direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×C40⋊C2, C40⋊10C23, C20.54C24, C23.60D20, Dic10⋊4C23, D20.20C23, (C2×C8)⋊35D10, C8⋊9(C22×D5), (C2×C10)⋊9SD16, C10⋊1(C2×SD16), C4.44(C2×D20), (C2×C4).99D20, (C22×C8)⋊10D5, C5⋊1(C22×SD16), (C22×C40)⋊14C2, (C2×C40)⋊46C22, C20.289(C2×D4), (C2×C20).390D4, C4.51(C23×D5), C2.23(C22×D20), C10.21(C22×D4), C22.69(C2×D20), (C2×C20).786C23, (C22×D20).10C2, (C22×C4).442D10, (C22×C10).144D4, (C22×Dic10)⋊11C2, (C2×Dic10)⋊56C22, (C2×D20).236C22, (C22×C20).525C22, (C2×C10).177(C2×D4), (C2×C4).735(C22×D5), SmallGroup(320,1411)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1342 in 298 conjugacy classes, 127 normal (15 characteristic)
C1, C2, C2 [×6], C2 [×4], C4, C4 [×3], C4 [×4], C22 [×7], C22 [×16], C5, C8 [×4], C2×C4 [×6], C2×C4 [×6], D4 [×10], Q8 [×10], C23, C23 [×10], D5 [×4], C10, C10 [×6], C2×C8 [×6], SD16 [×16], C22×C4, C22×C4, C2×D4 [×9], C2×Q8 [×9], C24, Dic5 [×4], C20, C20 [×3], D10 [×16], C2×C10 [×7], C22×C8, C2×SD16 [×12], C22×D4, C22×Q8, C40 [×4], Dic10 [×4], Dic10 [×6], D20 [×4], D20 [×6], C2×Dic5 [×6], C2×C20 [×6], C22×D5 [×10], C22×C10, C22×SD16, C40⋊C2 [×16], C2×C40 [×6], C2×Dic10 [×6], C2×Dic10 [×3], C2×D20 [×6], C2×D20 [×3], C22×Dic5, C22×C20, C23×D5, C2×C40⋊C2 [×12], C22×C40, C22×Dic10, C22×D20, C22×C40⋊C2
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, SD16 [×4], C2×D4 [×6], C24, D10 [×7], C2×SD16 [×6], C22×D4, D20 [×4], C22×D5 [×7], C22×SD16, C40⋊C2 [×4], C2×D20 [×6], C23×D5, C2×C40⋊C2 [×6], C22×D20, C22×C40⋊C2
Generators and relations
G = < a,b,c,d | a2=b2=c40=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c19 >
►On 160 points
Generators in S160
(1 132)(2 133)(3 134)(4 135)(5 136)(6 137)(7 138)(8 139)(9 140)(10 141)(11 142)(12 143)(13 144)(14 145)(15 146)(16 147)(17 148)(18 149)(19 150)(20 151)(21 152)(22 153)(23 154)(24 155)(25 156)(26 157)(27 158)(28 159)(29 160)(30 121)(31 122)(32 123)(33 124)(34 125)(35 126)(36 127)(37 128)(38 129)(39 130)(40 131)(41 109)(42 110)(43 111)(44 112)(45 113)(46 114)(47 115)(48 116)(49 117)(50 118)(51 119)(52 120)(53 81)(54 82)(55 83)(56 84)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)
(1 70)(2 71)(3 72)(4 73)(5 74)(6 75)(7 76)(8 77)(9 78)(10 79)(11 80)(12 41)(13 42)(14 43)(15 44)(16 45)(17 46)(18 47)(19 48)(20 49)(21 50)(22 51)(23 52)(24 53)(25 54)(26 55)(27 56)(28 57)(29 58)(30 59)(31 60)(32 61)(33 62)(34 63)(35 64)(36 65)(37 66)(38 67)(39 68)(40 69)(81 155)(82 156)(83 157)(84 158)(85 159)(86 160)(87 121)(88 122)(89 123)(90 124)(91 125)(92 126)(93 127)(94 128)(95 129)(96 130)(97 131)(98 132)(99 133)(100 134)(101 135)(102 136)(103 137)(104 138)(105 139)(106 140)(107 141)(108 142)(109 143)(110 144)(111 145)(112 146)(113 147)(114 148)(115 149)(116 150)(117 151)(118 152)(119 153)(120 154)
(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)
(2 20)(3 39)(4 18)(5 37)(6 16)(7 35)(8 14)(9 33)(10 12)(11 31)(13 29)(15 27)(17 25)(19 23)(22 40)(24 38)(26 36)(28 34)(30 32)(41 79)(42 58)(43 77)(44 56)(45 75)(46 54)(47 73)(48 52)(49 71)(51 69)(53 67)(55 65)(57 63)(59 61)(60 80)(62 78)(64 76)(66 74)(68 72)(81 95)(82 114)(83 93)(84 112)(85 91)(86 110)(87 89)(88 108)(90 106)(92 104)(94 102)(96 100)(97 119)(99 117)(101 115)(103 113)(105 111)(107 109)(116 120)(121 123)(122 142)(124 140)(125 159)(126 138)(127 157)(128 136)(129 155)(130 134)(131 153)(133 151)(135 149)(137 147)(139 145)(141 143)(144 160)(146 158)(148 156)(150 154)
