?
G = C10.112+ (1+4) order 320 = 26·5
11st non-split extension by C10 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.112+ (1+4), C20⋊7D4⋊28C2, C4⋊2D20⋊10C2, C4⋊C4.307D10, D20⋊8C4⋊10C2, C4.94(C4○D20), (C2×C10).58C24, C4.Dic10⋊10C2, D10.13D4⋊2C2, C20.196(C4○D4), (C2×C20).619C23, (C22×C4).183D10, C2.14(D4⋊6D10), C22.92(C23×D5), (C2×D20).142C22, C4⋊Dic5.194C22, (C2×Dic5).19C23, (C4×Dic5).74C22, (C22×D5).16C23, C23.229(C22×D5), (C22×C10).407C23, (C22×C20).220C22, C5⋊1(C22.47C24), C22.11(Q8⋊2D5), C23.D5.143C22, D10⋊C4.142C22, C10.D4.151C22, (C2×C4⋊C4)⋊23D5, (C10×C4⋊C4)⋊20C2, C4⋊C4⋊D5⋊2C2, (C4×C5⋊D4)⋊11C2, C4⋊C4⋊7D5⋊10C2, C10.25(C2×C4○D4), C2.27(C2×C4○D20), C2.9(C2×Q8⋊2D5), (C2×C4×D5).66C22, (C5×C4⋊C4).299C22, (C2×C4).146(C22×D5), (C2×C10).198(C4○D4), (C2×C5⋊D4).103C22, SmallGroup(320,1186)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 870 in 238 conjugacy classes, 99 normal (43 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×11], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×13], D4 [×10], C23, C23 [×3], D5 [×3], C10 [×3], C10 [×2], C42 [×3], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×6], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×D4 [×6], Dic5 [×5], C20 [×2], C20 [×5], D10 [×9], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C4⋊C4, C42⋊C2, C4×D4 [×4], C4⋊D4 [×4], C22.D4 [×2], C42.C2, C42⋊2C2 [×2], C4×D5 [×4], D20 [×4], C2×Dic5 [×3], C2×Dic5 [×2], C5⋊D4 [×6], C2×C20 [×2], C2×C20 [×4], C2×C20 [×4], C22×D5, C22×D5 [×2], C22×C10, C22.47C24, C4×Dic5, C4×Dic5 [×2], C10.D4, C10.D4 [×2], C4⋊Dic5, C4⋊Dic5 [×2], D10⋊C4, D10⋊C4 [×8], C23.D5, C5×C4⋊C4 [×2], C5×C4⋊C4 [×2], C2×C4×D5, C2×C4×D5 [×2], C2×D20, C2×D20 [×2], C2×C5⋊D4, C2×C5⋊D4 [×2], C22×C20, C22×C20 [×2], C4.Dic10, C4⋊C4⋊7D5, D20⋊8C4, D10.13D4 [×2], C4⋊2D20, C4⋊C4⋊D5 [×2], C4×C5⋊D4, C4×C5⋊D4 [×2], C20⋊7D4, C20⋊7D4 [×2], C10×C4⋊C4, C10.112+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2+ (1+4), C22×D5 [×7], C22.47C24, C4○D20 [×2], Q8⋊2D5 [×2], C23×D5, C2×C4○D20, D4⋊6D10, C2×Q8⋊2D5, C10.112+ (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, bd=db, 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 90 30 76)(2 89 21 75)(3 88 22 74)(4 87 23 73)(5 86 24 72)(6 85 25 71)(7 84 26 80)(8 83 27 79)(9 82 28 78)(10 81 29 77)(11 105 151 91)(12 104 152 100)(13 103 153 99)(14 102 154 98)(15 101 155 97)(16 110 156 96)(17 109 157 95)(18 108 158 94)(19 107 159 93)(20 106 160 92)(31 56 45 70)(32 55 46 69)(33 54 47 68)(34 53 48 67)(35 52 49 66)(36 51 50 65)(37 60 41 64)(38 59 42 63)(39 58 43 62)(40 57 44 61)(111 136 125 150)(112 135 126 149)(113 134 127 148)(114 133 128 147)(115 132 129 146)(116 131 130 145)(117 140 121 144)(118 139 122 143)(119 138 123 142)(120 137 124 141)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 31)(7 32)(8 33)(9 34)(10 35)(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 96 30 