metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.94D10, C10.512- 1+4, C4⋊C4.271D10, C42⋊D5⋊2C2, C20.6Q8⋊6C2, D10⋊2Q8⋊12C2, C4.95(C4○D20), C42⋊C2⋊13D5, (C2×C10).73C24, (C4×C20).24C22, C22⋊C4.97D10, C4.Dic10⋊12C2, D10.30(C4○D4), Dic5⋊3Q8⋊12C2, C20.197(C4○D4), C20.48D4⋊29C2, (C2×C20).148C23, (C22×C4).194D10, C23.D10⋊3C2, C4⋊Dic5.34C22, C23.85(C22×D5), D10.12D4.1C2, (C4×Dic5).77C22, (C2×Dic5).26C23, C22.102(C23×D5), C2.9(D4.10D10), C23.D5.96C22, D10⋊C4.96C22, (C22×C10).143C23, (C22×C20).231C22, C5⋊2(C22.46C24), C10.D4.97C22, (C22×D5).176C23, (C2×Dic10).149C22, (D5×C4⋊C4)⋊12C2, C2.12(D5×C4○D4), (C4×C5⋊D4).5C2, C2.32(C2×C4○D20), C10.30(C2×C4○D4), (C2×C4×D5).69C22, (C5×C42⋊C2)⋊15C2, (C5×C4⋊C4).309C22, (C2×C4).275(C22×D5), (C2×C5⋊D4).111C22, (C5×C22⋊C4).113C22, SmallGroup(320,1201)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.94D10
G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=dad-1=ab2, bc=cb, dbd-1=a2b, dcd-1=c9 >
Subgroups: 662 in 214 conjugacy classes, 97 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C22×D5, C22×C10, C22.46C24, C4×Dic5, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, D10⋊C4, C23.D5, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×C4×D5, C2×C5⋊D4, C22×C20, C20.6Q8, C42⋊D5, C23.D10, D10.12D4, Dic5⋊3Q8, C4.Dic10, D5×C4⋊C4, D10⋊2Q8, C20.48D4, C4×C5⋊D4, C5×C42⋊C2, C42.94D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2- 1+4, C22×D5, C22.46C24, C4○D20, C23×D5, C2×C4○D20, D5×C4○D4, D4.10D10, C42.94D10
►On 160 points
Generators in S160
(1 52 108 88)(2 43 109 99)(3 54 110 90)(4 45 111 81)(5 56 112 92)(6 47 113 83)(7 58 114 94)(8 49 115 85)(9 60 116 96)(10 51 117 87)(11 42 118 98)(12 53 119 89)(13 44 120 100)(14 55 101 91)(15 46 102 82)(16 57 103 93)(17 48 104 84)(18 59 105 95)(19 50 106 86)(20 41 107 97)(21 134 72 148)(22 125 73 159)(23 136 74 150)(24 127 75 141)(25 138 76 152)(26 129 77 143)(27 140 78 154)(28 131 79 145)(29 122 80 156)(30 133 61 147)(31 124 62 158)(32 135 63 149)(33 126 64 160)(34 137 65 151)(35 128 66 142)(36 139 67 153)(37 130 68 144)(38 121 69 155)(39 132 70 146)(40 123 71 157)
(1 151 11 141)(2 152 12 142)(3 153 13 143)(4 154 14 144)(5 155 15 145)(6 156 16 146)(7 157 17 147)(8 158 18 148)(9 159 19 149)(10 160 20 150)(21 49 31 59)(22 50 32 60)(23 51 33 41)(24 52 34 42)(25 53 35 43)(26 54 36 44)(27 55 37 45)(28 56 38 46)(29 57 39 47)(30 58 40 48)(61 94 71 84)(62 95 72 85)(63 96 73 86)(64 97 74 87)(65 98 75 88)(66 99 76 89)(67 100 77 90)(68 81 78 91)(69 82 79 92)(70 83 80 93)(101 130 111 140)(102 131 112 121)(103 132 113 122)(104 133 114 123)(105 134 115 124)(106 135 116 125)(107 136 117 126)(108 137 118 127)(109 138 119 128)(110 139 120 129)
(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 20 11 10)(2 9 12 19)(3 18 13 8)(4 7 14 17)(5 16 15 6)(21 67 31 77)(22 76 32 66)(23 65 33 75)(24 74 34 64)(25 63 35 73)(26 72 36 62)(27 61 37 71)(28 70 38 80)(29 79 39 69)(30 68 40 78)(41 52 51 42)(43 50 53 60)(44 59 54 49)(45 48 55 58)(46 57 56 47)(81 84 91 94)(82 93 92 83)(85 100 95 90)(86 89 96 99)(87 98 97 88)(101 104 111 114)(102 113 112 103)(105 120 115 110)(106 109 116 119)(107 118 117 108)(121 146 131 156)(122 155 132 145)(123 144 133 154)(124 153 134 143)(125 142 135 152)(126 151 136 141)(127 160 137 150)(128 149 138 159)(129 158 139 148)(130 147 140 157)
