?
G = C10.102+ (1+4) order 320 = 26·5
10th non-split extension by C10 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.102+ (1+4), C5⋊D4⋊4Q8, C20⋊Q8⋊10C2, C5⋊1(D4⋊3Q8), C4⋊C4.264D10, D10⋊Q8⋊2C2, D10⋊2Q8⋊9C2, C22.8(Q8×D5), D10.19(C2×Q8), Dic5⋊3Q8⋊9C2, C4.92(C4○D20), (C2×C10).55C24, Dic5.20(C2×Q8), C20.194(C4○D4), C20.48D4⋊18C2, C10.26(C22×Q8), (C2×C20).138C23, Dic5.Q8⋊1C2, (C22×C4).180D10, C2.13(D4⋊6D10), C22.89(C23×D5), C4⋊Dic5.192C22, C23.227(C22×D5), C23.D5.88C22, D10⋊C4.94C22, (C22×C20).103C22, (C22×C10).404C23, (C2×Dic5).201C23, (C4×Dic5).212C22, (C22×D5).170C23, (C2×Dic10).147C22, C10.D4.149C22, C2.9(C2×Q8×D5), (D5×C4⋊C4)⋊10C2, (C2×C4⋊C4)⋊20D5, (C10×C4⋊C4)⋊17C2, (C4×C5⋊D4).3C2, C10.22(C2×C4○D4), C2.24(C2×C4○D20), (C2×C10).95(C2×Q8), (C2×C4×D5).242C22, (C5×C4⋊C4).297C22, (C2×C4).573(C22×D5), (C2×C5⋊D4).159C22, SmallGroup(320,1183)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 726 in 228 conjugacy classes, 107 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×13], C22, C22 [×2], C22 [×6], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×15], D4 [×4], Q8 [×4], C23, C23, D5 [×2], C10 [×3], C10 [×2], C42 [×3], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×12], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×D4, C2×Q8 [×3], Dic5 [×2], Dic5 [×6], C20 [×2], C20 [×5], D10 [×2], D10 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C4⋊C4, C2×C4⋊C4, C4×D4 [×3], C4×Q8, C22⋊Q8 [×6], C42.C2 [×2], C4⋊Q8, Dic10 [×4], C4×D5 [×4], C2×Dic5 [×3], C2×Dic5 [×4], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×4], C22×D5, C22×C10, D4⋊3Q8, C4×Dic5, C4×Dic5 [×2], C10.D4, C10.D4 [×8], C4⋊Dic5, C4⋊Dic5 [×2], D10⋊C4, D10⋊C4 [×2], C23.D5, C23.D5 [×2], C5×C4⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10, C2×Dic10 [×2], C2×C4×D5, C2×C4×D5 [×2], C2×C5⋊D4, C22×C20, C22×C20 [×2], Dic5⋊3Q8, C20⋊Q8, Dic5.Q8 [×2], D5×C4⋊C4, D10⋊Q8 [×2], D10⋊2Q8, C20.48D4, C20.48D4 [×2], C4×C5⋊D4, C4×C5⋊D4 [×2], C10×C4⋊C4, C10.102+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C4○D4 [×2], C24, D10 [×7], C22×Q8, C2×C4○D4, 2+ (1+4), C22×D5 [×7], D4⋊3Q8, C4○D20 [×2], Q8×D5 [×2], C23×D5, C2×C4○D20, D4⋊6D10, C2×Q8×D5, C10.102+ (1+4)
Generators and relations
G = < a,b,c,d,e | a10=b4=c2=1, d2=b2, e2=a5, bab-1=a-1, ac=ca, ad=da, ae=ea, cbc=a5b-1, dbd-1=a5b, be=eb, cd=dc, ce=ec, 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 11 30 155)(2 20 21 154)(3 19 22 153)(4 18 23 152)(5 17 24 151)(6 16 25 160)(7 15 26 159)(8 14 27 158)(9 13 28 157)(10 12 29 156)(31 139 41 149)(32 138 42 148)(33 137 43 147)(34 136 44 146)(35 135 45 145)(36 134 46 144)(37 133 47 143)(38 132 48 142)(39 131 49 141)(40 140 50 150)(51 129 61 119)(52 128 62 118)(53 127 63 117)(54 126 64 116)(55 125 65 115)(56 124 66 114)(57 123 67 113)(58 122 68 112)(59 121 69 111)(60 130 70 120)(71 94 81 104)(72 93 82 103)(73 92 83 102)(74 91 84 101)(75 100 85 110)(76 99 86 109)(77 98 87 108)(78 97 88 107)(79 96 89 106)(80 95 90 105)
(1 40)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 145)(12 146)(13 147)(14 148)(15 149)(16 150)(17 141)(18 142)(19 143)(20 144)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(29 49)(30 50)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(61 81)(62 82)(63 83)(64 84)(65 85)(66 86)(67 87)(68 88)(69 89)(70 90)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)(101 121)(102 122)(103 123)(104 124)(105 125)(106 126)(107 127)(108 128)(109 129)(110 130)(131 151)(132 152)(133 153)(134 154)(135 155)(136 156)(137 157)(138 158)(139 159)(140 160)