110)(2 97 21 101)(3 98 22 102)(4 99 23 103)(5 100 24 104)(6 91 25 105)(7 92 26 106)(8 93 27 107)(9 94 28 108)(10 95 29 109)(11 71 151 85)(12 72 152 86)(13 73 153 87)(14 74 154 88)(15 75 155 89)(16 76 156 90)(17 77 157 81)(18 78 158 82)(19 79 159 83)(20 80 160 84)(31 111 45 125)(32 112 46 126)(33 113 47 127)(34 114 48 128)(35 115 49 129)(36 116 50 130)(37 117 41 121)(38 118 42 122)(39 119 43 123)(40 120 44 124)(51 131 65 145)(52 132 66 146)(53 133 67 147)(54 134 68 148)(55 135 69 149)(56 136 70 150)(57 137 61 141)(58 138 62 142)(59 139 63 143)(60 140 64 144)
(1 145 6 150)(2 146 7 141)(3 147 8 142)(4 148 9 143)(5 149 10 144)(11 31 16 36)(12 32 17 37)(13 33 18 38)(14 34 19 39)(15 35 20 40)(21 132 26 137)(22 133 27 138)(23 134 28 139)(24 135 29 140)(25 136 30 131)(41 152 46 157)(42 153 47 158)(43 154 48 159)(44 155 49 160)(45 156 50 151)(51 105 56 110)(52 106 57 101)(53 107 58 102)(54 108 59 103)(55 109 60 104)(61 97 66 92)(62 98 67 93)(63 99 68 94)(64 100 69 95)(65 91 70 96)(71 125 76 130)(72 126 77 121)(73 127 78 122)(74 128 79 123)(75 129 80 124)(81 117 86 112)(82 118 87 113)(83 119 88 114)(84 120 89 115)(85 111 90 116)
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), (1,90,30,76)(2,89,21,75)(3,88,22,74)(4,87,23,73)(5,86,24,72)(6,85,25,71)(7,84,26,80)(8,83,27,79)(9,82,28,78)(10,81,29,77)(11,105,151,91)(12,104,152,100)(13,103,153,99)(14,102,154,98)(15,101,155,97)(16,110,156,96)(17,109,157,95)(18,108,158,94)(19,107,159,93)(20,106,160,92)(31,56,45,70)(32,55,46,69)(33,54,47,68)(34,53,48,67)(35,52,49,66)(36,51,50,65)(37,60,41,64)(38,59,42,63)(39,58,43,62)(40,57,44,61)(111,136,125,150)(112,135,126,149)(113,134,127,148)(114,133,128,147)(115,132,129,146)(116,131,130,145)(117,140,121,144)(118,139,122,143)(119,138,123,142)(120,137,124,141), (1,36)(2,37)(3,38)(4,39)(5,40)(6,31)(7,32)(8,33)(9,34)(10,35)(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), (1,96,30,110)(2,97,21,101)(3,98,22,102)(4,99,23,103)(5,100,24,104)(6,91,25,105)(7,92,26,106)(8,93,27,107)(9,94,28,108)(10,95,29,109)(11,71,151,85)(12,72,152,86)(13,73,153,87)(14,74,154,88)(15,75,155,89)(16,76,156,90)(17,77,157,81)(18,78,158,82)(19,79,159,83)(20,80,160,84)(31,111,45,125)(32,112,46,126)(33,113,47,127)(34,114,48,128)(35,115,49,129)(36,116,50,130)(37,117,41,121)(38,118,42,122)(39,119,43,123)(40,120,44,124)(51,131,65,145)(52,132,66,146)(53,133,67,147)(54,134,68,148)(55,135,69,149)(56,136,70,150)(57,137,61,141)(58,138,62,142)(59,139,63,143)(60,140,64,144), (1,145,6,150)(2,146,7,141)(3,147,8,142)(4,148,9,143)(5,149,10,144)(11,31,16,36)(12,32,17,37)(13,33,18,38)(14,34,19,39)(15,35,20,40)(21,132,26,137)(22,133,27,138)(23,134,28,139)(24,135,29,140)(25,136,30,131)(41,152,46,157)(42,153,47,158)(43,154,48,159)(44,155,49,160)(45,156,50,151)(51,105,56,110)(52,106,57,101)(53,107,58,102)(54,108,59,103)(55,109,60,104)(61,97,66,92)(62,98,67,93)(63,99,68,94)(64,100,69,95)(65,91,70,96)(71,125,76,130)(72,126,77,121)(73,127,78,122)(74,128,79,123)(75,129,80,124)(81,117,86,112)(82,118,87,113)(83,119,88,114)(84,120,89,115)(85,111,90,116)>;
