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G = C22.58(D4×D5)  order 320 = 26·5

9th central extension by C22 of D4×D5

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C22.58(D4×D5), C2.C421D5, (C22×D5).102D4, (C22×C4).297D10, C2.8(C42⋊D5), C52(C23.34D4), C10.10C421C2, D10.34(C22⋊C4), C22.33(C4○D20), (C22×C20).12C22, (C23×D5).93C22, C23.254(C22×D5), C10.23(C42⋊C2), C2.2(D10.13D4), C2.3(D10.12D4), C22.35(D42D5), (C22×C10).289C23, C22.16(Q82D5), C10.8(C22.D4), (C22×Dic5).13C22, (C2×C4×D5)⋊12C4, C2.6(D5×C22⋊C4), C22.88(C2×C4×D5), (C2×C4).124(C4×D5), (D5×C22×C4).12C2, (C2×C20).315(C2×C4), C2.7(C4⋊C47D5), (C2×C10).198(C2×D4), C10.44(C2×C22⋊C4), (C2×D10⋊C4).3C2, (C22×D5).92(C2×C4), (C2×C10).182(C4○D4), (C5×C2.C42)⋊18C2, (C2×C10).149(C22×C4), (C2×Dic5).132(C2×C4), SmallGroup(320,291)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C22.58(D4×D5)
C1C5C10C2×C10C22×C10C23×D5D5×C22×C4 — C22.58(D4×D5)
C5C2×C10 — C22.58(D4×D5)
C1C23C2.C42

Generators and relations for C22.58(D4×D5)
 G = < a,b,c,d,e,f | a2=b2=c4=e5=f2=1, d2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fcf=bc=cb, bd=db, be=eb, bf=fb, dcd-1=abc-1, ce=ec, de=ed, df=fd, fef=e-1 >

Subgroups: 862 in 218 conjugacy classes, 71 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×8], C22 [×3], C22 [×4], C22 [×16], C5, C2×C4 [×2], C2×C4 [×26], C23, C23 [×10], D5 [×4], C10 [×3], C10 [×4], C22⋊C4 [×4], C22×C4, C22×C4 [×2], C22×C4 [×11], C24, Dic5 [×4], C20 [×4], D10 [×4], D10 [×12], C2×C10 [×3], C2×C10 [×4], C2.C42, C2.C42 [×3], C2×C22⋊C4 [×2], C23×C4, C4×D5 [×8], C2×Dic5 [×2], C2×Dic5 [×8], C2×C20 [×2], C2×C20 [×8], C22×D5 [×6], C22×D5 [×4], C22×C10, C23.34D4, D10⋊C4 [×4], C2×C4×D5 [×4], C2×C4×D5 [×4], C22×Dic5, C22×Dic5 [×2], C22×C20, C22×C20 [×2], C23×D5, C10.10C42, C10.10C42 [×2], C5×C2.C42, C2×D10⋊C4 [×2], D5×C22×C4, C22.58(D4×D5)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4○D4 [×4], D10 [×3], C2×C22⋊C4, C42⋊C2 [×2], C22.D4 [×4], C4×D5 [×2], C22×D5, C23.34D4, C2×C4×D5, C4○D20 [×2], D4×D5 [×2], D42D5, Q82D5, C42⋊D5, D5×C22⋊C4, D10.12D4 [×2], C4⋊C47D5, D10.13D4 [×2], C22.58(D4×D5)

