direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×C4○D20, C42.275D10, (C2×C42)⋊8D5, (C4×D20)⋊53C2, D20⋊35(C2×C4), C20⋊12(C4○D4), (D5×C42)⋊15C2, Dic10⋊33(C2×C4), (C4×Dic10)⋊55C2, C42⋊D5⋊38C2, C10.26(C23×C4), (C2×C10).18C24, C20.177(C22×C4), (C4×C20).333C22, (C2×C20).876C23, D10.10(C22×C4), (C22×C4).437D10, C22.15(C23×D5), (C2×D20).293C22, C4⋊Dic5.395C22, Dic5.10(C22×C4), C23.217(C22×D5), C23.21D10⋊40C2, (C22×C10).380C23, (C22×C20).564C22, (C2×Dic5).184C23, (C4×Dic5).330C22, (C22×D5).157C23, C23.D5.139C22, D10⋊C4.162C22, (C2×Dic10).322C22, C10.D4.174C22, C5⋊2(C4×C4○D4), (C2×C4×C20)⋊12C2, (C2×C4)⋊13(C4×D5), C4.117(C2×C4×D5), (C2×C20)⋊43(C2×C4), (C4×D5)⋊13(C2×C4), C5⋊D4⋊12(C2×C4), (C4×C5⋊D4)⋊62C2, C22.9(C2×C4×D5), C2.7(D5×C22×C4), C2.4(C2×C4○D20), C10.6(C2×C4○D4), (C2×C4○D20).28C2, (C2×C4×D5).370C22, (C2×C4).818(C22×D5), (C2×C10).249(C22×C4), (C2×C5⋊D4).157C22, SmallGroup(320,1146)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 894 in 310 conjugacy classes, 159 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×8], C4 [×10], C22, C22 [×2], C22 [×10], C5, C2×C4 [×2], C2×C4 [×8], C2×C4 [×26], D4 [×12], Q8 [×4], C23, C23 [×2], D5 [×4], C10, C10 [×2], C10 [×2], C42 [×2], C42 [×2], C42 [×6], C22⋊C4 [×6], C4⋊C4 [×6], C22×C4, C22×C4 [×2], C22×C4 [×6], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×4], Dic5 [×4], C20 [×8], C20 [×2], D10 [×4], D10 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C42, C2×C42 [×2], C42⋊C2 [×3], C4×D4 [×6], C4×Q8 [×2], C2×C4○D4, Dic10 [×4], C4×D5 [×8], C4×D5 [×8], D20 [×4], C2×Dic5 [×6], C5⋊D4 [×8], C2×C20 [×2], C2×C20 [×8], C2×C20 [×4], C22×D5 [×2], C22×C10, C4×C4○D4, C4×Dic5 [×6], C10.D4 [×4], C4⋊Dic5 [×2], D10⋊C4 [×4], C23.D5 [×2], C4×C20 [×2], C4×C20 [×2], C2×Dic10, C2×C4×D5 [×6], C2×D20, C4○D20 [×8], C2×C5⋊D4 [×2], C22×C20, C22×C20 [×2], C4×Dic10 [×2], D5×C42 [×2], C42⋊D5 [×2], C4×D20 [×2], C23.21D10, C4×C5⋊D4 [×4], C2×C4×C20, C2×C4○D20, C4×C4○D20
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C4○D4 [×4], C24, D10 [×7], C23×C4, C2×C4○D4 [×2], C4×D5 [×4], C22×D5 [×7], C4×C4○D4, C2×C4×D5 [×6], C4○D20 [×4], C23×D5, D5×C22×C4, C2×C4○D20 [×2], C4×C4○D20
Generators and relations
G = < a,b,c,d | a4=b4=d2=1, c10=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c9 >
►On 160 points
Generators in S160
(1 126 59 149)(2 127 60 150)(3 128 41 151)(4 129 42 152)(5 130 43 153)(6 131 44 154)(7 132 45 155)(8 133 46 156)(9 134 47 157)(10 135 48 158)(11 136 49 159)(12 137 50 160)(13 138 51 141)(14 139 52 142)(15 140 53 143)(16 121 54 144)(17 122 55 145)(18 123 56 146)(19 124 57 147)(20 125 58 148)(21 64 81 120)(22 65 82 101)(23 66 83 102)(24 67 84 103)(25 68 85 104)(26 69 86 105)(27 70 87 106)(28 71 88 107)(29 72 89 108)(30 73 90 109)(31 74 91 110)(32 75 92 111)(33 76 93 112)(34 77 94 113)(35 78 95 114)(36 79 96 115)(37 80 97 116)(38 61 98 117)(39 62 99 118)(40 63 100 119)
