?
G = C10.522+ (1+4) order 320 = 26·5
52nd non-split extension by C10 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.522+ (1+4), C5⋊D4⋊3Q8, C4⋊C4.97D10, C5⋊6(D4⋊3Q8), C22⋊Q8⋊13D5, C22.2(Q8×D5), (C2×Q8).75D10, D10.22(C2×Q8), D10⋊Q8⋊22C2, D10⋊3Q8⋊18C2, (C2×C20).59C23, C22⋊C4.61D10, Dic5.24(C2×Q8), Dic5⋊Q8⋊16C2, Dic5⋊3Q8⋊27C2, C10.38(C22×Q8), (C2×C10).180C24, Dic5⋊4D4.3C2, (C22×C4).242D10, C2.54(D4⋊6D10), Dic5.40(C4○D4), Dic5.Q8⋊19C2, C4⋊Dic5.217C22, (Q8×C10).111C22, (C2×Dic5).91C23, C22.201(C23×D5), C23.193(C22×D5), Dic5.14D4⋊26C2, D10⋊C4.25C22, (C22×C10).208C23, (C22×C20).380C22, (C4×Dic5).117C22, C10.D4.30C22, (C22×D5).212C23, C23.D5.120C22, (C2×Dic10).167C22, (C22×Dic5).121C22, (D5×C4⋊C4)⋊28C2, C2.21(C2×Q8×D5), C2.51(D5×C4○D4), (C2×C10).9(C2×Q8), (C4×C5⋊D4).18C2, (C5×C22⋊Q8)⋊16C2, C10.163(C2×C4○D4), (C2×C4×D5).109C22, (C2×C4).50(C22×D5), (C2×C10.D4)⋊41C2, (C5×C4⋊C4).162C22, (C2×C5⋊D4).135C22, (C5×C22⋊C4).35C22, SmallGroup(320,1308)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 742 in 228 conjugacy classes, 105 normal (91 characteristic)
C1, C2 [×3], C2 [×4], C4 [×15], C22, C22 [×2], C22 [×6], C5, C2×C4 [×6], C2×C4 [×15], D4 [×4], Q8 [×4], C23, C23, D5 [×2], C10 [×3], C10 [×2], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4 [×3], C4⋊C4 [×13], C22×C4, C22×C4 [×5], C2×D4, C2×Q8, C2×Q8 [×2], Dic5 [×4], Dic5 [×5], C20 [×6], D10 [×2], D10 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C4⋊C4 [×2], C4×D4 [×3], C4×Q8, C22⋊Q8, C22⋊Q8 [×5], C42.C2 [×2], C4⋊Q8, Dic10 [×3], C4×D5 [×4], C2×Dic5 [×7], C2×Dic5 [×3], C5⋊D4 [×4], C2×C20 [×6], C2×C20, C5×Q8, C22×D5, C22×C10, D4⋊3Q8, C4×Dic5 [×3], C10.D4 [×11], C4⋊Dic5 [×2], D10⋊C4 [×3], C23.D5, C5×C22⋊C4 [×2], C5×C4⋊C4 [×3], C2×Dic10 [×2], C2×C4×D5 [×3], C22×Dic5 [×2], C2×C5⋊D4, C22×C20, Q8×C10, Dic5.14D4 [×2], Dic5⋊4D4 [×2], Dic5⋊3Q8, Dic5.Q8 [×2], D5×C4⋊C4, D10⋊Q8 [×2], C2×C10.D4, C4×C5⋊D4, Dic5⋊Q8, D10⋊3Q8, C5×C22⋊Q8, C10.522+ (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, Q8×D5 [×2], C23×D5, D4⋊6D10, C2×Q8×D5, D5×C4○D4, C10.522+ (1+4)
Generators and relations
G = < a,b,c,d,e | a10=b4=e2=1, c2=a5, d2=b2, ab=ba, ac=ca, dad-1=eae=a-1, cbc-1=b-1, dbd-1=a5b, be=eb, cd=dc, ce=ec, ede=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 90 25 75)(2 81 26 76)(3 82 27 77)(4 83 28 78)(5 84 29 79)(6 85 30 80)(7 86 21 71)(8 87 22 72)(9 88 23 73)(10 89 24 74)(11 100 160 105)(12 91 151 106)(13 92 152 107)(14 93 153 108)(15 94 154 109)(16 95 155 110)(17 96 156 101)(18 97 157 102)(19 98 158 103)(20 99 159 104)(31 66 46 51)(32 67 47 52)(33 68 48 53)(34 69 49 54)(35 70 50 55)(36 61 41 56)(37 62 42 57)(38 63 43 58)(39 64 44 59)(40 65 45 60)(111 136 126 141)(112 137 127 142)(113 138 128 143)(114 139 129 144)(115 140 130 145)(116 131 121 146)(117 132 122 147)(118 133 123 148)(119 134 124 149)(120 135 125 150)