G:=sub<Sym(160)| (1,52,108,88)(2,43,109,99)(3,54,110,90)(4,45,111,81)(5,56,112,92)(6,47,113,83)(7,58,114,94)(8,49,115,85)(9,60,116,96)(10,51,117,87)(11,42,118,98)(12,53,119,89)(13,44,120,100)(14,55,101,91)(15,46,102,82)(16,57,103,93)(17,48,104,84)(18,59,105,95)(19,50,106,86)(20,41,107,97)(21,134,72,148)(22,125,73,159)(23,136,74,150)(24,127,75,141)(25,138,76,152)(26,129,77,143)(27,140,78,154)(28,131,79,145)(29,122,80,156)(30,133,61,147)(31,124,62,158)(32,135,63,149)(33,126,64,160)(34,137,65,151)(35,128,66,142)(36,139,67,153)(37,130,68,144)(38,121,69,155)(39,132,70,146)(40,123,71,157), (1,151,11,141)(2,152,12,142)(3,153,13,143)(4,154,14,144)(5,155,15,145)(6,156,16,146)(7,157,17,147)(8,158,18,148)(9,159,19,149)(10,160,20,150)(21,49,31,59)(22,50,32,60)(23,51,33,41)(24,52,34,42)(25,53,35,43)(26,54,36,44)(27,55,37,45)(28,56,38,46)(29,57,39,47)(30,58,40,48)(61,94,71,84)(62,95,72,85)(63,96,73,86)(64,97,74,87)(65,98,75,88)(66,99,76,89)(67,100,77,90)(68,81,78,91)(69,82,79,92)(70,83,80,93)(101,130,111,140)(102,131,112,121)(103,132,113,122)(104,133,114,123)(105,134,115,124)(106,135,116,125)(107,136,117,126)(108,137,118,127)(109,138,119,128)(110,139,120,129), (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), (1,20,11,10)(2,9,12,19)(3,18,13,8)(4,7,14,17)(5,16,15,6)(21,67,31,77)(22,76,32,66)(23,65,33,75)(24,74,34,64)(25,63,35,73)(26,72,36,62)(27,61,37,71)(28,70,38,80)(29,79,39,69)(30,68,40,78)(41,52,51,42)(43,50,53,60)(44,59,54,49)(45,48,55,58)(46,57,56,47)(81,84,91,94)(82,93,92,83)(85,100,95,90)(86,89,96,99)(87,98,97,88)(101,104,111,114)(102,113,112,103)(105,120,115,110)(106,109,116,119)(107,118,117,108)(121,146,131,156)(122,155,132,145)(123,144,133,154)(124,153,134,143)(125,142,135,152)(126,151,136,141)(127,160,137,150)(128,149,138,159)(129,158,139,148)(130,147,140,157)>;
G:=Group( (1,52,108,88)(2,43,109,99)(3,54,110,90)(4,45,111,81)(5,56,112,92)(6,47,113,83)(7,58,114,94)(8,49,115,85)(9,60,116,96)(10,51,117,87)(11,42,118,98)(12,53,119,89)(13,44,120,100)(14,55,101,91)(15,46,102,82)(16,57,103,93)(17,48,104,84)(18,59,105,95)(19,50,106,86)(20,41,107,97)(21,134,72,148)(22,125,73,159)(23,136,74,150)(24,127,75,141)(25,138,76,152)(26,129,77,143)(27,140,78,154)(28,131,79,145)(29,122,80,156)(30,133,61,147)(31,124,62,158)(32,135,63,149)(33,126,64,160)(34,137,65,151)(35,128,66,142)(36,139,67,153)(37,130,68,144)(38,121,69,155)(39,132,70,146)(40,123,71,157), (1,151,11,141)(2,152,12,142)(3,153,13,143)(4,154,14,144)(5,155,15,145)(6,156,16,146)(7,157,17,147)(8,158,18,148)(9,159,19,149)(10,160,20,150)(21,49,31,59)(22,50,32,60)(23,51,33,41)(24,52,34,42)(25,53,35,43)(26,54,36,44)(27,55,37,45)(28,56,38,46)(29,57,39,47)(30,58,40,48)(61,94,71,84)(62,95,72,85)(63,96,73,86)(64,97,74,87)(65,98,75,88)(66,99,76,89)(67,100,77,90)(68,81,78,91)(69,82,79,92)(70,83,80,93)(101,130,111,140)(102,131,112,121)(103,132,113,122)(104,133,114,123)(105,134,115,124)(106,135,116,125)(107,136,117,126)(108,137,118,127)(109,138,119,128)(110,139,120,129), (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), (1,20,11,10)(2,9,12,19)(3,18,13,8)(4,7,14,17)(5,16,15,6)(21,67,31,77)(22,76,32,66)(23,65,33,75)(24,74,34,64)(25,63,35,73)(26,72,36,62)(27,61,37,71)(28,70,38,80)(29,79,39,69)(30,68,40,78)(41,52,51,42)(43,50,53,60)(44,59,54,49)(45,48,55,58)(46,57,56,47)(81,84,91,94)(82,93,92,83)(85,100,95,90)(86,89,96,99)(87,98,97,88)(101,104,111,114)(102,113,112,103)(105,120,115,110)(106,109,116,119)(107,118,117,108)(121,146,131,156)(122,155,132,145)(123,144,133,154)(124,153,134,143)(125,142,135,152)(126,151,136,141)(127,160,137,150)(128,149,138,159)(129,158,139,148)(130,147,140,157) );