(1 40 11 30)(2 21 12 31)(3 22 13 32)(4 23 14 33)(5 24 15 34)(6 25 16 35)(7 26 17 36)(8 27 18 37)(9 28 19 38)(10 29 20 39)(41 82 51 92)(42 83 52 93)(43 84 53 94)(44 85 54 95)(45 86 55 96)(46 87 56 97)(47 88 57 98)(48 89 58 99)(49 90 59 100)(50 91 60 81)(61 134 71 124)(62 135 72 125)(63 136 73 126)(64 137 74 127)(65 138 75 128)(66 139 76 129)(67 140 77 130)(68 121 78 131)(69 122 79 132)(70 123 80 133)(101 141 111 151)(102 142 112 152)(103 143 113 153)(104 144 114 154)(105 145 115 155)(106 146 116 156)(107 147 117 157)(108 148 118 158)(109 149 119 159)(110 150 120 160)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(34 40)(35 39)(36 38)(41 51)(42 50)(43 49)(44 48)(45 47)(52 60)(53 59)(54 58)(55 57)(61 79)(62 78)(63 77)(64 76)(65 75)(66 74)(67 73)(68 72)(69 71)(81 93)(82 92)(83 91)(84 90)(85 89)(86 88)(94 100)(95 99)(96 98)(101 111)(102 110)(103 109)(104 108)(105 107)(112 120)(113 119)(114 118)(115 117)(121 125)(122 124)(126 140)(127 139)(128 138)(129 137)(130 136)(131 135)(132 134)(141 151)(142 150)(143 149)(144 148)(145 147)(152 160)(153 159)(154 158)(155 157)
G:=sub<Sym(160)| (1,126,59,149)(2,127,60,150)(3,128,41,151)(4,129,42,152)(5,130,43,153)(6,131,44,154)(7,132,45,155)(8,133,46,156)(9,134,47,157)(10,135,48,158)(11,136,49,159)(12,137,50,160)(13,138,51,141)(14,139,52,142)(15,140,53,143)(16,121,54,144)(17,122,55,145)(18,123,56,146)(19,124,57,147)(20,125,58,148)(21,64,81,120)(22,65,82,101)(23,66,83,102)(24,67,84,103)(25,68,85,104)(26,69,86,105)(27,70,87,106)(28,71,88,107)(29,72,89,108)(30,73,90,109)(31,74,91,110)(32,75,92,111)(33,76,93,112)(34,77,94,113)(35,78,95,114)(36,79,96,115)(37,80,97,116)(38,61,98,117)(39,62,99,118)(40,63,100,119), (1,40,11,30)(2,21,12,31)(3,22,13,32)(4,23,14,33)(5,24,15,34)(6,25,16,35)(7,26,17,36)(8,27,18,37)(9,28,19,38)(10,29,20,39)(41,82,51,92)(42,83,52,93)(43,84,53,94)(44,85,54,95)(45,86,55,96)(46,87,56,97)(47,88,57,98)(48,89,58,99)(49,90,59,100)(50,91,60,81)(61,134,71,124)(62,135,72,125)(63,136,73,126)(64,137,74,127)(65,138,75,128)(66,139,76,129)(67,140,77,130)(68,121,78,131)(69,122,79,132)(70,123,80,133)(101,141,111,151)(102,142,112,152)(103,143,113,153)(104,144,114,154)(105,145,115,155)(106,146,116,156)(107,147,117,157)(108,148,118,158)(109,149,119,159)(110,150,120,160), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