22,43)(9,33,23,42)(10,32,24,41)(11,131,156,150)(12,140,157,149)(13,139,158,148)(14,138,159,147)(15,137,160,146)(16,136,151,145)(17,135,152,144)(18,134,153,143)(19,133,154,142)(20,132,155,141)(51,90,70,71)(52,89,61,80)(53,88,62,79)(54,87,63,78)(55,86,64,77)(56,85,65,76)(57,84,66,75)(58,83,67,74)(59,82,68,73)(60,81,69,72)(91,130,110,111)(92,129,101,120)(93,128,102,119)(94,127,103,118)(95,126,104,117)(96,125,105,116)(97,124,106,115)(98,123,107,114)(99,122,108,113)(100,121,109,112), (1,71,6,76)(2,80,7,75)(3,79,8,74)(4,78,9,73)(5,77,10,72)(11,91,16,96)(12,100,17,95)(13,99,18,94)(14,98,19,93)(15,97,20,92)(21,84,26,89)(22,83,27,88)(23,82,28,87)(24,81,29,86)(25,90,30,85)(31,65,36,70)(32,64,37,69)(33,63,38,68)(34,62,39,67)(35,61,40,66)(41,55,46,60)(42,54,47,59)(43,53,48,58)(44,52,49,57)(45,51,50,56)(101,160,106,155)(102,159,107,154)(103,158,108,153)(104,157,109,152)(105,156,110,151)(111,131,116,136)(112,140,117,135)(113,139,118,134)(114,138,119,133)(115,137,120,132)(121,149,126,144)(122,148,127,143)(123,147,128,142)(124,146,129,141)(125,145,130,150), (1,151,30,11)(2,152,21,12)(3,153,22,13)(4,154,23,14)(5,155,24,15)(6,156,25,16)(7,157,26,17)(8,158,27,18)(9,159,28,19)(10,160,29,20)(31,150,45,136)(32,141,46,137)(33,142,47,138)(34,143,48,139)(35,144,49,140)(36,145,50,131)(37,146,41,132)(38,147,42,133)(39,148,43,134)(40,149,44,135)(51,116,65,130)(52,117,66,121)(53,118,67,122)(54,119,68,123)(55,120,69,124)(56,111,70,125)(57,112,61,126)(58,113,62,127)(59,114,63,128)(60,115,64,129)(71,110,85,96)(72,101,86,97)(73,102,87,98)(74,103,88,99)(75,104,89,100)(76,105,90,91)(77,106,81,92)(78,107,82,93)(79,108,83,94)(80,109,84,95), (1,45,30,31)(2,46,21,32)(3,47,22,33)(4,48,23,34)(5,49,24,35)(6,50,25,36)(7,41,26,37)(8,42,27,38)(9,43,28,39)(10,44,29,40)(11,136,151,150)(12,137,152,141)(13,138,153,142)(14,139,154,143)(15,140,155,144)(16,131,156,145)(17,132,157,146)(18,133,158,147)(19,134,159,148)(20,135,160,149)(51,90,65,76)(52,81,66,77)(53,82,67,78)(54,83,68,79)(55,84,69,80)(56,85,70,71)(57,86,61,72)(58,87,62,73)(59,88,63,74)(60,89,64,75)(91,116,105,130)(92,117,106,121)(93,118,107,122)(94,119,108,123)(95,120,109,124)(96,111,110,125)(97,112,101,126)(98,113,102,127)(99,114,103,128)(100,115,104,129) );
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)], [(1,31,25,50),(2,40,26,49),(3,39,27,48),(4,38,28,47),(5,37,29,46),(6,36,30,45),(7,35,21,44),(8,34,22,43),(9,33,23,42),(10,32,24,41),(11,131,156,150),(12,140,157,149),(13,139,158,148),(14,138,159,147),(15,137,160,146),(16,136,151,145),(17,135,152,144),(18,134,153,143),(19,133,154,142),(20,132,155,141),(51,90,70,71),(52,89,61,80),(53,88,62,79),(54,87,63,78),(55,86,64,77),(56,85,65,76),(57,84,66,75),(58,83,67,74),(59,82,68,73),(60,81,69,72),(91,130,110,111),(92,129,101,120),(93,128,102,119),(94,127,103,118),(95,126,104,117),(96,125,105,116),(97,124,106,115),(98,123,107,114),(99,122,108,113),(100,121,109,112)], [(1,71,6,76),(2,80,7,75),(3,79,8,74),(4,78,9,73),(5,77,10,72),(11,91,16,96),(12,100,17,95),(13,99,18,94),(14,98,19,93),(15,97,20,92),(21,84,26,89),(22,83,27,88),(23,82,28,87),(24,81,29,86),(25,90,30,85),(31,65,36,70),(32,64,37,69),(33,63,38,68),(34,62,39,67),(35,61,40,66),(41,55,46,60),(42,54,47,59),(43,53,48,58),(44,52,49,57),(45,51,50,56),(101,160,106,155),(102,159,107,154),(103,158,108,153),(104,157,109,152),(105,156,110,151),(111,131,116,136),(112,140,117,135),(113,139,118,134),(114,138,119,133),(115,137,120,132),(121,149,126,144),(122,148,127,143),(123