direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×C4.F5, C20⋊4M4(2), C42.10F5, C20.11C42, Dic5⋊6M4(2), D10.10C42, C4.4(C4×F5), C5⋊1(C4×M4(2)), (C4×C20).11C4, C10.3(C2×C42), (C4×Dic5).34C4, (D5×C42).24C2, C10.2(C2×M4(2)), C2.2(D5⋊M4(2)), C10.C42⋊16C2, C22.26(C22×F5), Dic5.27(C22×C4), (C4×Dic5).319C22, (C2×Dic5).313C23, (C4×C5⋊C8)⋊7C2, C5⋊C8⋊1(C2×C4), C2.6(C2×C4×F5), (C2×C4×D5).43C4, C2.1(C2×C4.F5), (C4×D5).64(C2×C4), (C2×C5⋊C8).15C22, (C2×C4).130(C2×F5), (C2×C4.F5).11C2, (C2×C20).120(C2×C4), (C2×C4×D5).389C22, (C2×C10).15(C22×C4), (C2×Dic5).163(C2×C4), (C22×D5).115(C2×C4), SmallGroup(320,1015)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 426 in 142 conjugacy classes, 74 normal (30 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×6], C22, C22 [×4], C5, C8 [×8], C2×C4 [×3], C2×C4 [×11], C23, D5 [×2], C10 [×3], C42, C42 [×3], C2×C8 [×4], M4(2) [×8], C22×C4 [×3], Dic5 [×4], Dic5, C20 [×4], C20, D10 [×2], D10 [×2], C2×C10, C4×C8 [×2], C8⋊C4 [×2], C2×C42, C2×M4(2) [×2], C5⋊C8 [×8], C4×D5 [×4], C4×D5 [×4], C2×Dic5 [×3], C2×C20 [×3], C22×D5, C4×M4(2), C4×Dic5 [×3], C4×C20, C4.F5 [×8], C2×C5⋊C8 [×4], C2×C4×D5 [×3], C4×C5⋊C8 [×2], C10.C42 [×2], D5×C42, C2×C4.F5 [×2], C4×C4.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], C4×F5 [×2], C22×F5, C2×C4.F5, D5⋊M4(2), C2×C4×F5, C4×C4.F5
Generators and relations
G = < a,b,c,d | a4=b4=c5=1, d4=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >
►On 160 points
Generators in S160
(1 81 148 21)(2 82 149 22)(3 83 150 23)(4 84 151 24)(5 85 152 17)(6 86 145 18)(7 87 146 19)(8 88 147 20)(9 44 96 35)(10 45 89 36)(11 46 90 37)(12 47 91 38)(13 48 92 39)(14 41 93 40)(15 42 94 33)(16 43 95 34)(25 114 108 56)(26 115 109 49)(27 116 110 50)(28 117 111 51)(29 118 112 52)(30 119 105 53)(31 120 106 54)(32 113 107 55)(57 124 142 102)(58 125 143 103)(59 126 144 104)(60 127 137 97)(61 128 138 98)(62 121 139 99)(63 122 140 100)(64 123 141 101)(65 160 134 80)(66 153 135 73)(67 154 136 74)(68 155 129 75)(69 156 130 76)(70 157 131 77)(71 158 132 78)(72 159 133 79)
(1 146 5 150)(2 151 6 147)(3 148 7 152)(4 145 8 149)(9 70 13 66)(10 67 14 71)(11 72 15 68)(12 69 16 65)(17 83 21 87)(18 88 22 84)(19 85 23 81)(20 82 24 86)(25 128 29 124)(26 125 30 121)(27 122 31 126)(28 127 32 123)(33 75 37 79)(34 80 38 76)(35 77 39 73)(36 74 40 78)(41 158 45 154)(42 155 46 159)(43 160 47 156)(44 157 48 153)(49 58 53 62)(50 63 54 59)(51 60 55 64)(52 57 56 61)(89 136 93 132)(90 133 94 129)(91 130 95 134)(92 135 96 131)(97 107 101 111)(98 112 102 108)(99 109 103 105)(100 106 104 110)(113 141 117 137)(114 138 118 142)(115 143 119 139)(116 140 120 144)
(1 155 99 28 39)(2 29 156 40 100)(3 33 30 101 157)(4 102 34 158 31)(5 159 103 32 35)(6 25 160 36 104)(7 37 26 97 153)(8 98 38 154 27)(9 85 133 58 113)(10 59 86 114 134)(11 115 60 135 87)(12 136 116 88 61)(13 81 129 62 117)(14 63 82 118 130)(15 119 64 131 83)(16 132 120 84 57)(17 72 143 55 96)(18 56 65 89 144)(19 90 49 137 66)(20 138 91 67 50)(21 68 139 51 92)(22 52 69 93 140)(23 94 53 141 70)(24 142 95 71 54)(41 122 149 112 76)(42 105 123 77 150)(43 78 106 151 124)(44 152 79 125 107)(45 126 145 108 80)(46 109 127 73 146)(47 74 110 147 128)(48 148 75 121 111)