G:=sub<Sym(160)| (1,132)(2,133)(3,134)(4,135)(5,136)(6,137)(7,138)(8,139)(9,140)(10,141)(11,142)(12,143)(13,144)(14,145)(15,146)(16,147)(17,148)(18,149)(19,150)(20,151)(21,152)(22,153)(23,154)(24,155)(25,156)(26,157)(27,158)(28,159)(29,160)(30,121)(31,122)(32,123)(33,124)(34,125)(35,126)(36,127)(37,128)(38,129)(39,130)(40,131)(41,109)(42,110)(43,111)(44,112)(45,113)(46,114)(47,115)(48,116)(49,117)(50,118)(51,119)(52,120)(53,81)(54,82)(55,83)(56,84)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108), (1,70)(2,71)(3,72)(4,73)(5,74)(6,75)(7,76)(8,77)(9,78)(10,79)(11,80)(12,41)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(21,50)(22,51)(23,52)(24,53)(25,54)(26,55)(27,56)(28,57)(29,58)(30,59)(31,60)(32,61)(33,62)(34,63)(35,64)(36,65)(37,66)(38,67)(39,68)(40,69)(81,155)(82,156)(83,157)(84,158)(85,159)(86,160)(87,121)(88,122)(89,123)(90,124)(91,125)(92,126)(93,127)(94,128)(95,129)(96,130)(97,131)(98,132)(99,133)(100,134)(101,135)(102,136)(103,137)(104,138)(105,139)(106,140)(107,141)(108,142)(109,143)(110,144)(111,145)(112,146)(113,147)(114,148)(115,149)(116,150)(117,151)(118,152)(119,153)(120,154), (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), (2,20)(3,39)(4,18)(5,37)(6,16)(7,35)(8,14)(9,33)(10,12)(11,31)(13,29)(15,27)(17,25)(19,23)(22,40)(24,38)(26,36)(28,34)(30,32)(41,79)(42,58)(43,77)(44,56)(45,75)(46,54)(47,73)(48,52)(49,71)(51,69)(53,67)(55,65)(57,63)(59,61)(60,80)(62,78)(64,76)(66,74)(68,72)(81,95)(82,114)(83,93)(84,112)(85,91)(86,110)(87,89)(88,108)(90,106)(92,104)(94,102)(96,100)(97,119)(99,117)(101,115)(103,113)(105,111)(107,109)(116,120)(121,123)(122,142)(124,140)(125,159)(126,138)(127,157)(128,136)(129,155)(130,134)(131,153)(133,151)(135,149)(137,147)(139,145)(141,143)(144,160)(146,158)(148,156)(150,154)>;
G:=Group( (1,132)(2,133)(3,134)(4,135)(5,136)(6,137)(7,138)(8,139)(9,140)(10,141)(11,142)(12,143)(13,144)(14,145)(15,146)(16,147)(17,148)(18,149)(19,150)(20,151)(21,152)(22,153)(23,154)(24,155)(25,156)(26,157)(27,158)(28,159)(29,160)(30,121)(31,122)(32,123)(33,124)(34,125)(35,126)(36,127)(37,128)(38,129)(39,130)(40,131)(41,109)(42,110)(43,111)(44,112)(45,113)(46,114)(47,115)(48,116)(49,117)(50,118)(51,119)(52,120)(53,81)(54,82)(55,83)(56,84)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108), (1,70)(2,71)(3,72)(4,73)(5,74)(6,75)(7,76)(8,77)(9,78)(10,79)(11,80)(12,41)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(21,50)(22,51)(23,52)(24,53)(25,54)(26,55)(27,56)(28,57)(29,58)(30,59)(31,60)(32,61)(33,62)(34,63)(35,64)(36,65)(37,66)(38,67)(39,68)(40,69)(81,155)(82,156)(83,157)(84,158)(85,159)(86,160)(87,121)(88,122)(89,123)(90,124)(91,125)(92,126)(93,127)(94,128)(95,129)(96,130)(97,131)(98,132)(99,133)(100,134)(101,135)(102,136)(103,137)(104,138)(105,139)(106,140)(107,141)(108,142)(109,143)(110,144)(111,145)(112,146)(113,147)(114,148)(115,149)(116,150)(117,151)(118,152)(119,153)(120,154), (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), (2,20)(3,39)(4,18)(5,37)(6,16)(7,35)(8,14)(9,33)(10,12)(11,31)(13,29)(15,27)(17,25)(19,23)(22,40)(24,38)(26,36)(28,34)(30,32)(41,79)(42,58)(43,77)(44,56)(45,75)(46,54)(47,73)(48,52)(49,71)(51,69)(53,67)(55,65)(57,63)(59,61)(60,80)(62,78)(64,76)(66,74)(68,72)(81,95)(82,114)(83,93)(84,112)(85,91)(86,110)(87,89)(88,108)(90,106)(92,104)(94,102)(96,100)(97,119)(99,117)(101,115)(103,113)(105,111)(107,109)(116,120)(121,123)(122,142)(124,140)(125,159)(126,138)(127,157)(128,136)(129,155)(130,134)(131,153)(133,151)(135,149)(137,147)(139,145)(141,143)(144,160)(146,158)(148,156)(150,154) );