(34,40)(35,39)(36,38)(41,51)(42,50)(43,49)(44,48)(45,47)(52,60)(53,59)(54,58)(55,57)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(81,93)(82,92)(83,91)(84,90)(85,89)(86,88)(94,100)(95,99)(96,98)(101,111)(102,110)(103,109)(104,108)(105,107)(112,120)(113,119)(114,118)(115,117)(121,125)(122,124)(126,140)(127,139)(128,138)(129,137)(130,136)(131,135)(132,134)(141,151)(142,150)(143,149)(144,148)(145,147)(152,160)(153,159)(154,158)(155,157)>;
G:=Group( (1,126,59,149)(2,127,60,150)(3,128,41,151)(4,129,42,152)(5,130,43,153)(6,131,44,154)(7,132,45,155)(8,133,46,156)(9,134,47,157)(10,135,48,158)(11,136,49,159)(12,137,50,160)(13,138,51,141)(14,139,52,142)(15,140,53,143)(16,121,54,144)(17,122,55,145)(18,123,56,146)(19,124,57,147)(20,125,58,148)(21,64,81,120)(22,65,82,101)(23,66,83,102)(24,67,84,103)(25,68,85,104)(26,69,86,105)(27,70,87,106)(28,71,88,107)(29,72,89,108)(30,73,90,109)(31,74,91,110)(32,75,92,111)(33,76,93,112)(34,77,94,113)(35,78,95,114)(36,79,96,115)(37,80,97,116)(38,61,98,117)(39,62,99,118)(40,63,100,119), (1,40,11,30)(2,21,12,31)(3,22,13,32)(4,23,14,33)(5,24,15,34)(6,25,16,35)(7,26,17,36)(8,27,18,37)(9,28,19,38)(10,29,20,39)(41,82,51,92)(42,83,52,93)(43,84,53,94)(44,85,54,95)(45,86,55,96)(46,87,56,97)(47,88,57,98)(48,89,58,99)(49,90,59,100)(50,91,60,81)(61,134,71,124)(62,135,72,125)(63,136,73,126)(64,137,74,127)(65,138,75,128)(66,139,76,129)(67,140,77,130)(68,121,78,131)(69,122,79,132)(70,123,80,133)(101,141,111,151)(102,142,112,152)(103,143,113,153)(104,144,114,154)(105,145,115,155)(106,146,116,156)(107,147,117,157)(108,148,118,158)(109,149,119,159)(110,150,120,160), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(34,40)(35,39)(36,38)(41,51)(42,50)(43,49)(44,48)(45,47)(52,60)(53,59)(54,58)(55,57)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(81,93)(82,92)(83,91)(84,90)(85,89)(86,88)(94,100)(95,99)(96,98)(101,111)(102,110)(103,109)(104,108)(105,107)(112,120)(113,119)(114,118)(115,117)(121,125)(122,124)(126,140)(127,139)(128,138)(129,137)(130,136)(131,135)(132,134)(141,151)(142,150)(143,149)(144,148)(145,147)(152,160)(153,159)(154,158)(155,157) );
G=PermutationGroup([(1,126,59,149),(2,127,60,150),(3,128,41,151),(4,129,42,152),(5,130,43,153),(6,131,44,154),(7,132,45,155),(8,133,46,156),(9,134,47,157),(10,135,48,158),(11,136,49,159),(12,137,50,160),(13,138,51,141),(14,139,52,142),(15,140,53,143),(16,121,54,144),(17,122,55,145),(18,123,56,146),(19,124,57,147),(20,125,58,148),(21,64,81,120),(22,65,82,101),(23,66,83,102),(24,67,84,103),(25,68,85,104),(26,69,86,105),(27,70,87,106),(28,71,88,107),(29,72,89,108),(30,73