(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)| (1,81,148,21)(2,82,149,22)(3,83,150,23)(4,84,151,24)(5,85,152,17)(6,86,145,18)(7,87,146,19)(8,88,147,20)(9,44,96,35)(10,45,89,36)(11,46,90,37)(12,47,91,38)(13,48,92,39)(14,41,93,40)(15,42,94,33)(16,43,95,34)(25,114,108,56)(26,115,109,49)(27,116,110,50)(28,117,111,51)(29,118,112,52)(30,119,105,53)(31,120,106,54)(32,113,107,55)(57,124,142,102)(58,125,143,103)(59,126,144,104)(60,127,137,97)(61,128,138,98)(62,121,139,99)(63,122,140,100)(64,123,141,101)(65,160,134,80)(66,153,135,73)(67,154,136,74)(68,155,129,75)(69,156,130,76)(70,157,131,77)(71,158,132,78)(72,159,133,79), (1,146,5,150)(2,151,6,147)(3,148,7,152)(4,145,8,149)(9,70,13,66)(10,67,14,71)(11,72,15,68)(12,69,16,65)(17,83,21,87)(18,88,22,84)(19,85,23,81)(20,82,24,86)(25,128,29,124)(26,125,30,121)(27,122,31,126)(28,127,32,123)(33,75,37,79)(34,80,38,76)(35,77,39,73)(36,74,40,78)(41,158,45,154)(42,155,46,159)(43,160,47,156)(44,157,48,153)(49,58,53,62)(50,63,54,59)(51,60,55,64)(52,57,56,61)(89,136,93,132)(90,133,94,129)(91,130,95,134)(92,135,96,131)(97,107,101,111)(98,112,102,108)(99,109,103,105)(100,106,104,110)(113,141,117,137)(114,138,118,142)(115,143,119,139)(116,140,120,144), (1,155,99,28,39)(2,29,156,40,100)(3,33,30,101,157)(4,102,34,158,31)(5,159,103,32,35)(6,25,160,36,104)(7,37,26,97,153)(8,98,38,154,27)(9,85,133,58,113)(10,59,86,114,134)(11,115,60,135,87)(12,136,116,88,61)(13,81,129,62,117)(14,63,82,118,130)(15,119,64,131,83)(16,132,120,84,57)(17,72,143,55,96)(18,56,65,89,144)(19,90,49,137,66)(20,138,91,67,50)(21,68,139,51,92)(22,52,69,93,140)(23,94,53,141,70)(24,142,95,71,54)(41,122,149,112,76)(42,105,123,77,150)(43,78,106,151,124)(44,152,79,125,107)(45,126,145,108,80)(46,109,127,73,146)(47,74,110,147,128)(48,148,75,121,111), (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( (1,81,148,21)(2,82,149,22)(3,83,150,23)(4,84,151,24)(5,85,152,17)(6,86,145,18)(7,87,146,19)(8,88,147,20)(9,44,96,35)(10,45,89,36)(11,46,90,37)(12,47,91,38)(13,48,92,39)(14,41,93,40)(15,42,94,33)(16,43,95,34)(25,114,108,56)(26,115,109,49)(27,116,110,50)(28,117,111,51)(29,118,112,52)(30,119,105,53)(31,120,106,54)(32,113,107,55)(57,124,142,102)(58,125,143,103)(59,126,144,104)(60,127,137,97)(61,128,138,98)(62,121,139,99)(63,122,140,100)(64,123,141,101)(65,160,134,80)(66,153,135,73)(67,154,136,74)(68,155,129,75)(69,156,130,76)(70,157,131,77)(71,158,132,78)(72,159,133,79), (1,146,5,150)(2,151,6,147)(3,148,7,152)(4,145,8,149)(9,70,13,66)(10,67,14,71)(11,72,15,68)(12,69,16,65)(17,83,21,87)(18,88,22,84)(19,85,23,81)(20,82,24,86)(25,128,29,124)(26,125,30,121)(27,122,31,126)(28,127,32,123)(33,75,37,79)(34,80,38,76)(35,77,39,73)(36,74,40,78)(41,158,45,154)(42,155,46,159)(43,160,47,156)(44,157,48,153)(49,58,53,62)(50,63,54,59)(51,60,55,64)(52,57,56,61)(89,136,93,132)(90,133,94,129)(91,130,95,134)(92,135,96,131)(97,107,101,111)(98,112,102,108)(99,109,103,105)(100,106,104,110)(113,141,117,137)(114,138,118,142)(115,143,119,139)(116,140,120,144), (1,155,99,28,39)(2,29,156,40,100)(3,33,30,101,157)(4,102,34,158,31)(5,159,103,32,35)(6,25,160,36,104)(7,37,26,97,153)(8,98,38,154,27)(9,85,133,58,113)(10,59,86,114,134)(11,115,60,135,87)(12,136,116,88,61)(13,81,129,62,117)(14,63,82,118,130)(15,119,64,131,83)(16,132,120,84,57)(17,72,143,55,96)(18,56,65,89,144)(19,90,49,137,66)(20,138,91,67,50)(21,68,139,51,92)(22,52,69,93,140)(23,94,53,141,70)(24,142,95,71,54)(41,122,149,112,76)(42,105,123,77,150)(43,78,106,151,124)(44,152,79,125,107)(45,126,145,108,80)(46,109,127,73,146)(47,74,110,147,128)(48,148,75,121,111), (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([(1,81,148,21),(2,82,149,22),(3,83,150,23),(4,84,151,24),(5,85,152,17),(6,86,145,18),(7,87,146,19),(8,88,147,20),(9,44,96,35),(10,45,89,36),(11,46,90