metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.113D10, C10.192+ 1+4, (C4×D4)⋊20D5, (D4×C20)⋊22C2, (C4×D20)⋊32C2, C20⋊2D4⋊9C2, D10⋊D4⋊9C2, C4⋊C4.318D10, (C2×D4).219D10, C4.65(C4○D20), C20.6Q8⋊16C2, (C22×C4).47D10, D10.52(C4○D4), C20.110(C4○D4), (C4×C20).157C22, (C2×C20).700C23, (C2×C10).102C24, C22⋊C4.115D10, C23.D10⋊8C2, C2.20(D4⋊6D10), C23.99(C22×D5), (D4×C10).262C22, (C2×D20).221C22, C4⋊Dic5.200C22, (C2×Dic5).43C23, (C22×D5).36C23, C22.127(C23×D5), C23.D5.14C22, D10⋊C4.86C22, C23.23D10⋊17C2, (C22×C20).364C22, (C22×C10).172C23, C5⋊4(C22.47C24), (C4×Dic5).225C22, C10.D4.100C22, (D5×C4⋊C4)⋊16C2, (C4×C5⋊D4)⋊44C2, C2.25(D5×C4○D4), C4⋊C4⋊7D5⋊15C2, C2.51(C2×C4○D20), (C2×C4×D5).75C22, C10.142(C2×C4○D4), (C5×C4⋊C4).331C22, (C2×C4).285(C22×D5), (C2×C5⋊D4).17C22, (C5×C22⋊C4).126C22, SmallGroup(320,1230)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.113D10
G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b-1, dcd-1=c9 >
Subgroups: 838 in 238 conjugacy classes, 97 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4⋊D4, C22.D4, C42.C2, C42⋊2C2, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C22.47C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, D10⋊C4, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C4×D5, C2×C4×D5, C2×D20, C2×C5⋊D4, C22×C20, D4×C10, C20.6Q8, C4×D20, C23.D10, D10⋊D4, D5×C4⋊C4, C4⋊C4⋊7D5, C4×C5⋊D4, C23.23D10, C20⋊2D4, D4×C20, C42.113D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.47C24, C4○D20, C23×D5, C2×C4○D20, D4⋊6D10, D5×C4○D4, C42.113D10
►On 160 points
Generators in S160
(1 104 23 58)(2 59 24 105)(3 106 25 60)(4 41 26 107)(5 108 27 42)(6 43 28 109)(7 110 29 44)(8 45 30 111)(9 112 31 46)(10 47 32 113)(11 114 33 48)(12 49 34 115)(13 116 35 50)(14 51 36 117)(15 118 37 52)(16 53 38 119)(17 120 39 54)(18 55 40 101)(19 102 21 56)(20 57 22 103)(61 85 131 158)(62 159 132 86)(63 87 133 160)(64 141 134 88)(65 89 135 142)(66 143 136 90)(67 91 137 144)(68 145 138 92)(69 93 139 146)(70 147 140 94)(71 95 121 148)(72 149 122 96)(73 97 123 150)(74 151 124 98)(75 99 125 152)(76 153 126 100)(77 81 127 154)(78 155 128 82)(79 83 129 156)(80 157 130 84)
(1 70 11 80)(2 71 12 61)(3 72 13 62)(4 73 14 63)(5 74 15 64)(6 75 16 65)(7 76 17 66)(8 77 18 67)(9 78 19 68)(10 79 20 69)(21 138 31 128)(22 139 32 129)(23 140 33 130)(24 121 34 131)(25 122 35 132)(26 123 36 133)(27 124 37 134)(28 125 38 135)(29 126 39 136)(30 127 40 137)(41 97 51 87)(42 98 52 88)(43 99 53 89)(44 100 54 90)(45 81 55 91)(46 82 56 92)(47 83 57 93)(48 84 58 94)(49 85 59 95)(50 86 60 96)(101 144 111 154)(102 145 112 155)(103 146 113 156)(104 147 114 157)(105 148 115 158)(106 149 116 159)(107 150 117 160)(108 151 118 141)(109 152 119 142)(110 153 120 143)
(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 93 11 83)(2 82 12 92)(3 91 13 81)(4 100 14 90)(5 89 15 99)(6 98 16 88)(7 87 17 97)(8 96 18 86)(9 85 19 95)(10 94 20 84)(21 148 31 158)(22 157 32 147)(23 146 33 156)(24 155 34 145)(25 144 35 154)(26 153 36 143)(27 142 37 152)(28 151 38 141)(29 160 39 150)(30 149 40 159)(41 126 51 136)(42 135 52 125)(43 124 53 134)(44 133 54 123)(45 122 55 132)(46 131 56 121)(47 140 57 130)(48 129 58 139)(49 138 59 128)(50 127 60 137)(61 102 71 112)(62 111 72 101)(63 120 73 110)(64 109 74 119)(65 118 75 108)(66 107 76 117)(67 116 77 106)(68 105 78 115)(69 114 79 104)(70 103 80 113)