(1 55 6 60)(2 56 7 51)(3 57 8 52)(4 58 9 53)(5 59 10 54)(11 125 16 130)(12 126 17 121)(13 127 18 122)(14 128 19 123)(15 129 20 124)(21 66 26 61)(22 67 27 62)(23 68 28 63)(24 69 29 64)(25 70 30 65)(31 76 36 71)(32 77 37 72)(33 78 38 73)(34 79 39 74)(35 80 40 75)(41 86 46 81)(42 87 47 82)(43 88 48 83)(44 89 49 84)(45 90 50 85)(91 136 96 131)(92 137 97 132)(93 138 98 133)(94 139 99 134)(95 140 100 135)(101 146 106 141)(102 147 107 142)(103 148 108 143)(104 149 109 144)(105 150 110 145)(111 156 116 151)(112 157 117 152)(113 158 118 153)(114 159 119 154)(115 160 120 155)
(1 135 25 150)(2 134 26 149)(3 133 27 148)(4 132 28 147)(5 131 29 146)(6 140 30 145)(7 139 21 144)(8 138 22 143)(9 137 23 142)(10 136 24 141)(11 35 160 50)(12 34 151 49)(13 33 152 48)(14 32 153 47)(15 31 154 46)(16 40 155 45)(17 39 156 44)(18 38 157 43)(19 37 158 42)(20 36 159 41)(51 99 66 104)(52 98 67 103)(53 97 68 102)(54 96 69 101)(55 95 70 110)(56 94 61 109)(57 93 62 108)(58 92 63 107)(59 91 64 106)(60 100 65 105)(71 119 86 124)(72 118 87 123)(73 117 88 122)(74 116 89 121)(75 115 90 130)(76 114 81 129)(77 113 82 128)(78 112 83 127)(79 111 84 126)(80 120 85 125)
(2 10)(3 9)(4 8)(5 7)(11 155)(12 154)(13 153)(14 152)(15 151)(16 160)(17 159)(18 158)(19 157)(20 156)(21 29)(22 28)(23 27)(24 26)(31 39)(32 38)(33 37)(34 36)(41 49)(42 48)(43 47)(44 46)(51 59)(52 58)(53 57)(54 56)(61 69)(62 68)(63 67)(64 66)(71 79)(72 78)(73 77)(74 76)(81 89)(82 88)(83 87)(84 86)(91 109)(92 108)(93 107)(94 106)(95 105)(96 104)(97 103)(98 102)(99 101)(100 110)(111 129)(112 128)(113 127)(114 126)(115 125)(116 124)(117 123)(118 122)(119 121)(120 130)(131 149)(132 148)(133 147)(134 146)(135 145)(136 144)(137 143)(138 142)(139 141)(140 150)
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), (1,90,25,75)(2,81,26,76)(3,82,27,77)(4,83,28,78)(5,84,29,79)(6,85,30,80)(7,86,21,71)(8,87,22,72)(9,88,23,73)(10,89,24,74)(11,100,160,105)(12,91,151,106)(13,92,152,107)(14,93,153,108)(15,94,154,109)(16,95,155,110)(17,96,156,101)(18,97,157,102)(19,98,158,103)(20,99,159,104)(31,66,46,51)(32,67,47,52)(33,68,48,53)(34,69,49,54)(35,70,50,55)(36,61,41,56)(37,62,42,57)(38,63,43,58)(39,64,44,59)(40,65,45,60)(111,136,126,141)(112,137,127,142)(113,138,128,143)(114,139,129,144)(115,140,130,145)(116,131,121,146)(117,132,122,147)(118,133,123,148)(119,134,124,149)(120,135,125,150), (1,55,6,60)(2,56,7,51)(3,57,8,52)(4,58,9,53)(5,59,10,54)(11,125,16,130)(12,126,17,121)(13,127,18,122)(14,128,19,123)(15,129,20,124)(21,66,26,61)(22,67,27,62)(23,68,28,63)(24,69,29,64)(25,70,30,65)(31,76,36,71)(32,77,37,72)(33,78,38,73)(34,79,39,74)(35,80,40,75)(41,86,46,81)(42,87,47,82)(43,88,48,83)(44,89,49,84)(45,90,50,