direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: SD16×C2×C10, C40⋊14C23, C20.79C24, C8⋊3(C22×C10), C4.19(D4×C10), (C22×C8)⋊10C10, (C22×C40)⋊24C2, (C2×C40)⋊52C22, (C2×C20).433D4, C20.326(C2×D4), C4.2(C23×C10), (C5×Q8)⋊11C23, Q8⋊1(C22×C10), (C22×Q8)⋊8C10, C23.61(C5×D4), (Q8×C10)⋊53C22, D4.1(C22×C10), (C5×D4).34C23, C22.66(D4×C10), (C2×C20).972C23, (C22×D4).12C10, (C22×C10).222D4, C10.200(C22×D4), (D4×C10).327C22, (C22×C20).602C22, (Q8×C2×C10)⋊20C2, C2.24(D4×C2×C10), (C2×C8)⋊14(C2×C10), (D4×C2×C10).25C2, (C2×C4).89(C5×D4), (C2×Q8)⋊13(C2×C10), (C2×D4).73(C2×C10), (C2×C10).687(C2×D4), (C22×C4).129(C2×C10), (C2×C4).142(C22×C10), SmallGroup(320,1572)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for SD16×C2×C10
G = < a,b,c,d | a2=b10=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c3 >
Subgroups: 498 in 298 conjugacy classes, 178 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C10, C10, C10, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C20, C20, C20, C2×C10, C2×C10, C22×C8, C2×SD16, C22×D4, C22×Q8, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×C10, C22×C10, C22×SD16, C2×C40, C5×SD16, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, Q8×C10, C23×C10, C22×C40, C10×SD16, D4×C2×C10, Q8×C2×C10, SD16×C2×C10
Quotients: C1, C2, C22, C5, D4, C23, C10, SD16, C2×D4, C24, C2×C10, C2×SD16, C22×D4, C5×D4, C22×C10, C22×SD16, C5×SD16, D4×C10, C23×C10, C10×SD16, D4×C2×C10, SD16×C2×C10
(1 94)(2 95)(3 96)(4 97)(5 98)(6 99)(7 100)(8 91)(9 92)(10 93)(11 146)(12 147)(13 148)(14 149)(15 150)(16 141)(17 142)(18 143)(19 144)(20 145)(21 112)(22 113)(23 114)(24 115)(25 116)(26 117)(27 118)(28 119)(29 120)(30 111)(31 140)(32 131)(33 132)(34 133)(35 134)(36 135)(37 136)(38 137)(39 138)(40 139)(41 73)(42 74)(43 75)(44 76)(45 77)(46 78)(47 79)(48 80)(49 71)(50 72)(51 103)(52 104)(53 105)(54 106)(55 107)(56 108)(57 109)(58 110)(59 101)(60 102)(61 88)(62 89)(63 90)(64 81)(65 82)(66 83)(67 84)(68 85)(69 86)(70 87)(121 153)(122 154)(123 155)(124 156)(125 157)(126 158)(127 159)(128 160)(129 151)(130 152)
(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)
(1 153 41 11 59 27 69 40)(2 154 42 12 60 28 70 31)(3 155 43 13 51 29 61 32)(4 156 44 14 52 30 62 33)(5 157 45 15 53 21 63 34)(6 158 46 16 54 22 64 35)(7 159 47 17 55 23 65 36)(8 160 48 18 56 24 66 37)(9 151 49 19 57 25 67 38)(10 152 50 20 58 26 68 39)(71 144 109 116 84 137 92 129)(72 145 110 117 85 138 93 130)(73 146 101 118 86 139 94 121)(74 147 102 119 87 140 95 122)(75 148 103 120 88 131 96 123)(76 149 104 111 89 132 97 124)(77 150 105 112 90 133 98 125)(78 141 106 113 81 134 99 126)(79 142 107 114 82 135 100 127)(80 143 108 115 83 136 91 128)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 158)(12 159)(13 160)(14 151)(15 152)(16 153)(17 154)(18 155)(19 156)(20 157)(21 39)(22 40)(23 31)(24 32)(25 33)(26 34)(27 35)(28 36)(29 37)(30 38)(41 64)(42 65)(43 66)(44 67)(45 68)(46 69)(47 70)(48 61)(49 62)(50 63)(51 56)(52 57)(53 58)(54 59)(55 60)(71 89)(72 90)(73 81)(74 82)(75 83)(76 84)(77 85)(78 86)(79 87)(80 88)(91 96)(92 97)(93 98)(94 99)(95 100)(101 106)(102 107)(103 108)(104 109)(105 110)(111 137)(112 138)(113 139)(114 140)(115 131)(116 132)(117 133)(118 134)(119 135)(120 136)(121 141)(122 142)(123 143)(124 144)(125 145)(126 146)(127 147)(128 148)(129 149)(130 150)
