direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×C22.F5, C20⋊9M4(2), Dic5⋊8M4(2), Dic5.14C42, C5⋊3(C4×M4(2)), C22.5(C4×F5), C23.39(C2×F5), (C22×C4).11F5, (C22×C20).33C4, (C2×C10).21C42, C10.18(C2×C42), (C4×Dic5).48C4, C2.6(D5⋊M4(2)), C10.24(C2×M4(2)), C10.C42⋊17C2, C22.44(C22×F5), Dic5.37(C22×C4), (C22×Dic5).30C4, (C4×Dic5).325C22, (C2×Dic5).341C23, (C22×Dic5).269C22, C5⋊C8⋊3(C2×C4), (C4×C5⋊C8)⋊17C2, C2.18(C2×C4×F5), (C2×C5⋊C8).36C22, (C2×C4).104(C2×F5), (C2×C4×Dic5).51C2, (C2×C20).104(C2×C4), C2.2(C2×C22.F5), (C2×C22.F5).7C2, (C2×C10).57(C22×C4), (C22×C10).57(C2×C4), (C2×Dic5).128(C2×C4), SmallGroup(320,1088)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 378 in 142 conjugacy classes, 74 normal (30 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×2], C5, C8 [×8], C2×C4 [×2], C2×C4 [×12], C23, C10 [×3], C10 [×2], C42 [×4], C2×C8 [×4], M4(2) [×8], C22×C4, C22×C4 [×2], Dic5 [×6], Dic5, C20 [×2], C20, C2×C10, C2×C10 [×2], C2×C10 [×2], C4×C8 [×2], C8⋊C4 [×2], C2×C42, C2×M4(2) [×2], C5⋊C8 [×8], C2×Dic5 [×4], C2×Dic5 [×4], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×2], C22×C10, C4×M4(2), C4×Dic5 [×4], C2×C5⋊C8 [×4], C22.F5 [×8], C22×Dic5 [×2], C22×C20, C4×C5⋊C8 [×2], C10.C42 [×2], C2×C4×Dic5, C2×C22.F5 [×2], C4×C22.F5
Quotients:
C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, C42 [×4], M4(2) [×4], C22×C4 [×3], F5, C2×C42, C2×M4(2) [×2], C2×F5 [×3], C4×M4(2), C4×F5 [×2], C22.F5 [×2], C22×F5, D5⋊M4(2), C2×C4×F5, C2×C22.F5, C4×C22.F5
Generators and relations
G = < a,b,c,d,e | a4=b2=c2=d5=1, e4=c, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d3 >
►On 160 points
Generators in S160
(1 78 121 18)(2 79 122 19)(3 80 123 20)(4 73 124 21)(5 74 125 22)(6 75 126 23)(7 76 127 24)(8 77 128 17)(9 116 160 44)(10 117 153 45)(11 118 154 46)(12 119 155 47)(13 120 156 48)(14 113 157 41)(15 114 158 42)(16 115 159 43)(25 49 60 40)(26 50 61 33)(27 51 62 34)(28 52 63 35)(29 53 64 36)(30 54 57 37)(31 55 58 38)(32 56 59 39)(65 146 102 91)(66 147 103 92)(67 148 104 93)(68 149 97 94)(69 150 98 95)(70 151 99 96)(71 152 100 89)(72 145 101 90)(81 106 130 137)(82 107 131 138)(83 108 132 139)(84 109 133 140)(85 110 134 141)(86 111 135 142)(87 112 136 143)(88 105 129 144)
(1 121)(2 126)(3 123)(4 128)(5 125)(6 122)(7 127)(8 124)(9 156)(10 153)(11 158)(12 155)(13 160)(14 157)(15 154)(16 159)(17 73)(18 78)(19 75)(20 80)(21 77)(22 74)(23 79)(24 76)(25 60)(26 57)(27 62)(28 59)(29 64)(30 61)(31 58)(32 63)(33 54)(34 51)(35 56)(36 53)(37 50)(38 55)(39 52)(40 49)(41 113)(42 118)(43 115)(44 120)(45 117)(46 114)(47 119)(48 116)(65 102)(66 99)(67 104)(68 101)(69 98)(70 103)(71 100)(72 97)(81 130)(82 135)(83 132)(84 129)(85 134)(86 131)(87 136)(88 133)(89 152)(90 149)(91 146)(92 151)(93 148)(94 145)(95 150)(96 147)(105 140)(106 137)(107 142)(108 139)(109 144)(110 141)(111 138)(112 143)