G:=sub<Sym(160)| (1,104,23,58)(2,59,24,105)(3,106,25,60)(4,41,26,107)(5,108,27,42)(6,43,28,109)(7,110,29,44)(8,45,30,111)(9,112,31,46)(10,47,32,113)(11,114,33,48)(12,49,34,115)(13,116,35,50)(14,51,36,117)(15,118,37,52)(16,53,38,119)(17,120,39,54)(18,55,40,101)(19,102,21,56)(20,57,22,103)(61,85,131,158)(62,159,132,86)(63,87,133,160)(64,141,134,88)(65,89,135,142)(66,143,136,90)(67,91,137,144)(68,145,138,92)(69,93,139,146)(70,147,140,94)(71,95,121,148)(72,149,122,96)(73,97,123,150)(74,151,124,98)(75,99,125,152)(76,153,126,100)(77,81,127,154)(78,155,128,82)(79,83,129,156)(80,157,130,84), (1,70,11,80)(2,71,12,61)(3,72,13,62)(4,73,14,63)(5,74,15,64)(6,75,16,65)(7,76,17,66)(8,77,18,67)(9,78,19,68)(10,79,20,69)(21,138,31,128)(22,139,32,129)(23,140,33,130)(24,121,34,131)(25,122,35,132)(26,123,36,133)(27,124,37,134)(28,125,38,135)(29,126,39,136)(30,127,40,137)(41,97,51,87)(42,98,52,88)(43,99,53,89)(44,100,54,90)(45,81,55,91)(46,82,56,92)(47,83,57,93)(48,84,58,94)(49,85,59,95)(50,86,60,96)(101,144,111,154)(102,145,112,155)(103,146,113,156)(104,147,114,157)(105,148,115,158)(106,149,116,159)(107,150,117,160)(108,151,118,141)(109,152,119,142)(110,153,120,143), (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), (1,93,11,83)(2,82,12,92)(3,91,13,81)(4,100,14,90)(5,89,15,99)(6,98,16,88)(7,87,17,97)(8,96,18,86)(9,85,19,95)(10,94,20,84)(21,148,31,158)(22,157,32,147)(23,146,33,156)(24,155,34,145)(25,144,35,154)(26,153,36,143)(27,142,37,152)(28,151,38,141)(29,160,39,150)(30,149,40,159)(41,126,51,136)(42,135,52,125)(43,124,53,134)(44,133,54,123)(45,122,55,132)(46,131,56,121)(47,140,57,130)(48,129,58,139)(49,138,59,128)(50,127,60,137)(61,102,71,112)(62,111,72,101)(63,120,73,110)(64,109,74,119)(65,118,75,108)(66,107,76,117)(67,116,77,106)(68,105,78,115)(69,114,79,104)(70,103,80,113)>;
G:=Group( (1,104,23,58)(2,59,24,105)(3,106,25,60)(4,41,26,107)(5,108,27,42)(6,43,28,109)(7,110,29,44)(8,45,30,111)(9,112,31,46)(10,47,32,113)(11,114,33,48)(12,49,34,115)(13,116,35,50)(14,51,36,117)(15,118,37,52)(16,53,38,119)(17,120,39,54)(18,55,40,101)(19,102,21,56)(20,57,22,103)(61,85,131,158)(62,159,132,86)(63,87,133,160)(64,141,134,88)(65,89,135,142)(66,143,136,90)(67,91,137,144)(68,145,138,92)(69,93,139,146)(70,147,140,94)(71,95,121,148)(72,149,122,96)(73,97,123,150)(74,151,124,98)(75,99,125,152)(76,153,126,100)(77,81,127,154)(78,155,128,82)(79,83,129,156)(80,157,130,84), (1,70,11,80)(2,71,12,61)(3,72,13,62)(4,73,14,63)(5,74,15,64)(6,75,16,65)(7,76,17,66)(8,77,18,67)(9,78,19,68)(10,79,20,69)(21,138,31,128)(22,139,32,129)(23,140,33,130)(24,121,34,131)(25,122,35,132)(26,123,36,133)(27,124,37,134)(28,125,38,135)(29,126,39,136)(30,127,40,137)(41,97,51,87)(42,98,52,88)(43,99,53,89)(44,100,54,90)(45,81,55,91)(46,82,56,92)(47,83,57,93)(48,84,58,94)(49,85,59,95)(50,86,60,96)(101,144,111,154)(102,145,112,155)(103,146,113,156)(104,147,114,157)(105,148,115,158)(106,149,116,159)(107,150,117,160)(108,151,118,141)(109,152,119,142)(110,153,120,143), (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), (1,93,11,83)(2,82,12,92)(3,91,13,81)(4,100,14,90)(5,89,15,99)(6,98,16,88)(7,87,17,97)(8,96,18,86)(9,85,19,95)(10,94,20,84)(21,148,31,158)(22,157,32,147)(23,146,33,156)(24,155,34,145)(25,144,35,154)(26,153,36,143)(27,142,37,152)(28,151,38,141)(29,160,39,150)(30,149,40,159)(41,126,51,136)(42,135,52,125)(43,124,53,134)(44,133,54,123)(45,122,55,132)(46,131,56,121)(47,140,57,130)(48,129,58,139)(49,138,59,128)(50,127,60,137)(61,102,71,112)(62,111,72,101)(63,120,73,110)(64,109,74,119)(65,118,75,108)(66,107,76,117)(67,116,77,106)(68,105,78,115)(69,114,79,104)(70,103,80,113) );
