metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.72D10, C4⋊1D4.4D5, (C2×D4).54D10, (C2×C20).291D4, C20.75(C4○D4), D4⋊Dic5⋊21C2, C20.6Q8⋊13C2, C4.23(D4⋊2D5), C10.93(C8⋊C22), (C2×C20).389C23, (C4×C20).119C22, (D4×C10).70C22, C42.D5⋊12C2, C10.44(C4.4D4), C4⋊Dic5.155C22, C2.14(D4.D10), C2.11(C20.17D4), C5⋊4(C42.29C22), (C5×C4⋊1D4).3C2, (C2×C10).520(C2×D4), (C2×C4).69(C5⋊D4), (C2×C4).487(C22×D5), C22.193(C2×C5⋊D4), (C2×C5⋊2C8).129C22, SmallGroup(320,698)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.72D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=a2bc-1 >
Subgroups: 366 in 110 conjugacy classes, 39 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, C23, C10, C10, C10, C42, C4⋊C4, C2×C8, C2×D4, C2×D4, Dic5, C20, C20, C2×C10, C2×C10, C8⋊C4, D4⋊C4, C42.C2, C4⋊1D4, C5⋊2C8, C2×Dic5, C2×C20, C2×C20, C5×D4, C22×C10, C42.29C22, C2×C5⋊2C8, C10.D4, C4⋊Dic5, C4×C20, D4×C10, D4×C10, C42.D5, D4⋊Dic5, C20.6Q8, C5×C4⋊1D4, C42.72D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4.4D4, C8⋊C22, C5⋊D4, C22×D5, C42.29C22, D4⋊2D5, C2×C5⋊D4, D4.D10, C20.17D4, C42.72D10
►On 160 points
Generators in S160
(1 6 24 29)(2 30 25 7)(3 8 26 21)(4 22 27 9)(5 10 28 23)(11 36 58 112)(12 113 59 37)(13 38 60 114)(14 115 51 39)(15 40 52 116)(16 117 53 31)(17 32 54 118)(18 119 55 33)(19 34 56 120)(20 111 57 35)(41 106 86 91)(42 92 87 107)(43 108 88 93)(44 94 89 109)(45 110 90 95)(46 96 81 101)(47 102 82 97)(48 98 83 103)(49 104 84 99)(50 100 85 105)(61 66 71 76)(62 77 72 67)(63 68 73 78)(64 79 74 69)(65 70 75 80)(121 157 131 147)(122 148 132 158)(123 159 133 149)(124 150 134 160)(125 151 135 141)(126 142 136 152)(127 153 137 143)(128 144 138 154)(129 155 139 145)(130 146 140 156)
(1 40 63 57)(2 58 64 31)(3 32 65 59)(4 60 66 33)(5 34 67 51)(6 52 68 35)(7 36 69 53)(8 54 70 37)(9 38 61 55)(10 56 62 39)(11 74 117 25)(12 26 118 75)(13 76 119 27)(14 28 120 77)(15 78 111 29)(16 30 112 79)(17 80 113 21)(18 22 114 71)(19 72 115 23)(20 24 116 73)(41 101 127 158)(42 159 128 102)(43 103 129 160)(44 151 130 104)(45 105 121 152)(46 153 122 106)(47 107 123 154)(48 155 124 108)(49 109 125 156)(50 157 126 110)(81 143 132 91)(82 92 133 144)(83 145 134 93)(84 94 135 146)(85 147 136 95)(86 96 137 148)(87 149 138 97)(88 98 139 150)(89 141 140 99)(90 100 131 142)
(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)
(1 109 24 94)(2 134 25 124)(3 107 26 92)(4 132 27 122)(5 105 28 100)(6 140 29 130)(7 103 30 98)(8 138 21 128)(9 101 22 96)(10 136 23 126)(11 155 58 145)(12 133 59 123)(13 153 60 143)(14 131 51 121)(15 151 52 141)(16 139 53 129)(17 159 54 149)(18 137 55 127)(19 157 56 147)(20 135 57 125)(31 93 117 108)(32 47 118 82)(33 91 119 106)(34 45 120 90)(35 99 111 104)(36 43 112 88)(37 97 113 102)(38 41 114 86)(39 95 115 110)(40 49 116 84)(42 70 87 80)(44 68 89 78)(46 66 81 76)(48 64 83 74)(50 62 85 72)(61 158 71 148)(63 156 73 146)(65 154 75 144)(67 152 77 142)(69 160 79 150)