85)(91,136,96,131)(92,137,97,132)(93,138,98,133)(94,139,99,134)(95,140,100,135)(101,146,106,141)(102,147,107,142)(103,148,108,143)(104,149,109,144)(105,150,110,145)(111,156,116,151)(112,157,117,152)(113,158,118,153)(114,159,119,154)(115,160,120,155), (1,135,25,150)(2,134,26,149)(3,133,27,148)(4,132,28,147)(5,131,29,146)(6,140,30,145)(7,139,21,144)(8,138,22,143)(9,137,23,142)(10,136,24,141)(11,35,160,50)(12,34,151,49)(13,33,152,48)(14,32,153,47)(15,31,154,46)(16,40,155,45)(17,39,156,44)(18,38,157,43)(19,37,158,42)(20,36,159,41)(51,99,66,104)(52,98,67,103)(53,97,68,102)(54,96,69,101)(55,95,70,110)(56,94,61,109)(57,93,62,108)(58,92,63,107)(59,91,64,106)(60,100,65,105)(71,119,86,124)(72,118,87,123)(73,117,88,122)(74,116,89,121)(75,115,90,130)(76,114,81,129)(77,113,82,128)(78,112,83,127)(79,111,84,126)(80,120,85,125), (2,10)(3,9)(4,8)(5,7)(11,155)(12,154)(13,153)(14,152)(15,151)(16,160)(17,159)(18,158)(19,157)(20,156)(21,29)(22,28)(23,27)(24,26)(31,39)(32,38)(33,37)(34,36)(41,49)(42,48)(43,47)(44,46)(51,59)(52,58)(53,57)(54,56)(61,69)(62,68)(63,67)(64,66)(71,79)(72,78)(73,77)(74,76)(81,89)(82,88)(83,87)(84,86)(91,109)(92,108)(93,107)(94,106)(95,105)(96,104)(97,103)(98,102)(99,101)(100,110)(111,129)(112,128)(113,127)(114,126)(115,125)(116,124)(117,123)(118,122)(119,121)(120,130)(131,149)(132,148)(133,147)(134,146)(135,145)(136,144)(137,143)(138,142)(139,141)(140,150)>;
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), (1,90,25,75)(2,81,26,76)(3,82,27,77)(4,83,28,78)(5,84,29,79)(6,85,30,80)(7,86,21,71)(8,87,22,72)(9,88,23,73)(10,89,24,74)(11,100,160,105)(12,91,151,106)(13,92,152,107)(14,93,153,108)(15,94,154,109)(16,95,155,110)(17,96,156,101)(18,97,157,102)(19,98,158,103)(20,99,159,104)(31,66,46,51)(32,67,47,52)(33,68,48,53)(34,69,49,54)(35,70,50,55)(36,61,41,56)(37,62,42,57)(38,63,43,58)(39,64,44,59)(40,65,45,60)(111,136,126,141)(112,137,127,142)(113,138,128,143)(114,139,129,144)(115,140,130,145)(116,131,121,146)(117,132,122,147)(118,133,123,148)(119,134,124,149)(120,135,125,150), (1,55,6,60)(2,56,7,51)(3,57,8,52)(4,58,9,53)(5,59,10,54)(11,125,16,130)(12,126,17,121)(13,127,18,122)(14,128,19,123)(15,129,20,124)(21,66,26,61)(22,67,27,62)(23,68,28,63)(24,69,29,64)(25,70,30,65)(31,76,36,71)(32,77,37,72)(33,78,38,73)(34,79,39,74)(35,80,40,75)(41,86,46,81)(42,87,47,82)(43,88,48,83)(44,89,49,84)(45,90,50,85)(91,136,96,131)(92,137,97,132)(93,138,98,133)(94,139,99,134)(95,140,100,135)(101,146,106,141)(102,147,107,142)(103,148,108,143)(104,149,109,144)(105,150,110,145)(111,156,116,151)(112,157,117,152)(113,158,118,153)(114,159,119,154)(115,160,120,155), (1,135,25,150)(2,134,26,149)(3,133,27,148)(4,132,28,147)(5,131,29,146)(6,140,30,145)(7,139,21,144)(8,138,22,143)(9,137,23,142)(10,136,24,141)(11,