,37),(12,47,91,38),(13,48,92,39),(14,41,93,40),(15,42,94,33),(16,43,95,34),(25,114,108,56),(26,115,109,49),(27,116,110,50),(28,117,111,51),(29,118,112,52),(30,119,105,53),(31,120,106,54),(32,113,107,55),(57,124,142,102),(58,125,143,103),(59,126,144,104),(60,127,137,97),(61,128,138,98),(62,121,139,99),(63,122,140,100),(64,123,141,101),(65,160,134,80),(66,153,135,73),(67,154,136,74),(68,155,129,75),(69,156,130,76),(70,157,131,77),(71,158,132,78),(72,159,133,79)], [(1,146,5,150),(2,151,6,147),(3,148,7,152),(4,145,8,149),(9,70,13,66),(10,67,14,71),(11,72,15,68),(12,69,16,65),(17,83,21,87),(18,88,22,84),(19,85,23,81),(20,82,24,86),(25,128,29,124),(26,125,30,121),(27,122,31,126),(28,127,32,123),(33,75,37,79),(34,80,38,76),(35,77,39,73),(36,74,40,78),(41,158,45,154),(42,155,46,159),(43,160,47,156),(44,157,48,153),(49,58,53,62),(50,63,54,59),(51,60,55,64),(52,57,56,61),(89,136,93,132),(90,133,94,129),(91,130,95,134),(92,135,96,131),(97,107,101,111),(98,112,102,108),(99,109,103,105),(100,106,104,110),(113,141,117,137),(114,138,118,142),(115,143,119,139),(116,140,120,144)], [(1,155,99,28,39),(2,29,156,40,100),(3,33,30,101,157),(4,102,34,158,31),(5,159,103,32,35),(6,25,160,36,104),(7,37,26,97,153),(8,98,38,154,27),(9,85,133,58,113),(10,59,86,114,134),(11,115,60,135,87),(12,136,116,88,61),(13,81,129,62,117),(14,63,82,118,130),(15,119,64,131,83),(16,132,120,84,57),(17,72,143,55,96),(18,56,65,89,144),(19,90,49,137,66),(20,138,91,67,50),(21,68,139,51,92),(22,52,69,93,140),(23,94,53,141,70),(24,142,95,71,54),(41,122,149,112,76),(42,105,123,77,150),(43,78,106,151,124),(44,152,79,125,107),(45,126,145,108,80),(46,109,127,73,146),(47,74,110,147,128),(48,148,75,121,111)], [(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 ⊆ GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 32 | 32 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 27 | 0 | 14 |
| 0 | 0 | 0 | 34 | 27 | 14 |
| 0 | 0 | 14 | 27 | 34 | 0 |
| 0 | 0 | 14 | 0 | 27 | 7 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 1 | 0 |
| 0 | 0 | 40 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 9 | 4 | 0 | 0 | 0 | 0 |
| 23 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 39 | 39 | 33 | 20 |
| 0 | 0 | 31 | 18 | 25 | 18 |
| 0 | 0 | 23 | 16 | 23 | 10 |
| 0 | 0 | 21 | 8 | 2 | 2 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[32,0,0,0,0,0,32,9,0,0,0,0,0,0,7,0,14,14,0,0,27,34,27,0,0,0,0,27,34,27,0,0,14,14,0,7],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,40,40,40,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[9,23,0,0,0,0,4,32,0,0,0,0,0,0,39,31,23,21,0,0,39,18,16,8,0,0,33,25,23,2,0,0,20,18,10,2] >;
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 4Q | 4R | 5 | 8A | ··· | 8P | 10A | 10B | 10C | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 5 | ··· | 5 | 10 | 10 | 4 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | M4(2) | M4(2) | F5 | C2×F5 | C4.F5 | C4×F5 | D5⋊M4(2) |
| kernel | C4×C4.F5 | C4×C5⋊C8 | C10.C42 | D5×C42 | C2×C4.F5 | C4×Dic5 | C4×C20 | C4.F5 | C2×C4×D5 | Dic5 | C20 | C42 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 16 | 4 | 4 | 4 | 1 | 3 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4\times C_4.F_5
% in TeX
G:=Group("C4xC4.F5"); // GroupNames label
G:=SmallGroup(320,1015);
// by ID
G=gap.SmallGroup(320,1015);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,758,184,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^5=1,d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