G:=sub<Sym(160)| (1,6,24,29)(2,30,25,7)(3,8,26,21)(4,22,27,9)(5,10,28,23)(11,36,58,112)(12,113,59,37)(13,38,60,114)(14,115,51,39)(15,40,52,116)(16,117,53,31)(17,32,54,118)(18,119,55,33)(19,34,56,120)(20,111,57,35)(41,106,86,91)(42,92,87,107)(43,108,88,93)(44,94,89,109)(45,110,90,95)(46,96,81,101)(47,102,82,97)(48,98,83,103)(49,104,84,99)(50,100,85,105)(61,66,71,76)(62,77,72,67)(63,68,73,78)(64,79,74,69)(65,70,75,80)(121,157,131,147)(122,148,132,158)(123,159,133,149)(124,150,134,160)(125,151,135,141)(126,142,136,152)(127,153,137,143)(128,144,138,154)(129,155,139,145)(130,146,140,156), (1,40,63,57)(2,58,64,31)(3,32,65,59)(4,60,66,33)(5,34,67,51)(6,52,68,35)(7,36,69,53)(8,54,70,37)(9,38,61,55)(10,56,62,39)(11,74,117,25)(12,26,118,75)(13,76,119,27)(14,28,120,77)(15,78,111,29)(16,30,112,79)(17,80,113,21)(18,22,114,71)(19,72,115,23)(20,24,116,73)(41,101,127,158)(42,159,128,102)(43,103,129,160)(44,151,130,104)(45,105,121,152)(46,153,122,106)(47,107,123,154)(48,155,124,108)(49,109,125,156)(50,157,126,110)(81,143,132,91)(82,92,133,144)(83,145,134,93)(84,94,135,146)(85,147,136,95)(86,96,137,148)(87,149,138,97)(88,98,139,150)(89,141,140,99)(90,100,131,142), (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), (1,109,24,94)(2,134,25,124)(3,107,26,92)(4,132,27,122)(5,105,28,100)(6,140,29,130)(7,103,30,98)(8,138,21,128)(9,101,22,96)(10,136,23,126)(11,155,58,145)(12,133,59,123)(13,153,60,143)(14,131,51,121)(15,151,52,141)(16,139,53,129)(17,159,54,149)(18,137,55,127)(19,157,56,147)(20,135,57,125)(31,93,117,108)(32,47,118,82)(33,91,119,106)(34,45,120,90)(35,99,111,104)(36,43,112,88)(37,97,113,102)(38,41,114,86)(39,95,115,110)(40,49,116,84)(42,70,87,80)(44,68,89,78)(46,66,81,76)(48,64,83,74)(50,62,85,72)(61,158,71,148)(63,156,73,146)(65,154,75,144)(67,152,77,142)(69,160,79,150)>;
G:=Group( (1,6,24,29)(2,30,25,7)(3,8,26,21)(4,22,27,9)(5,10,28,23)(11,36,58,112)(12,113,59,37)(13,38,60,114)(14,115,51,39)(15,40,52,116)(16,117,53,31)(17,32,54,118)(18,119,55,33)(19,34,56,120)(20,111,57,35)(41,106,86,91)(42,92,87,107)(43,108,88,93)(44,94,89,109)(45,110,90,95)(46,96,81,101)(47,102,82,97)(48,98,83,103)(49,104,84,99)(50,100,85,105)(61,66,71,76)(62,77,72,67)(63,68,73,78)(64,79,74,69)(65,70,75,80)(121,157,131,147)(122,148,132,158)(123,159,133,149)(124,150,134,160)(125,151,135,141)(126,142,136,152)(127,153,137,143)(128,144,138,154)(129,155,139,145)(130,146,140,156), (1,40,63,57)(2,58,64,31)(3,32,65,59)(4,60,66,33)(5,34,67,51)(6,52,68,35)(7,36,69,53)(8,54,70,37)(9,38,61,55)(10,56,62,39)(11,74,117,25)(12,26,118,75)(13,76,119,27)(14,28,120,77)(15,78,111,29)(16,30,112,79)(17,80,113,21)(18,22,114,71)(19,72,115,23)(20,24,116,73)(41,101,127,158)(42,159,128,102)(43,103,129,160)(44,151,130,104)(45,105,121,152)(46,153,122,106)(47,107,123,154)(48,155,124,108)(49,109,125,156)(50,157,126,110)(81,143,132,91)(82,92,133,144)(83,145,134,93)(84,94,135,146)(85,147,136,95)(86,96,137,148)(87,149,138,97)(88,98,139,150)(89,141,140,99)(90,100,131,142), (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), (1,109,24,94)(2,134,25,124)(3,107,26,92)(4,132,27,122)(5,105,28,100)(6,140,29,130)(7,103,30,98)(8,138,21,128)(9,101,22,96)(10,136,23,126)(11,155,58,145)(12,133,59,123)(13,153,60,143)(14,131,51,121)(15,151,52,141)(16,139,53,129)(17,159,54,149)(18,137,55,127)(19,157,56,147)(20,135,57,125)(31,93,117,108)(32,47,118,82)(33,91,119,106)(34,45,120,90)(35,99,111,104)(36,43,112,88)(37,97,113,102)(38,41,114,86)(39,95,115,110)(40,49,116,84)(42,70,87,80)(44,68,89,78)(46,66,81,76)(48,64,83,74)(50,62,85,72)(61,158,71,148)(63,156,73,146)(65,154,75,144)(67,152,77,142)(69,160,79,150) );
