metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22.F5⋊4C4, (C22×C4).3F5, C22.3(C4×F5), (C22×C20).9C4, C22.4(C4⋊F5), C23.35(C2×F5), Dic5.8(C4⋊C4), (C2×C10).15C42, (C2×Dic5).12Q8, C2.2(C23.F5), C5⋊2(C22.C42), (C2×Dic5).104D4, C10.2(C4.D4), (C22×Dic5).8C4, C10.5(C4.10D4), Dic5.6(C22⋊C4), C22.40(C22⋊F5), C2.19(D10.3Q8), C2.3(Dic5.D4), C10.19(C2.C42), (C22×Dic5).174C22, (C2×C10).18(C4⋊C4), (C2×C22.F5).2C2, (C22×C10).48(C2×C4), (C2×Dic5).45(C2×C4), (C2×C10.D4).2C2, (C2×C10).32(C22⋊C4), SmallGroup(320,257)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22.(C4×F5)
G = < a,b,c,d,e | a2=b2=c4=d5=1, e4=b, eae-1=ab=ba, ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ece-1=abc, ede-1=d3 >
Subgroups: 354 in 98 conjugacy classes, 38 normal (24 characteristic)
C1, C2 [×3], C2 [×2], C4 [×6], C22 [×3], C22 [×2], C5, C8 [×4], C2×C4 [×10], C23, C10 [×3], C10 [×2], C4⋊C4 [×2], C2×C8 [×2], M4(2) [×6], C22×C4, C22×C4 [×2], Dic5 [×4], Dic5, C20, C2×C10 [×3], C2×C10 [×2], C2×C4⋊C4, C2×M4(2) [×2], C5⋊C8 [×4], C2×Dic5 [×2], C2×Dic5 [×4], C2×Dic5 [×2], C2×C20 [×2], C22×C10, C22.C42, C10.D4 [×2], C2×C5⋊C8 [×2], C22.F5 [×4], C22.F5 [×2], C22×Dic5 [×2], C22×C20, C2×C10.D4, C2×C22.F5 [×2], C22.(C4×F5)
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], F5, C2.C42, C4.D4, C4.10D4, C2×F5, C22.C42, C4×F5, C4⋊F5, C22⋊F5, Dic5.D4, D10.3Q8, C23.F5, C22.(C4×F5)
►On 160 points
Generators in S160
(1 88)(2 85)(3 82)(4 87)(5 84)(6 81)(7 86)(8 83)(9 130)(10 135)(11 132)(12 129)(13 134)(14 131)(15 136)(16 133)(17 123)(18 128)(19 125)(20 122)(21 127)(22 124)(23 121)(24 126)(25 150)(26 147)(27 152)(28 149)(29 146)(30 151)(31 148)(32 145)(33 138)(34 143)(35 140)(36 137)(37 142)(38 139)(39 144)(40 141)(41 116)(42 113)(43 118)(44 115)(45 120)(46 117)(47 114)(48 119)(49 67)(50 72)(51 69)(52 66)(53 71)(54 68)(55 65)(56 70)(57 111)(58 108)(59 105)(60 110)(61 107)(62 112)(63 109)(64 106)(73 92)(74 89)(75 94)(76 91)(77 96)(78 93)(79 90)(80 95)(97 154)(98 159)(99 156)(100 153)(101 158)(102 155)(103 160)(104 157)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)(113 117)(114 118)(115 119)(116 120)(121 125)(122 126)(123 127)(124 128)(129 133)(130 134)(131 135)(132 136)(137 141)(138 142)(139 143)(140 144)(145 149)(146 150)(147 151)(148 152)(153 157)(154 158)(155 159)(156 160)
(1 78 88 93)(2 90 81 75)(3 76 82 91)(4 96 83 73)(5 74 84 89)(6 94 85 79)(7 80 86 95)(8 92 87 77)(9 110 130 60)(10 57 131 107)(11 108 132 58)(12 63 133 105)(13 106 134 64)(14 61 135 111)(15 112 136 62)(16 59 129 109)(17 35 127 144)(18 137 128 36)(19 33 121 142)(20 143 122 34)(21 39 123 140)(22 141 124 40)(23 37 125 138)(24 139 126 38)(25 65 150 55)(26 52 151 70)(27 71 152 53)(28 50 145 68)(29 69 146 51)(30 56 147 66)(31 67 148 49)(32 54 149 72)(41 156 120 103)(42 104 113 157)(43 154 114 101)(44 102 115 155)(45 160 116 99)(46 100 117 153)(47 158 118 97)(48 98 119 159)