G:=sub<Sym(160)| (1,94)(2,95)(3,96)(4,97)(5,98)(6,99)(7,100)(8,91)(9,92)(10,93)(11,146)(12,147)(13,148)(14,149)(15,150)(16,141)(17,142)(18,143)(19,144)(20,145)(21,112)(22,113)(23,114)(24,115)(25,116)(26,117)(27,118)(28,119)(29,120)(30,111)(31,140)(32,131)(33,132)(34,133)(35,134)(36,135)(37,136)(38,137)(39,138)(40,139)(41,73)(42,74)(43,75)(44,76)(45,77)(46,78)(47,79)(48,80)(49,71)(50,72)(51,103)(52,104)(53,105)(54,106)(55,107)(56,108)(57,109)(58,110)(59,101)(60,102)(61,88)(62,89)(63,90)(64,81)(65,82)(66,83)(67,84)(68,85)(69,86)(70,87)(121,153)(122,154)(123,155)(124,156)(125,157)(126,158)(127,159)(128,160)(129,151)(130,152), (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), (1,153,41,11,59,27,69,40)(2,154,42,12,60,28,70,31)(3,155,43,13,51,29,61,32)(4,156,44,14,52,30,62,33)(5,157,45,15,53,21,63,34)(6,158,46,16,54,22,64,35)(7,159,47,17,55,23,65,36)(8,160,48,18,56,24,66,37)(9,151,49,19,57,25,67,38)(10,152,50,20,58,26,68,39)(71,144,109,116,84,137,92,129)(72,145,110,117,85,138,93,130)(73,146,101,118,86,139,94,121)(74,147,102,119,87,140,95,122)(75,148,103,120,88,131,96,123)(76,149,104,111,89,132,97,124)(77,150,105,112,90,133,98,125)(78,141,106,113,81,134,99,126)(79,142,107,114,82,135,100,127)(80,143,108,115,83,136,91,128), (1,6)(2,7)(3,8)(4,9)(5,10)(11,158)(12,159)(13,160)(14,151)(15,152)(16,153)(17,154)(18,155)(19,156)(20,157)(21,39)(22,40)(23,31)(24,32)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(41,64)(42,65)(43,66)(44,67)(45,68)(46,69)(47,70)(48,61)(49,62)(50,63)(51,56)(52,57)(53,58)(54,59)(55,60)(71,89)(72,90)(73,81)(74,82)(75,83)(76,84)(77,85)(78,86)(79,87)(80,88)(91,96)(92,97)(93,98)(94,99)(95,100)(101,106)(102,107)(103,108)(104,109)(105,110)(111,137)(112,138)(113,139)(114,140)(115,131)(116,132)(117,133)(118,134)(119,135)(120,136)(121,141)(122,142)(123,143)(124,144)(125,145)(126,146)(127,147)(128,148)(129,149)(130,150)>;
G:=Group( (1,94)(2,95)(3,96)(4,97)(5,98)(6,99)(7,100)(8,91)(9,92)(10,93)(11,146)(12,147)(13,148)(14,149)(15,150)(16,141)(17,142)(18,143)(19,144)(20,145)(21,112)(22,113)(23,114)(24,115)(25,116)(26,117)(27,118)(28,119)(29,120)(30,111)(31,140)(32,131)(33,132)(34,133)(35,134)(36,135)(37,136)(38,137)(39,138)(40,139)(41,73)(42,74)(43,75)(44,76)(45,77)(46,78)(47,79)(48,80)(49,71)(50,72)(51,103)(52,104)(53,105)(54,106)(55,107)(56,108)(57,109)(58,110)(59,101)(60,102)(61,88)(62,89)(63,90)(64,81)(65,82)(66,83)(67,84)(68,85)(69,86)(70,87)(121,153)(122,154)(123,155)(124,156)(125,157)(126,158)(127,159)(128,160)(129,151)(130,152), (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), (1,153,41,11,59,27,69,40)(2,154,42,12,60,28,70,31)(3,155,43,13,51,29,61,32)(4,156,44,14,52,30,62,33)(5,157,45,15,53,21,63,34)(6,158,46,16,54,22,64,35)(7,159,47,17,55,23,65,36)(8,160,48,18,56,24,66,37)(9,151,49,19,57,25,67,38)(10,152,50,20,58,26,68,39)(71,144,109,116,84,137,92,129)(72,145,110,117,85,138,93,130)(73,146,101,118,86,139,94,121)(74,147,102,119,87,140,95,122)(75,148,103,120,88,131,96,123)(76,149,104,111,89,132,97,124)(77,150,105,112,90,133,98,125)(78,141,106,113,81,134,99,126)(79,142,107,114,82,135,100,127)(80,143,108,115,83,136,91,128), (1,6)(2,7)(3,8)(4,9)(5,10)(11,158)(12,159)(13,160)(14,151)(15,152)(16,153)(17,154)(18,155)(19,156)(20,157)(21,39)(22,40)(23,31)(24,32)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(41,64)(42,65)(43,66)(44,67)(45,68)(46,69)(47,70)(48,61)(49,62)(50,63)(51,56)(52,57)(53,58)(54,59)(55,60)(71,89)(72,90)(73,81)(74,82)(75,83)(76,84)(77,85)(78,86)(79,87)(80,88)(91,96)(92,97)(93,98)(94,99)(95,100)(101,106)(102,107)(103,108)(104,109)(105,110)(111,137)(112,138)(113,139)(114,140)(115,131)(116,132)(117,133)(118,134)(119,135)(120,136)(121,141)(122,142)(123,143)(124,144)(125,145)(126,146)(127,147)(128,148)(129,149)(130,150) );