(1 130 30 120)(2 121 21 111)(3 122 22 112)(4 123 23 113)(5 124 24 114)(6 125 25 115)(7 126 26 116)(8 127 27 117)(9 128 28 118)(10 129 29 119)(11 65 155 55)(12 66 156 56)(13 67 157 57)(14 68 158 58)(15 69 159 59)(16 70 160 60)(17 61 151 51)(18 62 152 52)(19 63 153 53)(20 64 154 54)(31 101 41 91)(32 102 42 92)(33 103 43 93)(34 104 44 94)(35 105 45 95)(36 106 46 96)(37 107 47 97)(38 108 48 98)(39 109 49 99)(40 110 50 100)(71 141 81 131)(72 142 82 132)(73 143 83 133)(74 144 84 134)(75 145 85 135)(76 146 86 136)(77 147 87 137)(78 148 88 138)(79 149 89 139)(80 150 90 140)
(1 145 6 150)(2 146 7 141)(3 147 8 142)(4 148 9 143)(5 149 10 144)(11 35 16 40)(12 36 17 31)(13 37 18 32)(14 38 19 33)(15 39 20 34)(21 136 26 131)(22 137 27 132)(23 138 28 133)(24 139 29 134)(25 140 30 135)(41 156 46 151)(42 157 47 152)(43 158 48 153)(44 159 49 154)(45 160 50 155)(51 101 56 106)(52 102 57 107)(53 103 58 108)(54 104 59 109)(55 105 60 110)(61 91 66 96)(62 92 67 97)(63 93 68 98)(64 94 69 99)(65 95 70 100)(71 121 76 126)(72 122 77 127)(73 123 78 128)(74 124 79 129)(75 125 80 130)(81 111 86 116)(82 112 87 117)(83 113 88 118)(84 114 89 119)(85 115 90 120)
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), (1,11,30,155)(2,20,21,154)(3,19,22,153)(4,18,23,152)(5,17,24,151)(6,16,25,160)(7,15,26,159)(8,14,27,158)(9,13,28,157)(10,12,29,156)(31,139,41,149)(32,138,42,148)(33,137,43,147)(34,136,44,146)(35,135,45,145)(36,134,46,144)(37,133,47,143)(38,132,48,142)(39,131,49,141)(40,140,50,150)(51,129,61,119)(52,128,62,118)(53,127,63,117)(54,126,64,116)(55,125,65,115)(56,124,66,114)(57,123,67,113)(58,122,68,112)(59,121,69,111)(60,130,70,120)(71,94,81,104)(72,93,82,103)(73,92,83,102)(74,91,84,101)(75,100,85,110)(76,99,86,109)(77,98,87,108)(78,97,88,107)(79,96,89,106)(80,95,90,105), (1,40)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,145)(12,146)(13,147)(14,148)(15,149)(16,150)(17,141)(18,142)(19,143)(20,144)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(101,121)(102,122)(103,123)(104,124)(105,125)(106,126)(107,127)(108,128)(109,129)(110,130)(131,151)(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160), (1,130,30,120)(2,121,21,111)(3,122,22,112)(4,123,23,113)(5,124,24,114)(6,125,25,115)(7,126,26,116)(8,127,27,117)(9,128,28,118)(10,129,29,119)(11,65,155,55)(12,66,156,56)(13,67,157,57)(14,68,158,58)(15,69,159,59)(16,70,160,60)(17,61,151,51)(18,62,152,52)(19,63,153,53)(20,64,154,54)(31,101,41,91)(32,102,42,92)(33,103,43,93)(34,104,44,94)(35,105,45,95)(36,106,46,96)(37,107,47,97)(38,108,48,98)(39,109,49,99)(40,110,50,100)(71,141,81,131)(72,142,82,132)(73,143,83,133)(74,144,84,134)(75,145,85,135)(76,146,86,136)(77,147,87,137)(78,148,88,138)(79,149,89,139)(80,150,90,140), (1,145,6,150)(2,146,7,141)(3,147,8,142)(4,148,9,143)(5,149,10,144)(11,35,16,40)(12,36,17,31)(13,37,18,32)(14,38,19,33)(15,39,20,34)(21,136,26,131)(22,137,27,132)(23,138,28,133)(24,139,29,134)(25,140,30,135)(41,156,46,151)(42,157,47,152)(43,158,48,153)(44,159,49,154)(45,160,50,155)(51,101,56,106)(52,102,57,107)(53,103,58,108)(54,104,59,109)(55,105,60,110)(61,91,66,96)(62,92,67,97)(63,93,68,98)(64,94,69,99)(65,95,70,100)(71,121,76,126)(72,122,77,127)(73,123,78,128)(74,124,79,129)(75,125,80,130)(81,111,86,116)(82,112,87,117)(83,113,88,118)(84,114,89,119)(85,115,90,120)>;
