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×C40)⋊24C2, (C22×C8)⋊10C10, (C2×C40)⋊52C22, C20.326(C2×D4), (C2×C20).433D4, C4.2(C23×C10), (C22×Q8)⋊8C10, Q8⋊1(C22×C10), (C5×Q8)⋊11C23, C23.61(C5×D4), (Q8×C10)⋊53C22, D4.1(C22×C10), (C5×D4).34C23, C22.66(D4×C10), (C2×C20).972C23, (C22×D4).12C10, C10.200(C22×D4), (C22×C10).222D4, (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
Subgroups: 498 in 298 conjugacy classes, 178 normal (18 characteristic)
C1, C2, C2 [×6], C2 [×4], C4, C4 [×3], C4 [×4], C22 [×7], C22 [×16], C5, C8 [×4], C2×C4 [×6], C2×C4 [×6], D4 [×4], D4 [×6], Q8 [×4], Q8 [×6], C23, C23 [×10], C10, C10 [×6], C10 [×4], C2×C8 [×6], SD16 [×16], C22×C4, C22×C4, C2×D4 [×6], C2×D4 [×3], C2×Q8 [×6], C2×Q8 [×3], C24, C20, C20 [×3], C20 [×4], C2×C10 [×7], C2×C10 [×16], C22×C8, C2×SD16 [×12], C22×D4, C22×Q8, C40 [×4], C2×C20 [×6], C2×C20 [×6], C5×D4 [×4], C5×D4 [×6], C5×Q8 [×4], C5×Q8 [×6], C22×C10, C22×C10 [×10], C22×SD16, C2×C40 [×6], C5×SD16 [×16], C22×C20, C22×C20, D4×C10 [×6], D4×C10 [×3], Q8×C10 [×6], Q8×C10 [×3], C23×C10, C22×C40, C10×SD16 [×12], D4×C2×C10, Q8×C2×C10, SD16×C2×C10
Quotients:
C1, C2 [×15], C22 [×35], C5, D4 [×4], C23 [×15], C10 [×15], SD16 [×4], C2×D4 [×6], C24, C2×C10 [×35], C2×SD16 [×6], C22×D4, C5×D4 [×4], C22×C10 [×15], C22×SD16, C5×SD16 [×4], D4×C10 [×6], C23×C10, C10×SD16 [×6], D4×C2×C10, SD16×C2×C10
Generators and relations
G = < a,b,c,d | a2=b10=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c3 >
►On 160 points
Generators in S160
(1 100)(2 91)(3 92)(4 93)(5 94)(6 95)(7 96)(8 97)(9 98)(10 99)(11 123)(12 124)(13 125)(14 126)(15 127)(16 128)(17 129)(18 130)(19 121)(20 122)(21 145)(22 146)(23 147)(24 148)(25 149)(26 150)(27 141)(28 142)(29 143)(30 144)(31 118)(32 119)(33 120)(34 111)(35 112)(36 113)(37 114)(38 115)(39 116)(40 117)(41 106)(42 107)(43 108)(44 109)(45 110)(46 101)(47 102)(48 103)(49 104)(50 105)(51 79)(52 80)(53 71)(54 72)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)(61 86)(62 87)(63 88)(64 89)(65 90)(66 81)(67 82)(68 83)(69 84)(70 85)(131 159)(132 160)(133 151)(134 152)(135 153)(136 154)(137 155)(138 156)(139 157)(140 158)
(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 19 41 31 55 27 61 153)(2 20 42 32 56 28 62 154)(3 11 43 33 57 29 63 155)(4 12 44 34 58 30 64 156)(5 13 45 35 59 21 65 157)(6 14 46 36 60 22 66 158)(7 15 47 37 51 23 67 159)(8 16 48 38 52 24 68 160)(9 17 49 39 53 25 69 151)(10 18 50 40 54 26 70 152)(71 149 84 133 98 129 104 116)(72 150 85 134 99 130 105 117)(73 141 86 135 100 121 106 118)(74 142 87 136 91 122 107 119)(75 143 88 137 92 123 108 120)(76 144 89 138 93 124 109 111)(77 145 90 139 94 125 110 112)(78 146 81 140 95 126 101 113)(79 147 82 131 96 127 102 114)(80 148 