(1 159 9 36 65)(2 37 160 66 10)(3 67 38 11 153)(4 12 68 154 39)(5 155 13 40 69)(6 33 156 70 14)(7 71 34 15 157)(8 16 72 158 35)(17 77 109 149 47)(18 150 78 48 110)(19 41 151 111 79)(20 112 42 80 152)(21 73 105 145 43)(22 146 74 44 106)(23 45 147 107 75)(24 108 46 76 148)(25 93 119 60 128)(26 61 94 121 120)(27 122 62 113 95)(28 114 123 96 63)(29 89 115 64 124)(30 57 90 125 116)(31 126 58 117 91)(32 118 127 92 59)(49 139 132 100 82)(50 101 140 83 133)(51 84 102 134 141)(52 135 85 142 103)(53 143 136 104 86)(54 97 144 87 129)(55 88 98 130 137)(56 131 81 138 99)
(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)
G:=sub<Sym(160)| (1,88)(2,85)(3,82)(4,87)(5,84)(6,81)(7,86)(8,83)(9,130)(10,135)(11,132)(12,129)(13,134)(14,131)(15,136)(16,133)(17,123)(18,128)(19,125)(20,122)(21,127)(22,124)(23,121)(24,126)(25,150)(26,147)(27,152)(28,149)(29,146)(30,151)(31,148)(32,145)(33,138)(34,143)(35,140)(36,137)(37,142)(38,139)(39,144)(40,141)(41,116)(42,113)(43,118)(44,115)(45,120)(46,117)(47,114)(48,119)(49,67)(50,72)(51,69)(52,66)(53,71)(54,68)(55,65)(56,70)(57,111)(58,108)(59,105)(60,110)(61,107)(62,112)(63,109)(64,106)(73,92)(74,89)(75,94)(76,91)(77,96)(78,93)(79,90)(80,95)(97,154)(98,159)(99,156)(100,153)(101,158)(102,155)(103,160)(104,157), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144)(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)(156,160), (1,78,88,93)(2,90,81,75)(3,76,82,91)(4,96,83,73)(5,74,84,89)(6,94,85,79)(7,80,86,95)(8,92,87,77)(9,110,130,60)(10,57,131,107)(11,108,132,58)(12,63,133,105)(13,106,134,64)(14,61,135,111)(15,112,136,62)(16,59,129,109)(17,35,127,144)(18,137,128,36)(19,33,121,142)(20,143,122,34)(21,39,123,140)(22,141,124,40)(23,37,125,138)(24,139,126,38)(25,65,150,55)(26,52,151,70)(27,71,152,53)(28,50,145,68)(29,69,146,51)(30,56,147,66)(31,67,148,49)(32,54,149,72)(41,156,120,103)(42,104,113,157)(43,154,114,101)(44,102,115,155)(45,160,116,99)(46,100,117,153)(47,158,118,97)(48,98,119,159), (1,159,9,36,65)(2,37,160,66,10)(3,67,38,11,153)(4,12,68,154,39)(5,155,13,40,69)(6,33,156,70,14)(7,71,34,15,157)(8,16,72,158,35)(17,77,109,149,47)(18,150,78,48,110)(19,41,151,111,79)(20,112,42,80,152)(21,73,105,145,43)(22,146,74,44,106)(23,45,147,107,75)(24,108,46,76,148)(25,93,119,60,128)(26,61,94,121,120)(27,122,62,113,95)(28,114,123,96,63)(29,89,115,64,124)(30,57,90,125,116)(31,126,58,117,91)(32,118,127,92,59)(49,139,132,100,82)(50,101,140,83,133)(51,84,102,134,141)(52,135,85,142,103)(53,143,136,104,86)(54,97,144,87,129)(55,88,98,130,137)(56,131,81,138,99), (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)>;
G:=Group( (1,88)(2,85)(3,82)(4,87)(5,84)(6,81)(7,86)(8,83)(9,130)(10,135)(11,132)(12,129)(13,134)(14,131)(15,136)(16,133)(17,123)(18,128)(19,125)(20,122)(21,127)(22,124)(23,121)(24,126)(25,150)(26,147)(27,152)(28,149)(29,146)(30,151)(31,148)(32,145)(33,138)(34,143)(35,140)(36,137)(37,142)(38,139)(39,144)(40,141)(41,116)(42,113)(43,118)(44,115)(45,120)(46,117)(47,114)(48,119)(49,67)(50,72)(51,69)(52,66)(53,71)(54,68)(55,65)(56,70)(57,111)(58,108)(59,105)(60,110)(61,107)(62,112)(63,109)(64,106)(73,92)(74,89)(75,94)(76,91)(77,96)(78,93)(79,90)(80,95)