Smallest permutation representation of C22.58(D4×D5)
On 160 points
Generators in S160
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(81 101)(82 102)(83 103)(84 104)(85 105)(86 106)(87 107)(88 108)(89 109)(90 110)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)(121 141)(122 142)(123 143)(124 144)(125 145)(126 146)(127 147)(128 148)(129 149)(130 150)(131 151)(132 152)(133 153)(134 154)(135 155)(136 156)(137 157)(138 158)(139 159)(140 160)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)(141 151)(142 152)(143 153)(144 154)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)
(1 156 36 121)(2 157 37 122)(3 158 38 123)(4 159 39 124)(5 160 40 125)(6 151 31 126)(7 152 32 127)(8 153 33 128)(9 154 34 129)(10 155 35 130)(11 146 26 131)(12 147 27 132)(13 148 28 133)(14 149 29 134)(15 150 30 135)(16 141 21 136)(17 142 22 137)(18 143 23 138)(19 144 24 139)(20 145 25 140)(41 96 76 101)(42 97 77 102)(43 98 78 103)(44 99 79 104)(45 100 80 105)(46 91 71 106)(47 92 72 107)(48 93 73 108)(49 94 74 109)(50 95 75 110)(51 86 66 111)(52 87 67 112)(53 88 68 113)(54 89 69 114)(55 90 70 115)(56 81 61 116)(57 82 62 117)(58 83 63 118)(59 84 64 119)(60 85 65 120)
(1 91 11 81)(2 92 12 82)(3 93 13 83)(4 94 14 84)(5 95 15 85)(6 96 16 86)(7 97 17 87)(8 98 18 88)(9 99 19 89)(10 100 20 90)(21 111 31 101)(22 112 32 102)(23 113 33 103)(24 114 34 104)(25 115 35 105)(26 116 36 106)(27 117 37 107)(28 118 38 108)(29 119 39 109)(30 120 40 110)(41 131 51 121)(42 132 52 122)(43 133 53 123)(44 134 54 124)(45 135 55 125)(46 136 56 126)(47 137 57 127)(48 138 58 128)(49 139 59 129)(50 140 60 130)(61 151 71 141)(62 152 72 142)(63 153 73 143)(64 154 74 144)(65 155 75 145)(66 156 76 146)(67 157 77 147)(68 158 78 148)(69 159 79 149)(70 160 80 150)
(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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 40)(12 39)(13 38)(14 37)(15 36)(16 35)(17 34)(18 33)(19 32)(20 31)(41 80)(42 79)(43 78)(44 77)(45 76)(46 75)(47 74)(48 73)(49 72)(50 71)(51 70)(52 69)(53 68)(54 67)(55 66)(56 65)(57 64)(58 63)(59 62)(60 61)(81 110)(82 109)(83 108)(84 107)(85 106)(86 105)(87 104)(88 103)(89 102)(90 101)(91 120)(92 119)(93 118)(94 117)(95 116)(96 115)(97 114)(98 113)(99 112)(100 111)(121 160)(122 159)(123 158)(124 157)(125 156)(126 155)(127 154)(128 153)(129 152)(130 151)(131 150)(132 149)(133 148)(134 147)(135 146)(136 145)(137 144)(138 143)(139 142)(140 141)

G:=sub<Sym(160)| (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(81,101)(82,102)(83,103)(84,104)(85,105)(86,106)(87,107)(88,108)(89,109)(90,110)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(121,141)(122,142)(123,143)(124,144)(125,145)(126,146)(127,147)(128,148)(129,149)(130,150)(131,151)(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(141,151)(142,152)(143,153)(144,154)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160), (1,156,36,121)(2,157,37,122)(3,158,38,123)(4,159,39,124)(5,160,40,125)(6,151,31,126)(7,152,32,127)(8,153,33,128)(9,154,34,129)(10,155,35,130)(11,146,26,131)(12,147,27,132)(13,148,28,133)(14,149,29,134)(15,150,30,135)(16,141,21,136)(17,142,22,137)(18,143,23,138)(19,144,24,139)(20,145,25,140)(41,96,76,101)(42,97,77,102)(43,98,78,103)(44,99,79,104)(45,100,80,105)(46,91,71,106)(47,92,72,107)(48,93,73,108)(49,94,74,109)(50,95,75,110)(51,86,66,111)(52,87,67,112)(53,88,68,113)(54,89,69,114)(55,90,70,115)(56,81,61,116)(57,82,62,117)(58,83,63,118)(59,84,64,119)(60,85,65,120), (1,91,11,81)(2,92,12,82)(3,93,13,83)(4,94,14,84)(5,95,15,85)(6,96,16,86)(7,97,17,87)(8,98,18,88)(9,99,19,89)(10,100,20,90)(21,111,31,101)(22,112,32,102)(23,113,33,103)(24,114,34,104)(25,115,35,105)(26,116,36,106)(27,117,37,107)(28,118,38,108)(29,119,39,109)(30,120,40,110)(41,131,51,121)(42,132,52,122)(43,133,53,123)(44,134,54,124)(45,135,55,125)(46,136,56,126)(47,137,57,127)(48,138,58,128)(49,139,59,129)(50,140,60,130)(61,151,71,141)(62,152,72,142)(63,153,73,143)(64,154,74,144)(65,155,75,145)(66,156,76,146)(67,157,77,147)(68,158,78,148)(69,159,79,149)(70,160,80,150), (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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)(81,110)(82,109)(83,108)(84,107)(85,106)(86,105)(87,104)(88,103)(89,102)(90,101)(91,120)(92,119)(93,118)(94,117)(95,116)(96,115)(97,114)(98,113)(99,112)(100,111)(121,160)(122,159)(123,158)(124,157)(125,156)(126,155)(127,154)(128,153)(129,152)(130,151)(131,150)(132,149)(133,148)(134,147)(135,146)(136,145)(137,144)(138,143)(139,142)(140,141)>;