G=PermutationGroup([(1,132),(2,133),(3,134),(4,135),(5,136),(6,137),(7,138),(8,139),(9,140),(10,141),(11,142),(12,143),(13,144),(14,145),(15,146),(16,147),(17,148),(18,149),(19,150),(20,151),(21,152),(22,153),(23,154),(24,155),(25,156),(26,157),(27,158),(28,159),(29,160),(30,121),(31,122),(32,123),(33,124),(34,125),(35,126),(36,127),(37,128),(38,129),(39,130),(40,131),(41,109),(42,110),(43,111),(44,112),(45,113),(46,114),(47,115),(48,116),(49,117),(50,118),(51,119),(52,120),(53,81),(54,82),(55,83),(56,84),(57,85),(58,86),(59,87),(60,88),(61,89),(62,90),(63,91),(64,92),(65,93),(66,94),(67,95),(68,96),(69,97),(70,98),(71,99),(72,100),(73,101),(74,102),(75,103),(76,104),(77,105),(78,106),(79,107),(80,108)], [(1,70),(2,71),(3,72),(4,73),(5,74),(6,75),(7,76),(8,77),(9,78),(10,79),(11,80),(12,41),(13,42),(14,43),(15,44),(16,45),(17,46),(18,47),(19,48),(20,49),(21,50),(22,51),(23,52),(24,53),(25,54),(26,55),(27,56),(28,57),(29,58),(30,59),(31,60),(32,61),(33,62),(34,63),(35,64),(36,65),(37,66),(38,67),(39,68),(40,69),(81,155),(82,156),(83,157),(84,158),(85,159),(86,160),(87,121),(88,122),(89,123),(90,124),(91,125),(92,126),(93,127),(94,128),(95,129),(96,130),(97,131),(98,132),(99,133),(100,134),(101,135),(102,136),(103,137),(104,138),(105,139),(106,140),(107,141),(108,142),(109,143),(110,144),(111,145),(112,146),(113,147),(114,148),(115,149),(116,150),(117,151),(118,152),(119,153),(120,154)], [(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)], [(2,20),(3,39),(4,18),(5,37),(6,16),(7,35),(8,14),(9,33),(10,12),(11,31),(13,29),(15,27),(17,25),(19,23),(22,40),(24,38),(26,36),(28,34),(30,32),(41,79),(42,58),(43,77),(44,56),(45,75),(46,54),(47,73),(48,52),(49,71),(51,69),(53,67),(55,65),(57,63),(59,61),(60,80),(62,78),(64,76),(66,74),(68,72),(81,95),(82,114),(83,93),(84,112),(85,91),(86,110),(87,89),(88,108),(90,106),(92,104),(94,102),(96,100),(97,119),(99,117),(101,115),(103,113),(105,111),(107,109),(116,120),(121,123),(122,142),(124,140),(125,159),(126,138),(127,157),(128,136),(129,155),(130,134),(131,153),(133,151),(135,149),(137,147),(139,145),(141,143),(144,160),(146,158),(148,156),(150,154)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 31 | 24 |
| 0 | 0 | 13 | 14 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 6 | 36 |
| 0 | 0 | 7 | 35 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,1,0,0,0,0,31,13,0,0,24,14],[40,0,0,0,0,40,0,0,0,0,6,7,0,0,36,35] >;
92 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10N | 20A | ··· | 20P | 40A | ··· | 40AF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | ··· | 1 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | SD16 | D10 | D10 | D20 | D20 | C40⋊C2 |
| kernel | C22×C40⋊C2 | C2×C40⋊C2 | C22×C40 | C22×Dic10 | C22×D20 | C2×C20 | C22×C10 | C22×C8 | C2×C10 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 12 | 1 | 1 | 1 | 3 | 1 | 2 | 8 | 12 | 2 | 12 | 4 | 32 |
In GAP, Magma, Sage, TeX
C_2^2\times C_{40}\rtimes C_2 % in TeX
G:=Group("C2^2xC40:C2"); // GroupNames label
G:=SmallGroup(320,1411);
// by ID
G=gap.SmallGroup(320,1411);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,675,80,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^40=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^19>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