G=PermutationGroup([[(1,94),(2,95),(3,96),(4,97),(5,98),(6,99),(7,100),(8,91),(9,92),(10,93),(11,146),(12,147),(13,148),(14,149),(15,150),(16,141),(17,142),(18,143),(19,144),(20,145),(21,112),(22,113),(23,114),(24,115),(25,116),(26,117),(27,118),(28,119),(29,120),(30,111),(31,140),(32,131),(33,132),(34,133),(35,134),(36,135),(37,136),(38,137),(39,138),(40,139),(41,73),(42,74),(43,75),(44,76),(45,77),(46,78),(47,79),(48,80),(49,71),(50,72),(51,103),(52,104),(53,105),(54,106),(55,107),(56,108),(57,109),(58,110),(59,101),(60,102),(61,88),(62,89),(63,90),(64,81),(65,82),(66,83),(67,84),(68,85),(69,86),(70,87),(121,153),(122,154),(123,155),(124,156),(125,157),(126,158),(127,159),(128,160),(129,151),(130,152)], [(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)], [(1,153,41,11,59,27,69,40),(2,154,42,12,60,28,70,31),(3,155,43,13,51,29,61,32),(4,156,44,14,52,30,62,33),(5,157,45,15,53,21,63,34),(6,158,46,16,54,22,64,35),(7,159,47,17,55,23,65,36),(8,160,48,18,56,24,66,37),(9,151,49,19,57,25,67,38),(10,152,50,20,58,26,68,39),(71,144,109,116,84,137,92,129),(72,145,110,117,85,138,93,130),(73,146,101,118,86,139,94,121),(74,147,102,119,87,140,95,122),(75,148,103,120,88,131,96,123),(76,149,104,111,89,132,97,124),(77,150,105,112,90,133,98,125),(78,141,106,113,81,134,99,126),(79,142,107,114,82,135,100,127),(80,143,108,115,83,136,91,128)], [(1,6),(2,7),(3,8),(4,9),(5,10),(11,158),(12,159),(13,160),(14,151),(15,152),(16,153),(17,154),(18,155),(19,156),(20,157),(21,39),(22,40),(23,31),(24,32),(25,33),(26,34),(27,35),(28,36),(29,37),(30,38),(41,64),(42,65),(43,66),(44,67),(45,68),(46,69),(47,70),(48,61),(49,62),(50,63),(51,56),(52,57),(53,58),(54,59),(55,60),(71,89),(72,90),(73,81),(74,82),(75,83),(76,84),(77,85),(78,86),(79,87),(80,88),(91,96),(92,97),(93,98),(94,99),(95,100),(101,106),(102,107),(103,108),(104,109),(105,110),(111,137),(112,138),(113,139),(114,140),(115,131),(116,132),(117,133),(118,134),(119,135),(120,136),(121,141),(122,142),(123,143),(124,144),(125,145),(126,146),(127,147),(128,148),(129,149),(130,150)]])
140 conjugacy classes
class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10AB | 10AC | ··· | 10AR | 20A | ··· | 20P | 20Q | ··· | 20AF | 40A | ··· | 40AF |
order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
140 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 | C5 | C10 | C10 | C10 | C10 | D4 | D4 | SD16 | C5×D4 | C5×D4 | C5×SD16 |
kernel | SD16×C2×C10 | C22×C40 | C10×SD16 | D4×C2×C10 | Q8×C2×C10 | C22×SD16 | C22×C8 | C2×SD16 | C22×D4 | C22×Q8 | C2×C20 | C22×C10 | C2×C10 | C2×C4 | C23 | C22 |
# reps | 1 | 1 | 12 | 1 | 1 | 4 | 4 | 48 | 4 | 4 | 3 | 1 | 8 | 12 | 4 | 32 |
Matrix representation of SD16×C2×C10 ►in GL4(𝔽41) generated by
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 | 10 | 0 |
0 | 0 | 0 | 10 |
1 | 0 | 0 | 0 |
0 | 1 | 0 | 0 |
0 | 0 | 15 | 26 |
0 | 0 | 15 | 15 |
1 | 0 | 0 | 0 |
0 | 40 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,1,0,0,0,0,10,0,0,0,0,10],[1,0,0,0,0,1,0,0,0,0,15,15,0,0,26,15],[1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,40] >;
SD16×C2×C10 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\times C_2\times C_{10}
% in TeX
G:=Group("SD16xC2xC10");
// GroupNames label
G:=SmallGroup(320,1572);
// by ID
G=gap.SmallGroup(320,1572);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1120,1149,10085,5052,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^10=c^8=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^3>;
// generators/relations