G=PermutationGroup([[(1,6,24,29),(2,30,25,7),(3,8,26,21),(4,22,27,9),(5,10,28,23),(11,36,58,112),(12,113,59,37),(13,38,60,114),(14,115,51,39),(15,40,52,116),(16,117,53,31),(17,32,54,118),(18,119,55,33),(19,34,56,120),(20,111,57,35),(41,106,86,91),(42,92,87,107),(43,108,88,93),(44,94,89,109),(45,110,90,95),(46,96,81,101),(47,102,82,97),(48,98,83,103),(49,104,84,99),(50,100,85,105),(61,66,71,76),(62,77,72,67),(63,68,73,78),(64,79,74,69),(65,70,75,80),(121,157,131,147),(122,148,132,158),(123,159,133,149),(124,150,134,160),(125,151,135,141),(126,142,136,152),(127,153,137,143),(128,144,138,154),(129,155,139,145),(130,146,140,156)], [(1,40,63,57),(2,58,64,31),(3,32,65,59),(4,60,66,33),(5,34,67,51),(6,52,68,35),(7,36,69,53),(8,54,70,37),(9,38,61,55),(10,56,62,39),(11,74,117,25),(12,26,118,75),(13,76,119,27),(14,28,120,77),(15,78,111,29),(16,30,112,79),(17,80,113,21),(18,22,114,71),(19,72,115,23),(20,24,116,73),(41,101,127,158),(42,159,128,102),(43,103,129,160),(44,151,130,104),(45,105,121,152),(46,153,122,106),(47,107,123,154),(48,155,124,108),(49,109,125,156),(50,157,126,110),(81,143,132,91),(82,92,133,144),(83,145,134,93),(84,94,135,146),(85,147,136,95),(86,96,137,148),(87,149,138,97),(88,98,139,150),(89,141,140,99),(90,100,131,142)], [(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)], [(1,109,24,94),(2,134,25,124),(3,107,26,92),(4,132,27,122),(5,105,28,100),(6,140,29,130),(7,103,30,98),(8,138,21,128),(9,101,22,96),(10,136,23,126),(11,155,58,145),(12,133,59,123),(13,153,60,143),(14,131,51,121),(15,151,52,141),(16,139,53,129),(17,159,54,149),(18,137,55,127),(19,157,56,147),(20,135,57,125),(31,93,117,108),(32,47,118,82),(33,91,119,106),(34,45,120,90),(35,99,111,104),(36,43,112,88),(37,97,113,102),(38,41,114,86),(39,95,115,110),(40,49,116,84),(42,70,87,80),(44,68,89,78),(46,66,81,76),(48,64,83,74),(50,62,85,72),(61,158,71,148),(63,156,73,146),(65,154,75,144),(67,152,77,142),(69,160,79,150)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 2 | 2 | 4 | 4 | 40 | 40 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | C8⋊C22 | D4⋊2D5 | D4.D10 |
| kernel | C42.72D10 | C42.D5 | D4⋊Dic5 | C20.6Q8 | C5×C4⋊1D4 | C2×C20 | C4⋊1D4 | C20 | C42 | C2×D4 | C2×C4 | C10 | C4 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 8 | 2 | 4 | 8 |
Matrix representation of C42.72D10 ►in GL6(𝔽41)
| 1 | 23 | 0 | 0 | 0 | 0 |
| 32 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 39 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 40 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 39 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 39 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 32 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 37 | 33 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 0 | 31 | 31 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 1 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 20 |
| 0 | 0 | 0 | 0 | 0 | 31 |
| 0 | 0 | 4 | 8 | 0 | 0 |
| 0 | 0 | 0 | 37 | 0 | 0 |
G:=sub<GL(6,GF(41))| [1,32,0,0,0,0,23,40,0,0,0,0,0,0,40,1,0,0,0,0,39,1,0,0,0,0,0,0,1,40,0,0,0,0,2,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,1,0,0,0,0,39,1,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[1,32,0,0,0,0,0,40,0,0,0,0,0,0,37,0,0,0,0,0,33,4,0,0,0,0,0,0,10,31,0,0,0,0,0,31],[9,1,0,0,0,0,0,32,0,0,0,0,0,0,0,0,4,0,0,0,0,0,8,37,0,0,10,0,0,0,0,0,20,31,0,0] >;
C42.72D10 in GAP, Magma, Sage, TeX
C_4^2._{72}D_{10} % in TeX
G:=Group("C4^2.72D10"); // GroupNames label
G:=SmallGroup(320,698);
// by ID
G=gap.SmallGroup(320,698);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,477,64,590,135,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=a^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*b*c^-1>;
// generators/relations