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), (1,90,30,76)(2,89,21,75)(3,88,22,74)(4,87,23,73)(5,86,24,72)(6,85,25,71)(7,84,26,80)(8,83,27,79)(9,82,28,78)(10,81,29,77)(11,105,151,91)(12,104,152,100)(13,103,153,99)(14,102,154,98)(15,101,155,97)(16,110,156,96)(17,109,157,95)(18,108,158,94)(19,107,159,93)(20,106,160,92)(31,56,45,70)(32,55,46,69)(33,54,47,68)(34,53,48,67)(35,52,49,66)(36,51,50,65)(37,60,41,64)(38,59,42,63)(39,58,43,62)(40,57,44,61)(111,136,125,150)(112,135,126,149)(113,134,127,148)(114,133,128,147)(115,132,129,146)(116,131,130,145)(117,140,121,144)(118,139,122,143)(119,138,123,142)(120,137,124,141), (1,36)(2,37)(3,38)(4,39)(5,40)(6,31)(7,32)(8,33)(9,34)(10,35)(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), (1,96,30,110)(2,97,21,101)(3,98,22,102)(4,99,23,103)(5,100,24,104)(6,91,25,105)(7,92,26,106)(8,93,27,107)(9,94,28,108)(10,95,29,109)(11,71,151,85)(12,72,152,86)(13,73,153,87)(14,74,154,88)(15,75,155,89)(16,76,156,90)(17,77,157,81)(18,78,158,82)(19,79,159,83)(20,80,160,84)(31,111,45,125)(32,112,46,126)(33,113,47,127)(34,114,48,128)(35,115,49,129)(36,116,50,130)(37,117,41,121)(38,118,42,122)(39,119,43,123)(40,120,44,124)(51,131,65,145)(52,132,66,146)(53,133,67,147)(54,134,68,148)(55,135,69,149)(56,136,70,150)(57,137,61,141)(58,138,62,142)(59,139,63,143)(60,140,64,144), (1,145,6,150)(2,146,7,141)(3,147,8,142)(4,148,9,143)(5,149,10,144)(11,31,16,36)(12,32,17,37)(13,33,18,38)(14,34,19,39)(15,35,20,40)(21,132,26,137)(22,133,27,138)(23,134,28,139)(24,135,29,140)(25,136,30,131)(41,152,46,157)(42,153,47,158)(43,154,48,159)(44,155,49,160)(45,156,50,151)(51,105,56,110)(52,106,57,101)(53,107,58,102)(54,108,59,103)(55,109,60,104)(61,97,66,92)(62,98,67,93)(63,99,68,94)(64,100,69,95)(65,91,70,96)(71,125,76,130)(72,126,77,121)(73,127,78,122)(74,128,79,123)(75,129,80,124)(81,117,86,112)(82,118,87,113)(83,119,88,114)(84,120,89,115)(85,111,90,116) );
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)], [(1,90,30,76),(2,89,21,75),(3,88,22,74),(4,87,23,73),(5,86,24,72),(6,85,25,71),(7,84,26,80),(8,83,27,79),(9,82,28,78),(10,81,29,77),(11,105,151,91),(12,104,152,100),(13,103,153,99),(14,102,154,98),(15,101,155,97),(16,110,156,96),(17,109,157,95),(18,108,158,94),(19,107,159,93),(20,106,160,92),(31,56,45,70),(32,55,46,69),(33,54,47,68),(34,53,48,67),(35,52,49,66),(36,51,50,65),(37,60,41,64),(38,59,42,63),(39,58,43,62),(40,57,44,61),(111,136,125,150),(112,135,126,149),(113,134,127,148),(114,133,128,147),(115,132,129,146),(116,131,130,145),(117,140,121,144),(118,139,122,143),(119,138,123,142),(120,137,124,141)], [(1,36),(2,37),(3,38),(4,39),(5,40),(6,31),(7,32),(8,33),(9,34),(10,35),(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