(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)
(1 159 36 83 65)(2 84 160 66 37)(3 67 85 38 153)(4 39 68 154 86)(5 155 40 87 69)(6 88 156 70 33)(7 71 81 34 157)(8 35 72 158 82)(9 103 54 122 133)(10 123 104 134 55)(11 135 124 56 97)(12 49 136 98 125)(13 99 50 126 129)(14 127 100 130 51)(15 131 128 52 101)(16 53 132 102 121)(17 63 90 114 138)(18 115 64 139 91)(19 140 116 92 57)(20 93 141 58 117)(21 59 94 118 142)(22 119 60 143 95)(23 144 120 96 61)(24 89 137 62 113)(25 112 150 74 47)(26 75 105 48 151)(27 41 76 152 106)(28 145 42 107 77)(29 108 146 78 43)(30 79 109 44 147)(31 45 80 148 110)(32 149 46 111 73)
(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)| (1,78,121,18)(2,79,122,19)(3,80,123,20)(4,73,124,21)(5,74,125,22)(6,75,126,23)(7,76,127,24)(8,77,128,17)(9,116,160,44)(10,117,153,45)(11,118,154,46)(12,119,155,47)(13,120,156,48)(14,113,157,41)(15,114,158,42)(16,115,159,43)(25,49,60,40)(26,50,61,33)(27,51,62,34)(28,52,63,35)(29,53,64,36)(30,54,57,37)(31,55,58,38)(32,56,59,39)(65,146,102,91)(66,147,103,92)(67,148,104,93)(68,149,97,94)(69,150,98,95)(70,151,99,96)(71,152,100,89)(72,145,101,90)(81,106,130,137)(82,107,131,138)(83,108,132,139)(84,109,133,140)(85,110,134,141)(86,111,135,142)(87,112,136,143)(88,105,129,144), (1,121)(2,126)(3,123)(4,128)(5,125)(6,122)(7,127)(8,124)(9,156)(10,153)(11,158)(12,155)(13,160)(14,157)(15,154)(16,159)(17,73)(18,78)(19,75)(20,80)(21,77)(22,74)(23,79)(24,76)(25,60)(26,57)(27,62)(28,59)(29,64)(30,61)(31,58)(32,63)(33,54)(34,51)(35,56)(36,53)(37,50)(38,55)(39,52)(40,49)(41,113)(42,118)(43,115)(44,120)(45,117)(46,114)(47,119)(48,116)(65,102)(66,99)(67,104)(68,101)(69,98)(70,103)(71,100)(72,97)(81,130)(82,135)(83,132)(84,129)(85,134)(86,131)(87,136)(88,133)(89,152)(90,149)(91,146)(92,151)(93,148)(94,145)(95,150)(96,147)(105,140)(106,137)(107,142)(108,139)(109,144)(110,141)(111,138)(112,143), (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), (1,159,36,83,65)(2,84,160,66,37)(3,67,85,38,153)(4,39,68,154,86)(5,155,40,87,69)(6,88,156,70,33)(7,71,81,34,157)(8,35,72,158,82)(9,103,54,122,133)(10,123,104,134,55)(11,135,124,56,97)(12,49,136,98,125)(13,99,50,126,129)(14,127,100,130,51)(15,131,128,52,101)(16,53,132,102,121)(17,63,90,114,138)(18,115,64,139,91)(19,140,116,92,57)(20,93,141,58,117)(21,59,94,118,142)(22,119,60,143,95)(23,144,120,96,61)(24,89,137,62,113)(25,112,150,74,47)(26,75,105,48,151)(27,41,76,152,106)(28,145,42,107,77)(29,108,146,78,43)(30,79,109,44,147)(31,45,80,148,110)(32,149,46,111,73), (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( (1,78,121,18)(2,79,122,19)(3,80,123,20)(4,73,124,21)(5,74,125,22)(6,75,126,23)(7,