(97,154)(98,159)(99,156)(100,153)(101,158)(102,155)(103,160)(104,157), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144)(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)(156,160), (1,78,88,93)(2,90,81,75)(3,76,82,91)(4,96,83,73)(5,74,84,89)(6,94,85,79)(7,80,86,95)(8,92,87,77)(9,110,130,60)(10,57,131,107)(11,108,132,58)(12,63,133,105)(13,106,134,64)(14,61,135,111)(15,112,136,62)(16,59,129,109)(17,35,127,144)(18,137,128,36)(19,33,121,142)(20,143,122,34)(21,39,123,140)(22,141,124,40)(23,37,125,138)(24,139,126,38)(25,65,150,55)(26,52,151,70)(27,71,152,53)(28,50,145,68)(29,69,146,51)(30,56,147,66)(31,67,148,49)(32,54,149,72)(41,156,120,103)(42,104,113,157)(43,154,114,101)(44,102,115,155)(45,160,116,99)(46,100,117,153)(47,158,118,97)(48,98,119,159), (1,159,9,36,65)(2,37,160,66,10)(3,67,38,11,153)(4,12,68,154,39)(5,155,13,40,69)(6,33,156,70,14)(7,71,34,15,157)(8,16,72,158,35)(17,77,109,149,47)(18,150,78,48,110)(19,41,151,111,79)(20,112,42,80,152)(21,73,105,145,43)(22,146,74,44,106)(23,45,147,107,75)(24,108,46,76,148)(25,93,119,60,128)(26,61,94,121,120)(27,122,62,113,95)(28,114,123,96,63)(29,89,115,64,124)(30,57,90,125,116)(31,126,58,117,91)(32,118,127,92,59)(49,139,132,100,82)(50,101,140,83,133)(51,84,102,134,141)(52,135,85,142,103)(53,143,136,104,86)(54,97,144,87,129)(55,88,98,130,137)(56,131,81,138,99), (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) );
G=PermutationGroup([(1,88),(2,85),(3,82),(4,87),(5,84),(6,81),(7,86),(8,83),(9,130),(10,135),(11,132),(12,129),(13,134),(14,131),(15,136),(16,133),(17,123),(18,128),(19,125),(20,122),(21,127),(22,124),(23,121),(24,126),(25,150),(26,147),(27,152),(28,149),(29,146),(30,151),(31,148),(32,145),(33,138),(34,143),(35,140),(36,137),(37,142),(38,139),(39,144),(40,141),(41,116),(42,113),(43,118),(44,115),(45,120),(46,117),(47,114),(48,119),(49,67),(50,72),(51,69),(52,66),(53,71),(54,68),(55,65),(56,70),(57,111),(58,108),(59,105),(60,110),(61,107),(62,112),(63,109),(64,106),(73,92),(74,89),(75,94),(76,91),(77,96),(78,93),(79,90),(80,95),(97,154),(98,159),(99,156),(100,153),(101,158),(102,155),(103,160),(104,157)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64),(65,69),(66,70),(67,71),(68,72),(73,77),(74,78),(75,79),(76,80),(81,85),(82,86),(83,87),(84,88),(89,93),(90,94),(91,95),(92,96),(97,101),(98,102),(99,103),(100,104),(105,109),(106,110),(107,111),(108,112),(113,117),(114,118),(115,119),(116,120),(121,125),(122,126),(123,127),(124,128),(129,133),(130,134),(131,135),(132,136),(137,141),(138,142),(139,143),(140,144),(145,149),(146,150),(147,151),(148,152),(153,157),(154,158),(155,159),(156,160)], [(1,78,88,93),(2,90,81,75),(3,76,82,91),(4,96,83,73),(5,74,84,89),(6,94,85,79),(7,80,86,95),(8,92,87,77),(9,110,130,60),(10,57,131,107),(11,108,132,58),(12,63,133,105),(13,106,134,64),(14,61,135,111),(15,112,136,62),(16,59,129,109),(17,35,127,144),(18,137,128,36),(19,33,121,142),(20,143,122,34),(21,39,123,140),(22,141,124,40),(23,37,125,138),(24,139,126,38),(25,65,150,55),(26,52,151