G=PermutationGroup([[(1,52,108,88),(2,43,109,99),(3,54,110,90),(4,45,111,81),(5,56,112,92),(6,47,113,83),(7,58,114,94),(8,49,115,85),(9,60,116,96),(10,51,117,87),(11,42,118,98),(12,53,119,89),(13,44,120,100),(14,55,101,91),(15,46,102,82),(16,57,103,93),(17,48,104,84),(18,59,105,95),(19,50,106,86),(20,41,107,97),(21,134,72,148),(22,125,73,159),(23,136,74,150),(24,127,75,141),(25,138,76,152),(26,129,77,143),(27,140,78,154),(28,131,79,145),(29,122,80,156),(30,133,61,147),(31,124,62,158),(32,135,63,149),(33,126,64,160),(34,137,65,151),(35,128,66,142),(36,139,67,153),(37,130,68,144),(38,121,69,155),(39,132,70,146),(40,123,71,157)], [(1,151,11,141),(2,152,12,142),(3,153,13,143),(4,154,14,144),(5,155,15,145),(6,156,16,146),(7,157,17,147),(8,158,18,148),(9,159,19,149),(10,160,20,150),(21,49,31,59),(22,50,32,60),(23,51,33,41),(24,52,34,42),(25,53,35,43),(26,54,36,44),(27,55,37,45),(28,56,38,46),(29,57,39,47),(30,58,40,48),(61,94,71,84),(62,95,72,85),(63,96,73,86),(64,97,74,87),(65,98,75,88),(66,99,76,89),(67,100,77,90),(68,81,78,91),(69,82,79,92),(70,83,80,93),(101,130,111,140),(102,131,112,121),(103,132,113,122),(104,133,114,123),(105,134,115,124),(106,135,116,125),(107,136,117,126),(108,137,118,127),(109,138,119,128),(110,139,120,129)], [(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)], [(1,20,11,10),(2,9,12,19),(3,18,13,8),(4,7,14,17),(5,16,15,6),(21,67,31,77),(22,76,32,66),(23,65,33,75),(24,74,34,64),(25,63,35,73),(26,72,36,62),(27,61,37,71),(28,70,38,80),(29,79,39,69),(30,68,40,78),(41,52,51,42),(43,50,53,60),(44,59,54,49),(45,48,55,58),(46,57,56,47),(81,84,91,94),(82,93,92,83),(85,100,95,90),(86,89,96,99),(87,98,97,88),(101,104,111,114),(102,113,112,103),(105,120,115,110),(106,109,116,119),(107,118,117,108),(121,146,131,156),(122,155,132,145),(123,144,133,154),(124,153,134,143),(125,142,135,152),(126,151,136,141),(127,160,137,150),(128,149,138,159),(129,158,139,148),(130,147,140,157)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4R | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 10 | 10 | 2 | ··· | 2 | 4 | 4 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | D10 | C4○D20 | 2- 1+4 | D5×C4○D4 | D4.10D10 |
| kernel | C42.94D10 | C20.6Q8 | C42⋊D5 | C23.D10 | D10.12D4 | Dic5⋊3Q8 | C4.Dic10 | D5×C4⋊C4 | D10⋊2Q8 | C20.48D4 | C4×C5⋊D4 | C5×C42⋊C2 | C42⋊C2 | C20 | D10 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C4 | C10 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 2 | 16 | 1 | 4 | 4 |
Matrix representation of C42.94D10 ►in GL4(𝔽41) generated by
| 34 | 39 | 0 | 0 |
| 24 | 7 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 23 | 35 |
| 0 | 0 | 6 | 18 |
| 13 | 35 | 0 | 0 |
| 1 | 28 | 0 | 0 |
| 0 | 0 | 40 | 35 |
| 0 | 0 | 6 | 35 |
| 28 | 6 | 0 | 0 |
| 40 | 13 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 6 | 1 |
G:=sub<GL(4,GF(41))| [34,24,0,0,39,7,0,0,0,0,9,0,0,0,0,9],[32,0,0,0,0,32,0,0,0,0,23,6,0,0,35,18],[13,1,0,0,35,28,0,0,0,0,40,6,0,0,35,35],[28,40,0,0,6,13,0,0,0,0,40,6,0,0,0,1] >;
C42.94D10 in GAP, Magma, Sage, TeX
C_4^2._{94}D_{10} % in TeX
G:=Group("C4^2.94D10"); // GroupNames label
G:=SmallGroup(320,1201);
// by ID
G=gap.SmallGroup(320,1201);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,100,675,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^9>;
// generators/relations