,147,128,142),(124,146,129,141),(125,145,130,150)], [(1,151,30,11),(2,152,21,12),(3,153,22,13),(4,154,23,14),(5,155,24,15),(6,156,25,16),(7,157,26,17),(8,158,27,18),(9,159,28,19),(10,160,29,20),(31,150,45,136),(32,141,46,137),(33,142,47,138),(34,143,48,139),(35,144,49,140),(36,145,50,131),(37,146,41,132),(38,147,42,133),(39,148,43,134),(40,149,44,135),(51,116,65,130),(52,117,66,121),(53,118,67,122),(54,119,68,123),(55,120,69,124),(56,111,70,125),(57,112,61,126),(58,113,62,127),(59,114,63,128),(60,115,64,129),(71,110,85,96),(72,101,86,97),(73,102,87,98),(74,103,88,99),(75,104,89,100),(76,105,90,91),(77,106,81,92),(78,107,82,93),(79,108,83,94),(80,109,84,95)], [(1,45,30,31),(2,46,21,32),(3,47,22,33),(4,48,23,34),(5,49,24,35),(6,50,25,36),(7,41,26,37),(8,42,27,38),(9,43,28,39),(10,44,29,40),(11,136,151,150),(12,137,152,141),(13,138,153,142),(14,139,154,143),(15,140,155,144),(16,131,156,145),(17,132,157,146),(18,133,158,147),(19,134,159,148),(20,135,160,149),(51,90,65,76),(52,81,66,77),(53,82,67,78),(54,83,68,79),(55,84,69,80),(56,85,70,71),(57,86,61,72),(58,87,62,73),(59,88,63,74),(60,89,64,75),(91,116,105,130),(92,117,106,121),(93,118,107,122),(94,119,108,123),(95,120,109,124),(96,111,110,125),(97,112,101,126),(98,113,102,127),(99,114,103,128),(100,115,104,129)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 35 | 0 | 0 |
| 0 | 0 | 7 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 35 |
| 0 | 0 | 0 | 0 | 7 | 35 |
| 40 | 40 | 0 | 0 | 0 | 0 |
| 2 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 5 | 0 | 0 |
| 0 | 0 | 34 | 6 | 0 | 0 |
| 0 | 0 | 12 | 31 | 6 | 36 |
| 0 | 0 | 14 | 29 | 7 | 35 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 39 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 38 | 0 | 0 |
| 0 | 0 | 38 | 20 | 0 | 0 |
| 0 | 0 | 40 | 6 | 20 | 3 |
| 0 | 0 | 6 | 1 | 3 | 21 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 27 | 40 | 0 |
| 0 | 0 | 30 | 33 | 0 | 40 |
| 0 | 0 | 27 | 28 | 35 | 14 |
| 0 | 0 | 22 | 14 | 11 | 8 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 39 | 0 | 40 | 0 |
| 0 | 0 | 0 | 39 | 0 | 40 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,7,0,0,0,0,35,35,0,0,0,0,0,0,0,7,0,0,0,0,35,35],[40,2,0,0,0,0,40,1,0,0,0,0,0,0,35,34,12,14,0,0,5,6,31,29,0,0,0,0,6,7,0,0,0,0,36,35],[1,39,0,0,0,0,1,40,0,0,0,0,0,0,21,38,40,6,0,0,38,20,6,1,0,0,0,0,20,3,0,0,0,0,3,21],[1,0,0,0,0,0,1,40,0,0,0,0,0,0,6,30,27,22,0,0,27,33,28,14,0,0,40,0,35,11,0,0,0,40,14,8],[1,0,0,0,0,0,1,40,0,0,0,0,0,0,1,0,39,0,0,0,0,1,0,39,0,0,1,0,40,0,0,0,0,1,0,40] >;
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 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 | 20 | 20 | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | D10 | D10 | C5⋊D4 | 2- (1+4) | Q8.10D10 |
| kernel | C10.442- (1+4) | C23.21D10 | C23.23D10 | Dic5⋊Q8 | D10⋊3Q8 | C20.23D4 | C2×C4○D20 | Q8×C2×C10 | C2×C20 | C22×Q8 | C22×C4 | C2×Q8 | C2×C4 | C10 | C2 |
| # reps | 1 | 1 | 4 | 2 | 4 | 2 | 1 | 1 | 4 | 2 | 6 | 8 | 16 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_{10}._{44}2_-^{(1+4)} % in TeX
G:=Group("C10.44ES-(2,2)"); // GroupNames label
G:=SmallGroup(320,1488);
// by ID
G=gap.SmallGroup(320,1488);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,675,570,136,12550]);
// 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=a^-1,a*d=d*a,a*e=e*a,c*b*c^-1=a^5*b^-1,d*b*d^-1=e*b*e^-1=a^5*b,d*c*d^-1=e*c*e^-1=a^5*c,e*d*e^-1=a^5*b^2*d>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