metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.136D10, C10.1132+ (1+4), (C4×Q8)⋊18D5, (C4×D20)⋊42C2, (Q8×C20)⋊20C2, C4⋊C4.303D10, D20⋊8C4⋊18C2, (C4×Dic10)⋊42C2, C4.19(C4○D20), C4.D20⋊21C2, C4⋊D20.11C2, (C2×Q8).184D10, C20.123(C4○D4), C20.23D4⋊10C2, (C2×C10).129C24, (C2×C20).592C23, (C4×C20).181C22, C4.51(Q8⋊2D5), (C2×D20).31C22, D10.13D4⋊10C2, C2.25(D4⋊8D10), C4⋊Dic5.401C22, (Q8×C10).229C22, (C2×Dic5).59C23, (C4×Dic5).96C22, (C22×D5).51C23, C22.150(C23×D5), C5⋊2(C22.53C24), D10⋊C4.145C22, (C2×Dic10).252C22, C10.D4.116C22, C10.58(C2×C4○D4), C2.68(C2×C4○D20), C2.14(C2×Q8⋊2D5), (C2×C4×D5).258C22, (C5×C4⋊C4).357C22, (C2×C4).291(C22×D5), SmallGroup(320,1257)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 934 in 236 conjugacy classes, 99 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×9], C22, C22 [×12], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×8], D4 [×10], Q8 [×4], C23 [×4], D5 [×4], C10 [×3], C42, C42 [×2], C42 [×2], C22⋊C4 [×12], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×3], C22×C4 [×4], C2×D4 [×6], C2×Q8, C2×Q8, Dic5 [×4], C20 [×4], C20 [×5], D10 [×12], C2×C10, C4×D4 [×4], C4×Q8, C4×Q8, C22.D4 [×4], C4.4D4 [×4], C4⋊1D4, Dic10 [×2], C4×D5 [×4], D20 [×10], C2×Dic5 [×4], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×2], C22×D5 [×4], C22.53C24, C4×Dic5 [×2], C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×12], C4×C20, C4×C20 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×Dic10, C2×C4×D5 [×4], C2×D20 [×6], Q8×C10, C4×Dic10, C4×D20 [×2], C4⋊D20, C4.D20 [×2], D20⋊8C4 [×2], D10.13D4 [×4], C20.23D4 [×2], Q8×C20, C42.136D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2+ (1+4), C22×D5 [×7], C22.53C24, C4○D20 [×2], Q8⋊2D5 [×2], C23×D5, C2×C4○D20, C2×Q8⋊2D5, D4⋊8D10, C42.136D10
Generators and relations
G = < a,b,c,d | a4=b4=d2=1, c10=a2, ab=ba, cac-1=dad=a-1b2, bc=cb, dbd=b-1, dcd=a2c9 >
►On 160 points
Generators in S160
(1 26 11 36)(2 60 12 50)(3 28 13 38)(4 42 14 52)(5 30 15 40)(6 44 16 54)(7 32 17 22)(8 46 18 56)(9 34 19 24)(10 48 20 58)(21 76 31 66)(23 78 33 68)(25 80 35 70)(27 62 37 72)(29 64 39 74)(41 63 51 73)(43 65 53 75)(45 67 55 77)(47 69 57 79)(49 71 59 61)(81 136 91 126)(82 101 92 111)(83 138 93 128)(84 103 94 113)(85 140 95 130)(86 105 96 115)(87 122 97 132)(88 107 98 117)(89 124 99 134)(90 109 100 119)(102 146 112 156)(104 148 114 158)(106 150 116 160)(108 152 118 142)(110 154 120 144)(121 149 131 159)(123 151 133 141)(125 153 135 143)(127 155 137 145)(129 157 139 147)
(1 135 61 109)(2 136 62 110)(3 137 63 111)(4 138 64 112)(5 139 65 113)(6 140 66 114)(7 121 67 115)(8 122 68 116)(9 123 69 117)(10 124 70 118)(11 125 71 119)(12 126 72 120)(13 127 73 101)(14 128 74 102)(15 129 75 103)(16 130 76 104)(17 131 77 105)(18 132 78 106)(19 133 79 107)(20 134 80 108)(21 158 44 95)(22 159 45 96)(23 160 46 97)(24 141 47 98)(25 142 48 99)(26 143 49 100)(27 144 50 81)(28 145 51 82)(29 146 52 83)(30 147 53 84)(31 148 54 85)(32 149 55 86)(33 150 56 87)(34 151 57 88)(35 152 58 89)(36 153 59 90)(37 154 60 91)(38 155 41 92)(39 156 42 93)(40 157 43 94)
(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 5)(2 4)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(21 48)(22 47)(23 46)(24 45)(25 44)(26 43)(27 42)(28 41)(29 60)(30 59)(31 58)(32 57)(33 56)(34 55)(35 54)(36 53)(37 52)(38 51)(39 50)(40 49)(61 65)(62 64)(66 80)(67 79)(68 78)(69 77)(70 76)(71 75)(72 74)(81 93)(82 92)(83 91)(84 90)(85 89)(86 88)(94 100)(95 99)(96 98)(101 127)(102 126)(103 125)(104 124)(105 123)(106 122)(107 121)(108 140)(109 139)(110 138)(111 137)(112 136)(113 135)(114 134)(115 133)(116 132)(117 131)(118 130)(119 129)(120 128)(141 159)(142 158)(143 157)(144 156)(145 155)(146 154)(147 153)(148 152)(149 151)