35,160,50)(12,34,151,49)(13,33,152,48)(14,32,153,47)(15,31,154,46)(16,40,155,45)(17,39,156,44)(18,38,157,43)(19,37,158,42)(20,36,159,41)(51,99,66,104)(52,98,67,103)(53,97,68,102)(54,96,69,101)(55,95,70,110)(56,94,61,109)(57,93,62,108)(58,92,63,107)(59,91,64,106)(60,100,65,105)(71,119,86,124)(72,118,87,123)(73,117,88,122)(74,116,89,121)(75,115,90,130)(76,114,81,129)(77,113,82,128)(78,112,83,127)(79,111,84,126)(80,120,85,125), (2,10)(3,9)(4,8)(5,7)(11,155)(12,154)(13,153)(14,152)(15,151)(16,160)(17,159)(18,158)(19,157)(20,156)(21,29)(22,28)(23,27)(24,26)(31,39)(32,38)(33,37)(34,36)(41,49)(42,48)(43,47)(44,46)(51,59)(52,58)(53,57)(54,56)(61,69)(62,68)(63,67)(64,66)(71,79)(72,78)(73,77)(74,76)(81,89)(82,88)(83,87)(84,86)(91,109)(92,108)(93,107)(94,106)(95,105)(96,104)(97,103)(98,102)(99,101)(100,110)(111,129)(112,128)(113,127)(114,126)(115,125)(116,124)(117,123)(118,122)(119,121)(120,130)(131,149)(132,148)(133,147)(134,146)(135,145)(136,144)(137,143)(138,142)(139,141)(140,150) );
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)], [(1,90,25,75),(2,81,26,76),(3,82,27,77),(4,83,28,78),(5,84,29,79),(6,85,30,80),(7,86,21,71),(8,87,22,72),(9,88,23,73),(10,89,24,74),(11,100,160,105),(12,91,151,106),(13,92,152,107),(14,93,153,108),(15,94,154,109),(16,95,155,110),(17,96,156,101),(18,97,157,102),(19,98,158,103),(20,99,159,104),(31,66,46,51),(32,67,47,52),(33,68,48,53),(34,69,49,54),(35,70,50,55),(36,61,41,56),(37,62,42,57),(38,63,43,58),(39,64,44,59),(40,65,45,60),(111,136,126,141),(112,137,127,142),(113,138,128,143),(114,139,129,144),(115,140,130,145),(116,131,121,146),(117,132,122,147),(118,133,123,148),(119,134,124,149),(120,135,125,150)], [(1,55,6,60),(2,56,7,51),(3,57,8,52),(4,58,9,53),(5,59,10,54),(11,125,16,130),(12,126,17,121),(13,127,18,122),(14,128,19,123),(15,129,20,124),(21,66,26,61),(22,67,27,62),(23,68,28,63),(24,69,29,64),(25,70,30,65),(31,76,36,71),(32,77,37,72),(33,78,38,73),(34,79,39,74),(35,80,40,75),(41,86,46,81),(42,87,47,82),(43,88,48,83),(44,89,49,84),(45,90,50,85),(91,136,96,131),(92,137,97,132),(93,138,98,133),(94,139,99,134),(95,140,100,135),(101,146,106,141),(102,147,107,142),(103,148,108,143),(104,149,109,144),(105,150,110,145),(111,156,116,151),(112,157,117,152),(113,158,118,153),(114,159,119,154),(115,160,120,155)], [(1,135,25,150),(2,134,26,149),(3,133,27,148),(4,132,28,147),(5,131,29,146),(6,140,30,145),(7,139,21,144),(8,138,22,143),(9,137,23,142),(10,136,24,141),(11,35,160,50),(12,34,151,49),(13,33,152,48),(14,32,153,47),(15,31,154,46),(16,40,155,45),(17,39,156,44),(18,38,157,43),(19,37,158,42),(20,36,159,41),(51,99,66,104),(52,98,67,103),(53,97,68,102),(54,96,69,101),(55,95,70,110),(56,94,61,109),(57,93,62,108),(58,92,63,107),(59,91,64,106),(60,100,65,105),(71,