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), (1,11,30,155)(2,20,21,154)(3,19,22,153)(4,18,23,152)(5,17,24,151)(6,16,25,160)(7,15,26,159)(8,14,27,158)(9,13,28,157)(10,12,29,156)(31,139,41,149)(32,138,42,148)(33,137,43,147)(34,136,44,146)(35,135,45,145)(36,134,46,144)(37,133,47,143)(38,132,48,142)(39,131,49,141)(40,140,50,150)(51,129,61,119)(52,128,62,118)(53,127,63,117)(54,126,64,116)(55,125,65,115)(56,124,66,114)(57,123,67,113)(58,122,68,112)(59,121,69,111)(60,130,70,120)(71,94,81,104)(72,93,82,103)(73,92,83,102)(74,91,84,101)(75,100,85,110)(76,99,86,109)(77,98,87,108)(78,97,88,107)(79,96,89,106)(80,95,90,105), (1,40)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,145)(12,146)(13,147)(14,148)(15,149)(16,150)(17,141)(18,142)(19,143)(20,144)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(101,121)(102,122)(103,123)(104,124)(105,125)(106,126)(107,127)(108,128)(109,129)(110,130)(131,151)(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160), (1,130,30,120)(2,121,21,111)(3,122,22,112)(4,123,23,113)(5,124,24,114)(6,125,25,115)(7,126,26,116)(8,127,27,117)(9,128,28,118)(10,129,29,119)(11,65,155,55)(12,66,156,56)(13,67,157,57)(14,68,158,58)(15,69,159,59)(16,70,160,60)(17,61,151,51)(18,62,152,52)(19,63,153,53)(20,64,154,54)(31,101,41,91)(32,102,42,92)(33,103,43,93)(34,104,44,94)(35,105,45,95)(36,106,46,96)(37,107,47,97)(38,108,48,98)(39,109,49,99)(40,110,50,100)(71,141,81,131)(72,142,82,132)(73,143,83,133)(74,144,84,134)(75,145,85,135)(76,146,86,136)(77,147,87,137)(78,148,88,138)(79,149,89,139)(80,150,90,140), (1,145,6,150)(2,146,7,141)(3,147,8,142)(4,148,9,143)(5,149,10,144)(11,35,16,40)(12,36,17,31)(13,37,18,32)(14,38,19,33)(15,39,20,34)(21,136,26,131)(22,137,27,132)(23,138,28,133)(24,139,29,134)(25,140,30,135)(41,156,46,151)(42,157,47,152)(43,158,48,153)(44,159,49,154)(45,160,50,155)(51,101,56,106)(52,102,57,107)(53,103,58,108)(54,104,59,109)(55,105,60,110)(61,91,66,96)(62,92,67,97)(63,93,68,98)(64,94,69,99)(65,95,70,100)(71,121,76,126)(72,122,77,127)(73,123,78,128)(74,124,79,129)(75,125,80,130)(81,111,86,116)(82,112,87,117)(83,113,88,118)(84,114,89,119)(85,115,90,120) );
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)], [(1,11,30,155),(2,20,21,154),(3,19,22,153),(4,18,23,152),(5,17,24,151),(6,16,25,160),(7,15,26,159),(8,14,27,158),(9,13,28,157),(10,12,29,156),(31,139,41,149),(32,138,42,148),(33,137,43,147),(34,136,44,146),(35,135,45,145),(36,134,46,144),(37,133,47,143),(38,132,48,142),(39,131,49,141),(40,140,50,150),(51,129,61,119),(52,128,62,118),(53,127,63,117),(54,126,64,116),(55,125,65,115),(56,124,66,114),(57,123,67,113),(58,122,68,112),(59,121,69,111),(60,130,70,120),(71,94,81,104),(72,93,82,103),(73,92,83,102),(74,91,84,101),(75,100,85,110),(76,99,86,109),(77,98,87,108),(78,97,88,107),(79,96,89,106),(80,95,90,105)], [(1,40),(2,31),(3,32),(4,33),(5,34),(6,35),(7,36),(8,37),(9,38),(10,39),(11,145),(12,146),(13,147),(14,148),(15,149),(16,150),(17,141),(18,142),(19,143),(20,144),(21,41),(22,42),(23,43),(24,44),(25,45),(26,46),(27,47),(28,48),(29,49),(30,50),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80),(61,81),(62,82),(63,83),(64,84),(65,85),(66,86),(67,87),(68,88),(69,89),(70,90),(91,111),(92,112),(93,113),(94,114),(95,115),(96,116),(97,117),(98,118),(99,119),(100,120),(101,121),(102,122),(103,123),(104,124),(105,125),(106,126),(107,127),(108,128),(109,129),(110,130),(131,151),(132,152),(133,153),(134