G:=Group( (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(81,101)(82,102)(83,103)(84,104)(85,105)(86,106)(87,107)(88,108)(89,109)(90,110)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(121,141)(122,142)(123,143)(124,144)(125,145)(126,146)(127,147)(128,148)(129,149)(130,150)(131,151)(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(141,151)(142,152)(143,153)(144,154)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160), (1,156,36,121)(2,157,37,122)(3,158,38,123)(4,159,39,124)(5,160,40,125)(6,151,31,126)(7,152,32,127)(8,153,33,128)(9,154,34,129)(10,155,35,130)(11,146,26,131)(12,147,27,132)(13,148,28,133)(14,149,29,134)(15,150,30,135)(16,141,21,136)(17,142,22,137)(18,143,23,138)(19,144,24,139)(20,145,25,140)(41,96,76,101)(42,97,77,102)(43,98,78,103)(44,99,79,104)(45,100,80,105)(46,91,71,106)(47,92,72,107)(48,93,73,108)(49,94,74,109)(50,95,75,110)(51,86,66,111)(52,87,67,112)(53,88,68,113)(54,89,69,114)(55,90,70,115)(56,81,61,116)(57,82,62,117)(58,83,63,118)(59,84,64,119)(60,85,65,120), (1,91,11,81)(2,92,12,82)(3,93,13,83)(4,94,14,84)(5,95,15,85)(6,96,16,86)(7,97,17,87)(8,98,18,88)(9,99,19,89)(10,100,20,90)(21,111,31,101)(22,112,32,102)(23,113,33,103)(24,114,34,104)(25,115,35,105)(26,116,36,106)(27,117,37,107)(28,118,38,108)(29,119,39,109)(30,120,40,110)(41,131,51,121)(42,132,52,122)(43,133,53,123)(44,134,54,124)(45,135,55,125)(46,136,56,126)(47,137,57,127)(48,138,58,128)(49,139,59,129)(50,140,60,130)(61,151,71,141)(62,152,72,142)(63,153,73,143)(64,154,74,144)(65,155,75,145)(66,156,76,146)(67,157,77,147)(68,158,78,148)(69,159,79,149)(70,160,80,150), (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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)(81,110)(82,109)(83,108)(84,107)(85,106)(86,105)(87,104)(88,103)(89,102)(90,101)(91,120)(92,119)(93,118)(94,117)(95,116)(96,115)(97,114)(98,113)(99,112)(100,111)(121,160)(122,159)(123,158)(124,157)(125,156)(126,155)(127,154)(128,153)(129,152)(130,151)(131,150)(132,149)(133,148)(134,147)(135,146)(136,145)(137,144)(138,143)(139,142)(140,141) );