G=PermutationGroup([[(1,104,23,58),(2,59,24,105),(3,106,25,60),(4,41,26,107),(5,108,27,42),(6,43,28,109),(7,110,29,44),(8,45,30,111),(9,112,31,46),(10,47,32,113),(11,114,33,48),(12,49,34,115),(13,116,35,50),(14,51,36,117),(15,118,37,52),(16,53,38,119),(17,120,39,54),(18,55,40,101),(19,102,21,56),(20,57,22,103),(61,85,131,158),(62,159,132,86),(63,87,133,160),(64,141,134,88),(65,89,135,142),(66,143,136,90),(67,91,137,144),(68,145,138,92),(69,93,139,146),(70,147,140,94),(71,95,121,148),(72,149,122,96),(73,97,123,150),(74,151,124,98),(75,99,125,152),(76,153,126,100),(77,81,127,154),(78,155,128,82),(79,83,129,156),(80,157,130,84)], [(1,70,11,80),(2,71,12,61),(3,72,13,62),(4,73,14,63),(5,74,15,64),(6,75,16,65),(7,76,17,66),(8,77,18,67),(9,78,19,68),(10,79,20,69),(21,138,31,128),(22,139,32,129),(23,140,33,130),(24,121,34,131),(25,122,35,132),(26,123,36,133),(27,124,37,134),(28,125,38,135),(29,126,39,136),(30,127,40,137),(41,97,51,87),(42,98,52,88),(43,99,53,89),(44,100,54,90),(45,81,55,91),(46,82,56,92),(47,83,57,93),(48,84,58,94),(49,85,59,95),(50,86,60,96),(101,144,111,154),(102,145,112,155),(103,146,113,156),(104,147,114,157),(105,148,115,158),(106,149,116,159),(107,150,117,160),(108,151,118,141),(109,152,119,142),(110,153,120,143)], [(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)], [(1,93,11,83),(2,82,12,92),(3,91,13,81),(4,100,14,90),(5,89,15,99),(6,98,16,88),(7,87,17,97),(8,96,18,86),(9,85,19,95),(10,94,20,84),(21,148,31,158),(22,157,32,147),(23,146,33,156),(24,155,34,145),(25,144,35,154),(26,153,36,143),(27,142,37,152),(28,151,38,141),(29,160,39,150),(30,149,40,159),(41,126,51,136),(42,135,52,125),(43,124,53,134),(44,133,54,123),(45,122,55,132),(46,131,56,121),(47,140,57,130),(48,129,58,139),(49,138,59,128),(50,127,60,137),(61,102,71,112),(62,111,72,101),(63,120,73,110),(64,109,74,119),(65,118,75,108),(66,107,76,117),(67,116,77,106),(68,105,78,115),(69,114,79,104),(70,103,80,113)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4H | 4I | 4J | 4K | 4L | ··· | 4P | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 10 | 10 | 20 | 2 | ··· | 2 | 4 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | D10 | D10 | C4○D20 | 2+ 1+4 | D4⋊6D10 | D5×C4○D4 |
| kernel | C42.113D10 | C20.6Q8 | C4×D20 | C23.D10 | D10⋊D4 | D5×C4⋊C4 | C4⋊C4⋊7D5 | C4×C5⋊D4 | C23.23D10 | C20⋊2D4 | D4×C20 | C4×D4 | C20 | D10 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 2 | 4 | 4 | 2 | 4 | 2 | 4 | 2 | 16 | 1 | 4 | 4 |
Matrix representation of C42.113D10 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 32 |
| 39 | 28 | 0 | 0 |
| 13 | 2 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 28 | 28 | 0 | 0 |
| 13 | 32 | 0 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 0 | 9 | 0 |
| 21 | 20 | 0 | 0 |
| 23 | 20 | 0 | 0 |
| 0 | 0 | 0 | 32 |
| 0 | 0 | 32 | 0 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,32],[39,13,0,0,28,2,0,0,0,0,9,0,0,0,0,9],[28,13,0,0,28,32,0,0,0,0,0,9,0,0,9,0],[21,23,0,0,20,20,0,0,0,0,0,32,0,0,32,0] >;
C42.113D10 in GAP, Magma, Sage, TeX
C_4^2._{113}D_{10} % in TeX
G:=Group("C4^2.113D10"); // GroupNames label
G:=SmallGroup(320,1230);
// by ID
G=gap.SmallGroup(320,1230);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,100,1571,570,12550]);
// 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^-1,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^9>;
// generators/relations