,154),(135,155),(136,156),(137,157),(138,158),(139,159),(140,160)], [(1,130,30,120),(2,121,21,111),(3,122,22,112),(4,123,23,113),(5,124,24,114),(6,125,25,115),(7,126,26,116),(8,127,27,117),(9,128,28,118),(10,129,29,119),(11,65,155,55),(12,66,156,56),(13,67,157,57),(14,68,158,58),(15,69,159,59),(16,70,160,60),(17,61,151,51),(18,62,152,52),(19,63,153,53),(20,64,154,54),(31,101,41,91),(32,102,42,92),(33,103,43,93),(34,104,44,94),(35,105,45,95),(36,106,46,96),(37,107,47,97),(38,108,48,98),(39,109,49,99),(40,110,50,100),(71,141,81,131),(72,142,82,132),(73,143,83,133),(74,144,84,134),(75,145,85,135),(76,146,86,136),(77,147,87,137),(78,148,88,138),(79,149,89,139),(80,150,90,140)], [(1,145,6,150),(2,146,7,141),(3,147,8,142),(4,148,9,143),(5,149,10,144),(11,35,16,40),(12,36,17,31),(13,37,18,32),(14,38,19,33),(15,39,20,34),(21,136,26,131),(22,137,27,132),(23,138,28,133),(24,139,29,134),(25,140,30,135),(41,156,46,151),(42,157,47,152),(43,158,48,153),(44,159,49,154),(45,160,50,155),(51,101,56,106),(52,102,57,107),(53,103,58,108),(54,104,59,109),(55,105,60,110),(61,91,66,96),(62,92,67,97),(63,93,68,98),(64,94,69,99),(65,95,70,100),(71,121,76,126),(72,122,77,127),(73,123,78,128),(74,124,79,129),(75,125,80,130),(81,111,86,116),(82,112,87,117),(83,113,88,118),(84,114,89,119),(85,115,90,120)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 35 | 0 | 0 |
| 0 | 0 | 6 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 9 | 1 | 0 | 0 | 0 | 0 |
| 2 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 6 | 0 | 0 |
| 0 | 0 | 1 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 21 |
| 0 | 0 | 0 | 0 | 21 | 38 |
| 40 | 9 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 32 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 38 |
| 0 | 0 | 0 | 0 | 38 | 20 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 38 | 20 |
| 0 | 0 | 0 | 0 | 20 | 3 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,6,0,0,0,0,35,35,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,2,0,0,0,0,1,32,0,0,0,0,0,0,35,1,0,0,0,0,6,6,0,0,0,0,0,0,3,21,0,0,0,0,21,38],[40,0,0,0,0,0,9,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,32,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,21,38,0,0,0,0,38,20],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,38,20,0,0,0,0,20,3] >;
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | ··· | 4Q | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 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 | C2 | C2 | Q8 | D5 | C4○D4 | D10 | D10 | C4○D20 | 2+ (1+4) | Q8×D5 | D4⋊6D10 |
| kernel | C10.102+ (1+4) | Dic5⋊3Q8 | C20⋊Q8 | Dic5.Q8 | D5×C4⋊C4 | D10⋊Q8 | D10⋊2Q8 | C20.48D4 | C4×C5⋊D4 | C10×C4⋊C4 | C5⋊D4 | C2×C4⋊C4 | C20 | C4⋊C4 | C22×C4 | C4 | C10 | C22 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 3 | 3 | 1 | 4 | 2 | 4 | 8 | 6 | 16 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_{10}._{10}2_+^{(1+4)} % in TeX
G:=Group("C10.10ES+(2,2)"); // GroupNames label
G:=SmallGroup(320,1183);
// by ID
G=gap.SmallGroup(320,1183);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,100,1571,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=c^2=1,d^2=b^2,e^2=a^5,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=a^5*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=b^2*d>;
// generators/relations
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ℤ
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