G:=sub<Sym(160)| (1,26,11,36)(2,60,12,50)(3,28,13,38)(4,42,14,52)(5,30,15,40)(6,44,16,54)(7,32,17,22)(8,46,18,56)(9,34,19,24)(10,48,20,58)(21,76,31,66)(23,78,33,68)(25,80,35,70)(27,62,37,72)(29,64,39,74)(41,63,51,73)(43,65,53,75)(45,67,55,77)(47,69,57,79)(49,71,59,61)(81,136,91,126)(82,101,92,111)(83,138,93,128)(84,103,94,113)(85,140,95,130)(86,105,96,115)(87,122,97,132)(88,107,98,117)(89,124,99,134)(90,109,100,119)(102,146,112,156)(104,148,114,158)(106,150,116,160)(108,152,118,142)(110,154,120,144)(121,149,131,159)(123,151,133,141)(125,153,135,143)(127,155,137,145)(129,157,139,147), (1,135,61,109)(2,136,62,110)(3,137,63,111)(4,138,64,112)(5,139,65,113)(6,140,66,114)(7,121,67,115)(8,122,68,116)(9,123,69,117)(10,124,70,118)(11,125,71,119)(12,126,72,120)(13,127,73,101)(14,128,74,102)(15,129,75,103)(16,130,76,104)(17,131,77,105)(18,132,78,106)(19,133,79,107)(20,134,80,108)(21,158,44,95)(22,159,45,96)(23,160,46,97)(24,141,47,98)(25,142,48,99)(26,143,49,100)(27,144,50,81)(28,145,51,82)(29,146,52,83)(30,147,53,84)(31,148,54,85)(32,149,55,86)(33,150,56,87)(34,151,57,88)(35,152,58,89)(36,153,59,90)(37,154,60,91)(38,155,41,92)(39,156,42,93)(40,157,43,94), (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), (1,5)(2,4)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(21,48)(22,47)(23,46)(24,45)(25,44)(26,43)(27,42)(28,41)(29,60)(30,59)(31,58)(32,57)(33,56)(34,55)(35,54)(36,53)(37,52)(38,51)(39,50)(40,49)(61,65)(62,64)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(81,93)(82,92)(83,91)(84,90)(85,89)(86,88)(94,100)(95,99)(96,98)(101,127)(102,126)(103,125)(104,124)(105,123)(106,122)(107,121)(108,140)(109,139)(110,138)(111,137)(112,136)(113,135)(114,134)(115,133)(116,132)(117,131)(118,130)(119,129)(120,128)(141,159)(142,158)(143,157)(144,156)(145,155)(146,154)(147,153)(148,152)(149,151)>;
G:=Group( (1,26,11,36)(2,60,12,50)(3,28,13,38)(4,42,14,52)(5,30,15,40)(6,44,16,54)(7,32,17,22)(8,46,18,56)(9,34,19,24)(10,48,20,58)(21,76,31,66)(23,78,33,68)(25,80,35,70)(27,62,37,72)(29,64,39,74)(41,63,51,73)(43,65,53,75)(45,67,55,77)(47,69,57,79)(49,71,59,61)(81,136,91,126)(82,101,92,111)(83,138,93,128)(84,103,94,113)(85,140,95,130)(86,105,96,115)(87,122,97,132)(88,107,98,117)(89,124,99,134)(90,109,100,119)(102,146,112,156)(104,148,114,158)(106,150,116,160)(108,152,118,142)(110,154,120,144)(121,149,131,159)(123,151,133,141)(125,153,135,143)(127,155,137,145)(129,157,139,147), (1,135,61,109)(2,136,62,110)(3,137,63,111)(4,138,64,112)(5,139,65,113)(6,140,66,114)(7,121,67,115)(8,122,68,116)(9,123,69,117)(10,124,70,118)(11,125,71,119)(12,126,72,120)(13,127,73,101)(14,128,74,102)(15,129,75,103)(16,130,76,104)(17,131,77,105)(18,132,78,106)(19,133,79,107)(20,134,80,108)(21,158,44,95)(22,159,45,96)(23,160,46,97)(24,141,47,98)(25,142,48,99)(26,143,49,100)(27,144,50,81)(28,145,51,82)(29,146,52,83)(30,147,53,84)(31,148,54,85)(32,149,55,86)(33,150,56,87)(34,151,57,88)(35,152,58,89)(36,153,59,90)(37,154,60,91)(38,155,41,92)(39,156,42,93)(40,157,43,94), (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), (1,5)(2,4)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(21,48)(22,47)(23,46)(24,45)(25,44)(26,43)(27,42)(28,41)(29,60)(30,59)(31,58)(32,57)(33,56)(34,55)(35,54)(36,53)(37,52)(38,51)(39,50)(40,49)(61,65)(62,64)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(81,93)(82,92)(83,91)(84,90)(85,89)(86,88)(94,100)(95,99)(96,98)(101,127)(102,126)(103,125)(104,124)(105,123)(106,122)(107,121)(108,140)(109,139)(110,138)(111,137)(112,136)(113,135)(114,134)(115,133)(116,132)(117,131)(118,130)(119,129)(120,128)(141,159)(142,158)(143,157)(144,156)(145,155)(146,154)(147,153)(148,152)(149,151) );