119,86,124),(72,118,87,123),(73,117,88,122),(74,116,89,121),(75,115,90,130),(76,114,81,129),(77,113,82,128),(78,112,83,127),(79,111,84,126),(80,120,85,125)], [(2,10),(3,9),(4,8),(5,7),(11,155),(12,154),(13,153),(14,152),(15,151),(16,160),(17,159),(18,158),(19,157),(20,156),(21,29),(22,28),(23,27),(24,26),(31,39),(32,38),(33,37),(34,36),(41,49),(42,48),(43,47),(44,46),(51,59),(52,58),(53,57),(54,56),(61,69),(62,68),(63,67),(64,66),(71,79),(72,78),(73,77),(74,76),(81,89),(82,88),(83,87),(84,86),(91,109),(92,108),(93,107),(94,106),(95,105),(96,104),(97,103),(98,102),(99,101),(100,110),(111,129),(112,128),(113,127),(114,126),(115,125),(116,124),(117,123),(118,122),(119,121),(120,130),(131,149),(132,148),(133,147),(134,146),(135,145),(136,144),(137,143),(138,142),(139,141),(140,150)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 35 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 6 |
| 0 | 0 | 0 | 0 | 19 | 7 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 38 |
| 0 | 0 | 0 | 0 | 38 | 20 |
| 6 | 39 | 0 | 0 | 0 | 0 |
| 38 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 7 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 3 |
| 0 | 0 | 0 | 0 | 3 | 21 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 6 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,7,34,0,0,0,0,7,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,35,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,19,0,0,0,0,6,7],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,21,38,0,0,0,0,38,20],[6,38,0,0,0,0,39,35,0,0,0,0,0,0,40,7,0,0,0,0,0,1,0,0,0,0,0,0,20,3,0,0,0,0,3,21],[1,6,0,0,0,0,0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | ··· | 4G | 4H | ··· | 4M | 4N | 4O | 4P | 4Q | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 2 | 2 | 4 | ··· | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | C4○D4 | D10 | D10 | D10 | D10 | 2+ (1+4) | Q8×D5 | D4⋊6D10 | D5×C4○D4 |
| kernel | C10.522+ (1+4) | Dic5.14D4 | Dic5⋊4D4 | Dic5⋊3Q8 | Dic5.Q8 | D5×C4⋊C4 | D10⋊Q8 | C2×C10.D4 | C4×C5⋊D4 | Dic5⋊Q8 | D10⋊3Q8 | C5×C22⋊Q8 | C5⋊D4 | C22⋊Q8 | Dic5 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C10 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 4 | 4 | 6 | 2 | 2 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_{10}._{52}2_+^{(1+4)} % in TeX
G:=Group("C10.52ES+(2,2)"); // GroupNames label
G:=SmallGroup(320,1308);
// by ID
G=gap.SmallGroup(320,1308);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,100,570,409,80,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=e^2=1,c^2=a^5,d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^5*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=a^5*b^2*d>;
// generators/relations
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