,70),(27,71,152,53),(28,50,145,68),(29,69,146,51),(30,56,147,66),(31,67,148,49),(32,54,149,72),(41,156,120,103),(42,104,113,157),(43,154,114,101),(44,102,115,155),(45,160,116,99),(46,100,117,153),(47,158,118,97),(48,98,119,159)], [(1,159,9,36,65),(2,37,160,66,10),(3,67,38,11,153),(4,12,68,154,39),(5,155,13,40,69),(6,33,156,70,14),(7,71,34,15,157),(8,16,72,158,35),(17,77,109,149,47),(18,150,78,48,110),(19,41,151,111,79),(20,112,42,80,152),(21,73,105,145,43),(22,146,74,44,106),(23,45,147,107,75),(24,108,46,76,148),(25,93,119,60,128),(26,61,94,121,120),(27,122,62,113,95),(28,114,123,96,63),(29,89,115,64,124),(30,57,90,125,116),(31,126,58,117,91),(32,118,127,92,59),(49,139,132,100,82),(50,101,140,83,133),(51,84,102,134,141),(52,135,85,142,103),(53,143,136,104,86),(54,97,144,87,129),(55,88,98,130,137),(56,131,81,138,99)], [(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)])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5 | 8A | ··· | 8H | 10A | ··· | 10G | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 4 | 20 | ··· | 20 | 4 | ··· | 4 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | + | - | + | + | - | ||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | F5 | C4.D4 | C4.10D4 | C2×F5 | C4×F5 | C4⋊F5 | C22⋊F5 | Dic5.D4 | C23.F5 |
| kernel | C22.(C4×F5) | C2×C10.D4 | C2×C22.F5 | C22.F5 | C22×Dic5 | C22×C20 | C2×Dic5 | C2×Dic5 | C22×C4 | C10 | C10 | C23 | C22 | C22 | C22 | C2 | C2 |
| # reps | 1 | 1 | 2 | 8 | 2 | 2 | 3 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of C22.(C4×F5) ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 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 |
| 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 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 25 | 18 | 0 | 0 | 0 | 0 |
| 29 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 38 | 32 | 0 | 0 |
| 0 | 0 | 33 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 9 |
| 0 | 0 | 0 | 0 | 8 | 38 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 1 | 0 | 0 |
| 0 | 0 | 10 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 34 |
| 0 | 0 | 0 | 0 | 12 | 19 |
| 13 | 13 | 0 | 0 | 0 | 0 |
| 9 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 14 | 9 | 0 | 0 |
| 0 | 0 | 10 | 27 | 0 | 0 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[25,29,0,0,0,0,18,16,0,0,0,0,0,0,38,33,0,0,0,0,32,3,0,0,0,0,0,0,3,8,0,0,0,0,9,38],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,31,10,0,0,0,0,1,3,0,0,0,0,0,0,28,12,0,0,0,0,34,19],[13,9,0,0,0,0,13,28,0,0,0,0,0,0,0,0,14,10,0,0,0,0,9,27,0,0,40,0,0,0,0,0,0,40,0,0] >;
C22.(C4×F5) in GAP, Magma, Sage, TeX
C_2^2.(C_4\times F_5)
% in TeX
G:=Group("C2^2.(C4xF5)"); // GroupNames label
G:=SmallGroup(320,257);
// by ID
G=gap.SmallGroup(320,257);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,387,184,1684,6278,3156]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^5=1,e^4=b,e*a*e^-1=a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=a*b*c,e*d*e^-1=d^3>;
// generators/relations