,90,109),(31,74,91,110),(32,75,92,111),(33,76,93,112),(34,77,94,113),(35,78,95,114),(36,79,96,115),(37,80,97,116),(38,61,98,117),(39,62,99,118),(40,63,100,119)], [(1,40,11,30),(2,21,12,31),(3,22,13,32),(4,23,14,33),(5,24,15,34),(6,25,16,35),(7,26,17,36),(8,27,18,37),(9,28,19,38),(10,29,20,39),(41,82,51,92),(42,83,52,93),(43,84,53,94),(44,85,54,95),(45,86,55,96),(46,87,56,97),(47,88,57,98),(48,89,58,99),(49,90,59,100),(50,91,60,81),(61,134,71,124),(62,135,72,125),(63,136,73,126),(64,137,74,127),(65,138,75,128),(66,139,76,129),(67,140,77,130),(68,121,78,131),(69,122,79,132),(70,123,80,133),(101,141,111,151),(102,142,112,152),(103,143,113,153),(104,144,114,154),(105,145,115,155),(106,146,116,156),(107,147,117,157),(108,148,118,158),(109,149,119,159),(110,150,120,160)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,33),(22,32),(23,31),(24,30),(25,29),(26,28),(34,40),(35,39),(36,38),(41,51),(42,50),(43,49),(44,48),(45,47),(52,60),(53,59),(54,58),(55,57),(61,79),(62,78),(63,77),(64,76),(65,75),(66,74),(67,73),(68,72),(69,71),(81,93),(82,92),(83,91),(84,90),(85,89),(86,88),(94,100),(95,99),(96,98),(101,111),(102,110),(103,109),(104,108),(105,107),(112,120),(113,119),(114,118),(115,117),(121,125),(122,124),(126,140),(127,139),(128,138),(129,137),(130,136),(131,135),(132,134),(141,151),(142,150),(143,149),(144,148),(145,147),(152,160),(153,159),(154,158),(155,157)])
Matrix representation ►G ⊆ GL3(𝔽41) generated by
| 9 | 0 | 0 |
| 0 | 40 | 0 |
| 0 | 0 | 40 |
| 1 | 0 | 0 |
| 0 | 32 | 0 |
| 0 | 0 | 32 |
| 40 | 0 | 0 |
| 0 | 30 | 39 |
| 0 | 16 | 14 |
| 40 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 8 | 40 |
G:=sub<GL(3,GF(41))| [9,0,0,0,40,0,0,0,40],[1,0,0,0,32,0,0,0,32],[40,0,0,0,30,16,0,39,14],[40,0,0,0,1,8,0,0,40] >;
104 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4L | 4M | ··· | 4R | 4S | ··· | 4AD | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 1 | ··· | 1 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
104 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D5 | C4○D4 | D10 | D10 | C4×D5 | C4○D20 |
| kernel | C4×C4○D20 | C4×Dic10 | D5×C42 | C42⋊D5 | C4×D20 | C23.21D10 | C4×C5⋊D4 | C2×C4×C20 | C2×C4○D20 | C4○D20 | C2×C42 | C20 | C42 | C22×C4 | C2×C4 | C4 |
| # reps | 1 | 2 | 2 | 2 | 2 | 1 | 4 | 1 | 1 | 16 | 2 | 8 | 8 | 6 | 16 | 32 |
In GAP, Magma, Sage, TeX
C_4\times C_4\circ D_{20} % in TeX
G:=Group("C4xC4oD20"); // GroupNames label
G:=SmallGroup(320,1146);
// by ID
G=gap.SmallGroup(320,1146);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^10=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