G=PermutationGroup([(1,26,11,36),(2,60,12,50),(3,28,13,38),(4,42,14,52),(5,30,15,40),(6,44,16,54),(7,32,17,22),(8,46,18,56),(9,34,19,24),(10,48,20,58),(21,76,31,66),(23,78,33,68),(25,80,35,70),(27,62,37,72),(29,64,39,74),(41,63,51,73),(43,65,53,75),(45,67,55,77),(47,69,57,79),(49,71,59,61),(81,136,91,126),(82,101,92,111),(83,138,93,128),(84,103,94,113),(85,140,95,130),(86,105,96,115),(87,122,97,132),(88,107,98,117),(89,124,99,134),(90,109,100,119),(102,146,112,156),(104,148,114,158),(106,150,116,160),(108,152,118,142),(110,154,120,144),(121,149,131,159),(123,151,133,141),(125,153,135,143),(127,155,137,145),(129,157,139,147)], [(1,135,61,109),(2,136,62,110),(3,137,63,111),(4,138,64,112),(5,139,65,113),(6,140,66,114),(7,121,67,115),(8,122,68,116),(9,123,69,117),(10,124,70,118),(11,125,71,119),(12,126,72,120),(13,127,73,101),(14,128,74,102),(15,129,75,103),(16,130,76,104),(17,131,77,105),(18,132,78,106),(19,133,79,107),(20,134,80,108),(21,158,44,95),(22,159,45,96),(23,160,46,97),(24,141,47,98),(25,142,48,99),(26,143,49,100),(27,144,50,81),(28,145,51,82),(29,146,52,83),(30,147,53,84),(31,148,54,85),(32,149,55,86),(33,150,56,87),(34,151,57,88),(35,152,58,89),(36,153,59,90),(37,154,60,91),(38,155,41,92),(39,156,42,93),(40,157,43,94)], [(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)], [(1,5),(2,4),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14),(21,48),(22,47),(23,46),(24,45),(25,44),(26,43),(27,42),(28,41),(29,60),(30,59),(31,58),(32,57),(33,56),(34,55),(35,54),(36,53),(37,52),(38,51),(39,50),(40,49),(61,65),(62,64),(66,80),(67,79),(68,78),(69,77),(70,76),(71,75),(72,74),(81,93),(82,92),(83,91),(84,90),(85,89),(86,88),(94,100),(95,99),(96,98),(101,127),(102,126),(103,125),(104,124),(105,123),(106,122),(107,121),(108,140),(109,139),(110,138),(111,137),(112,136),(113,135),(114,134),(115,133),(116,132),(117,131),(118,130),(119,129),(120,128),(141,159),(142,158),(143,157),(144,156),(145,155),(146,154),(147,153),(148,152),(149,151)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 9 | 23 |
| 0 | 0 | 0 | 32 |
| 2 | 4 | 0 | 0 |
| 9 | 39 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 25 | 12 | 0 | 0 |
| 27 | 13 | 0 | 0 |
| 0 | 0 | 1 | 39 |
| 0 | 0 | 1 | 40 |
| 6 | 5 | 0 | 0 |
| 34 | 35 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 40 |
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,23,32],[2,9,0,0,4,39,0,0,0,0,40,0,0,0,0,40],[25,27,0,0,12,13,0,0,0,0,1,1,0,0,39,40],[6,34,0,0,5,35,0,0,0,0,1,1,0,0,0,40] >;
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | C4○D20 | 2+ (1+4) | Q8⋊2D5 | D4⋊8D10 |
| kernel | C42.136D10 | C4×Dic10 | C4×D20 | C4⋊D20 | C4.D20 | D20⋊8C4 | D10.13D4 | C20.23D4 | Q8×C20 | C4×Q8 | C20 | C42 | C4⋊C4 | C2×Q8 | C4 | C10 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 1 | 2 | 8 | 6 | 6 | 2 | 16 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{136}D_{10} % in TeX
G:=Group("C4^2.136D10"); // GroupNames label
G:=SmallGroup(320,1257);
// by ID
G=gap.SmallGroup(320,1257);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,184,1571,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^10=a^2,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^2,b*c=c*b,d*b*d=b^-1,d*c*d=a^2*c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