76,127,24)(8,77,128,17)(9,116,160,44)(10,117,153,45)(11,118,154,46)(12,119,155,47)(13,120,156,48)(14,113,157,41)(15,114,158,42)(16,115,159,43)(25,49,60,40)(26,50,61,33)(27,51,62,34)(28,52,63,35)(29,53,64,36)(30,54,57,37)(31,55,58,38)(32,56,59,39)(65,146,102,91)(66,147,103,92)(67,148,104,93)(68,149,97,94)(69,150,98,95)(70,151,99,96)(71,152,100,89)(72,145,101,90)(81,106,130,137)(82,107,131,138)(83,108,132,139)(84,109,133,140)(85,110,134,141)(86,111,135,142)(87,112,136,143)(88,105,129,144), (1,121)(2,126)(3,123)(4,128)(5,125)(6,122)(7,127)(8,124)(9,156)(10,153)(11,158)(12,155)(13,160)(14,157)(15,154)(16,159)(17,73)(18,78)(19,75)(20,80)(21,77)(22,74)(23,79)(24,76)(25,60)(26,57)(27,62)(28,59)(29,64)(30,61)(31,58)(32,63)(33,54)(34,51)(35,56)(36,53)(37,50)(38,55)(39,52)(40,49)(41,113)(42,118)(43,115)(44,120)(45,117)(46,114)(47,119)(48,116)(65,102)(66,99)(67,104)(68,101)(69,98)(70,103)(71,100)(72,97)(81,130)(82,135)(83,132)(84,129)(85,134)(86,131)(87,136)(88,133)(89,152)(90,149)(91,146)(92,151)(93,148)(94,145)(95,150)(96,147)(105,140)(106,137)(107,142)(108,139)(109,144)(110,141)(111,138)(112,143), (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), (1,159,36,83,65)(2,84,160,66,37)(3,67,85,38,153)(4,39,68,154,86)(5,155,40,87,69)(6,88,156,70,33)(7,71,81,34,157)(8,35,72,158,82)(9,103,54,122,133)(10,123,104,134,55)(11,135,124,56,97)(12,49,136,98,125)(13,99,50,126,129)(14,127,100,130,51)(15,131,128,52,101)(16,53,132,102,121)(17,63,90,114,138)(18,115,64,139,91)(19,140,116,92,57)(20,93,141,58,117)(21,59,94,118,142)(22,119,60,143,95)(23,144,120,96,61)(24,89,137,62,113)(25,112,150,74,47)(26,75,105,48,151)(27,41,76,152,106)(28,145,42,107,77)(29,108,146,78,43)(30,79,109,44,147)(31,45,80,148,110)(32,149,46,111,73), (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([(1,78,121,18),(2,79,122,19),(3,80,123,20),(4,73,124,21),(5,74,125,22),(6,75,126,23),(7,76,127,24),(8,77,128,17),(9,116,160,44),(10,117,153,45),(11,118,154,46),(12,119,155,47),(13,120,156,48),(14,113,157,41),(15,114,158,42),(16,115,159,43),(25,49,60,40),(26,50,61,33),(27,51,62,34),(28,52,63,35),(29,53,64,36),(30,54,57,37),(31,55,58,38),(32,56,59,39),(65,146,102,91),(66,147,103,92),(67,148,104,93),(68,149,97,94),(69,150,98,95),(70,151,99,96),(71,152,100,89),(72,145,101,90),(81,106,130,137),(82,107,131,138),(83,108,132,139),(84,109,133,140),(85,110,134,141),(86,111,135,142),(87,112,136,143),(88,105,129,144)], [(1,121),(2,126),(3,123),(4,128),(5,125),(6,122),(7,127),(8,124),(9,156),(10,153),(11,158),(12,155),(13,160),(14,157),(15,154),(16,159),(17,73),(18,78),(19,75),(20,80),(21,77),(22,74),(23,79),(24,76),(25,60),(26,57),(27,62),(28,59),(29,64),(30,61),(31,58),(32,63),(33,54),(34,51),(35,56),(36,53),(37,50),(38,55),(39,52),(40,49