G=PermutationGroup([(1,21),(2,22),(3,23),(4,24),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68),(49,69),(50,70),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80),(81,101),(82,102),(83,103),(84,104),(85,105),(86,106),(87,107),(88,108),(89,109),(90,110),(91,111),(92,112),(93,113),(94,114),(95,115),(96,116),(97,117),(98,118),(99,119),(100,120),(121,141),(122,142),(123,143),(124,144),(125,145),(126,146),(127,147),(128,148),(129,149),(130,150),(131,151),(132,152),(133,153),(134,154),(135,155),(136,156),(137,157),(138,158),(139,159),(140,160)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80),(81,91),(82,92),(83,93),(84,94),(85,95),(86,96),(87,97),(88,98),(89,99),(90,100),(101,111),(102,112),(103,113),(104,114),(105,115),(106,116),(107,117),(108,118),(109,119),(110,120),(121,131),(122,132),(123,133),(124,134),(125,135),(126,136),(127,137),(128,138),(129,139),(130,140),(141,151),(142,152),(143,153),(144,154),(145,155),(146,156),(147,157),(148,158),(149,159),(150,160)], [(1,156,36,121),(2,157,37,122),(3,158,38,123),(4,159,39,124),(5,160,40,125),(6,151,31,126),(7,152,32,127),(8,153,33,128),(9,154,34,129),(10,155,35,130),(11,146,26,131),(12,147,27,132),(13,148,28,133),(14,149,29,134),(15,150,30,135),(16,141,21,136),(17,142,22,137),(18,143,23,138),(19,144,24,139),(20,145,25,140),(41,96,76,101),(42,97,77,102),(43,98,78,103),(44,99,79,104),(45,100,80,105),(46,91,71,106),(47,92,72,107),(48,93,73,108),(49,94,74,109),(50,95,75,110),(51,86,66,111),(52,87,67,112),(53,88,68,113),(54,89,69,114),(55,90,70,115),(56,81,61,116),(57,82,62,117),(58,83,63,118),(59,84,64,119),(60,85,65,120)], [(1,91,11,81),(2,92,12,82),(3,93,13,83),(4,94,14,84),(5,95,15,85),(6,96,16,86),(7,97,17,87),(8,98,18,88),(9,99,19,89),(10,100,20,90),(21,111,31,101),(22,112,32,102),(23,113,33,103),(24,114,34,104),(25,115,35,105),(26,116,36,106),(27,117,37,107),(28,118,38,108),(29,119,39,109),(30,120,40,110),(41,131,51,121),(42,132,52,122),(43,133,53,123),(44,134,54,124),(45,135,55,125),(46,136,56,126),(47,137,57,127),(48,138,58,128),(49,139,59,129),(50,140,60,130),(61,151,71,141),(62,152,72,142),(63,153,73,143),(64,154,74,144),(65,155,75,145),(66,156,76,146),(67,157,77,147),(68,158,78,148),(69,159,79,149),(70,160,80,150)], [(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)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,21),(11,40),(12,39),(13,38),(14,37),(15,36),(16,35),(17,34),(18,33),(19,32),(20,31),(41,80),(42,79),(43,78),(44,77),(45,76),(46,75),(47,74),(48,73),(49,72),(50,71),(51,70),(52,69),(53,68),(54,67),(55,66),(56,65),(57,64),(58,63),(59,62),(60,61),(81,110),(82,109),(83,108),(84,107),(85,106),(86,105),(87,104),(88,103),(89,102),(90,101),(91,120),(92,119),(93,118),(94,117),(95,116),(96,115),(97,114),(98,113),(99,112),(100,111),(121,160),(122,159),(123,158),(124,157),(125,156),(126,155),(127,154),(128,153),(129,152),(130,151),(131,150),(132,149),(133,148),(134,147),(135,146),(136,145),(137,144),(138,143),(139,142),(140,141)])

68 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P5A5B10A···10N20A···20X
order12···2222244444444444444445510···1020···20
size11···110101010222244441010101020202020222···24···4

68 irreducible representations

dim111111222222444
type+++++++++-+
imageC1C2C2C2C2C4D4D5C4○D4D10C4×D5C4○D20D4×D5D42D5Q82D5
kernelC22.58(D4×D5)C10.10C42C5×C2.C42C2×D10⋊C4D5×C22×C4C2×C4×D5C22×D5C2.C42C2×C10C22×C4C2×C4C22C22C22C22
# reps1312184286816422

Matrix representation of C22.58(D4×D5) in GL6(𝔽41)

4000000
0400000
001000
000100
000010
000001
,
100000
010000
0040000
0004000
000010
000001
,
3200000
0320000
000900
009000
00002015
00001721
,
4000000
0400000
009000
000900
00004012
000001
,
6400000
100000
001000
000100
000010
000001
,
3510000
660000
001000
0004000
0000400
0000040

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,9,0,0,0,0,9,0,0,0,0,0,0,0,20,17,0,0,0,0,15,21],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,12,1],[6,1,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[35,6,0,0,0,0,1,6,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;

C22.58(D4×D5) in GAP, Magma, Sage, TeX

C_2^2._{58}(D_4\times D_5)
% in TeX

G:=Group("C2^2.58(D4xD5)");
// GroupNames label

G:=SmallGroup(320,291);
// by ID

G=gap.SmallGroup(320,291);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,422,387,58,12550]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^4=e^5=f^2=1,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,f*c*f=b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d^-1=a*b*c^-1,c*e=e*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations

׿
×
𝔽