)], [(1,96,30,110),(2,97,21,101),(3,98,22,102),(4,99,23,103),(5,100,24,104),(6,91,25,105),(7,92,26,106),(8,93,27,107),(9,94,28,108),(10,95,29,109),(11,71,151,85),(12,72,152,86),(13,73,153,87),(14,74,154,88),(15,75,155,89),(16,76,156,90),(17,77,157,81),(18,78,158,82),(19,79,159,83),(20,80,160,84),(31,111,45,125),(32,112,46,126),(33,113,47,127),(34,114,48,128),(35,115,49,129),(36,116,50,130),(37,117,41,121),(38,118,42,122),(39,119,43,123),(40,120,44,124),(51,131,65,145),(52,132,66,146),(53,133,67,147),(54,134,68,148),(55,135,69,149),(56,136,70,150),(57,137,61,141),(58,138,62,142),(59,139,63,143),(60,140,64,144)], [(1,145,6,150),(2,146,7,141),(3,147,8,142),(4,148,9,143),(5,149,10,144),(11,31,16,36),(12,32,17,37),(13,33,18,38),(14,34,19,39),(15,35,20,40),(21,132,26,137),(22,133,27,138),(23,134,28,139),(24,135,29,140),(25,136,30,131),(41,152,46,157),(42,153,47,158),(43,154,48,159),(44,155,49,160),(45,156,50,151),(51,105,56,110),(52,106,57,101),(53,107,58,102),(54,108,59,103),(55,109,60,104),(61,97,66,92),(62,98,67,93),(63,99,68,94),(64,100,69,95),(65,91,70,96),(71,125,76,130),(72,126,77,121),(73,127,78,122),(74,128,79,123),(75,129,80,124),(81,117,86,112),(82,118,87,113),(83,119,88,114),(84,120,89,115),(85,111,90,116)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 34 | 34 | 0 | 0 |
| 7 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 11 | 9 | 0 | 0 |
| 14 | 30 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 17 | 40 | 0 | 0 |
| 1 | 24 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 4 |
| 0 | 0 | 20 | 1 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 32 | 36 |
| 0 | 0 | 0 | 9 |
G:=sub<GL(4,GF(41))| [34,7,0,0,34,1,0,0,0,0,40,0,0,0,0,40],[11,14,0,0,9,30,0,0,0,0,9,0,0,0,0,9],[17,1,0,0,40,24,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,40,20,0,0,4,1],[9,0,0,0,0,9,0,0,0,0,32,0,0,0,36,9] >;
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 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 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 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 | D5 | C4○D4 | C4○D4 | D10 | D10 | C4○D20 | 2+ (1+4) | Q8⋊2D5 | D4⋊6D10 |
| kernel | C10.112+ (1+4) | C4.Dic10 | C4⋊C4⋊7D5 | D20⋊8C4 | D10.13D4 | C4⋊2D20 | C4⋊C4⋊D5 | C4×C5⋊D4 | C20⋊7D4 | C10×C4⋊C4 | C2×C4⋊C4 | C20 | C2×C10 | C4⋊C4 | C22×C4 | C4 | C10 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 3 | 3 | 1 | 2 | 4 | 4 | 8 | 6 | 16 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_{10}._{11}2_+^{(1+4)} % in TeX
G:=Group("C10.11ES+(2,2)"); // GroupNames label
G:=SmallGroup(320,1186);
// by ID
G=gap.SmallGroup(320,1186);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,387,100,1571,12550]);
// 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,b*d=d*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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