),(41,113),(42,118),(43,115),(44,120),(45,117),(46,114),(47,119),(48,116),(65,102),(66,99),(67,104),(68,101),(69,98),(70,103),(71,100),(72,97),(81,130),(82,135),(83,132),(84,129),(85,134),(86,131),(87,136),(88,133),(89,152),(90,149),(91,146),(92,151),(93,148),(94,145),(95,150),(96,147),(105,140),(106,137),(107,142),(108,139),(109,144),(110,141),(111,138),(112,143)], [(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)], [(1,159,36,83,65),(2,84,160,66,37),(3,67,85,38,153),(4,39,68,154,86),(5,155,40,87,69),(6,88,156,70,33),(7,71,81,34,157),(8,35,72,158,82),(9,103,54,122,133),(10,123,104,134,55),(11,135,124,56,97),(12,49,136,98,125),(13,99,50,126,129),(14,127,100,130,51),(15,131,128,52,101),(16,53,132,102,121),(17,63,90,114,138),(18,115,64,139,91),(19,140,116,92,57),(20,93,141,58,117),(21,59,94,118,142),(22,119,60,143,95),(23,144,120,96,61),(24,89,137,62,113),(25,112,150,74,47),(26,75,105,48,151),(27,41,76,152,106),(28,145,42,107,77),(29,108,146,78,43),(30,79,109,44,147),(31,45,80,148,110),(32,149,46,111,73)], [(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)])
Matrix representation ►G ⊆ GL8(𝔽41)
| 32 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 8 | 40 | 34 |
| 0 | 0 | 0 | 0 | 8 | 32 | 7 | 7 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 39 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 16 | 1 | 0 |
| 0 | 0 | 0 | 0 | 17 | 16 | 0 | 1 |
| 0 | 0 | 0 | 0 | 10 | 31 | 24 | 25 |
| 0 | 0 | 0 | 0 | 19 | 8 | 24 | 25 |
G:=sub<GL(8,GF(41))| [32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,3,8,0,0,0,0,1,34,8,32,0,0,0,0,0,0,40,7,0,0,0,0,0,0,34,7],[0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,16,30,0,0,0,0,0,0,39,25,0,0,0,0,0,0,0,0,17,17,10,19,0,0,0,0,16,16,31,8,0,0,0,0,1,0,24,24,0,0,0,0,0,1,25,25] >;
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 4O | 4P | 4Q | 4R | 5 | 8A | ··· | 8P | 10A | ··· | 10G | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 4 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | M4(2) | M4(2) | F5 | C2×F5 | C2×F5 | C22.F5 | C4×F5 | D5⋊M4(2) |
| kernel | C4×C22.F5 | C4×C5⋊C8 | C10.C42 | C2×C4×Dic5 | C2×C22.F5 | C4×Dic5 | C22.F5 | C22×Dic5 | C22×C20 | Dic5 | C20 | C22×C4 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 2 | 4 | 16 | 2 | 2 | 4 | 4 | 1 | 2 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4\times C_2^2.F_5
% in TeX
G:=Group("C4xC2^2.F5"); // GroupNames label
G:=SmallGroup(320,1088);
// by ID
G=gap.SmallGroup(320,1088);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,100,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^2=c^2=d^5=1,e^4=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^3>;
// generators/relations
×
⋊
ℤ
𝔽
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≀