83 132 97 128 103 115)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 38)(12 39)(13 40)(14 31)(15 32)(16 33)(17 34)(18 35)(19 36)(20 37)(21 152)(22 153)(23 154)(24 155)(25 156)(26 157)(27 158)(28 159)(29 160)(30 151)(41 66)(42 67)(43 68)(44 69)(45 70)(46 61)(47 62)(48 63)(49 64)(50 65)(51 56)(52 57)(53 58)(54 59)(55 60)(71 76)(72 77)(73 78)(74 79)(75 80)(81 106)(82 107)(83 108)(84 109)(85 110)(86 101)(87 102)(88 103)(89 104)(90 105)(91 96)(92 97)(93 98)(94 99)(95 100)(111 129)(112 130)(113 121)(114 122)(115 123)(116 124)(117 125)(118 126)(119 127)(120 128)(131 142)(132 143)(133 144)(134 145)(135 146)(136 147)(137 148)(138 149)(139 150)(140 141)
G:=sub<Sym(160)| (1,100)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,123)(12,124)(13,125)(14,126)(15,127)(16,128)(17,129)(18,130)(19,121)(20,122)(21,145)(22,146)(23,147)(24,148)(25,149)(26,150)(27,141)(28,142)(29,143)(30,144)(31,118)(32,119)(33,120)(34,111)(35,112)(36,113)(37,114)(38,115)(39,116)(40,117)(41,106)(42,107)(43,108)(44,109)(45,110)(46,101)(47,102)(48,103)(49,104)(50,105)(51,79)(52,80)(53,71)(54,72)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78)(61,86)(62,87)(63,88)(64,89)(65,90)(66,81)(67,82)(68,83)(69,84)(70,85)(131,159)(132,160)(133,151)(134,152)(135,153)(136,154)(137,155)(138,156)(139,157)(140,158), (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), (1,19,41,31,55,27,61,153)(2,20,42,32,56,28,62,154)(3,11,43,33,57,29,63,155)(4,12,44,34,58,30,64,156)(5,13,45,35,59,21,65,157)(6,14,46,36,60,22,66,158)(7,15,47,37,51,23,67,159)(8,16,48,38,52,24,68,160)(9,17,49,39,53,25,69,151)(10,18,50,40,54,26,70,152)(71,149,84,133,98,129,104,116)(72,150,85,134,99,130,105,117)(73,141,86,135,100,121,106,118)(74,142,87,136,91,122,107,119)(75,143,88,137,92,123,108,120)(76,144,89,138,93,124,109,111)(77,145,90,139,94,125,110,112)(78,146,81,140,95,126,101,113)(79,147,82,131,96,127,102,114)(80,148,83,132,97,128,103,115), (1,6)(2,7)(3,8)(4,9)(5,10)(11,38)(12,39)(13,40)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,37)(21,152)(22,153)(23,154)(24,155)(25,156)(26,157)(27,158)(28,159)(29,160)(30,151)(41,66)(42,67)(43,68)(44,69)(45,70)(46,61)(47,62)(48,63)(49,64)(50,65)(51,56)(52,57)(53,58)(54,59)(55,60)(71,76)(72,77)(73,78)(74,79)(75,80)(81,106)(82,107)(83,108)(84,109)(85,110)(86,101)(87,102)(88,103)(89,104)(90,105)(91,96)(92,97)(93,98)(94,99)(95,100)(111,129)(112,130)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)(120,128)(131,142)(132,143)(133,144)(134,145)(135,146)(136,147)(137,148)(138,149)(139,150)(140,141)>;
G:=Group( (1,100)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,123)(12,124)(13,125)(14,126)(15,127)(16,128)(17,129)(18,130)(19,121)(20,122)(21,145)(22,146)(23,147)(24,148)(25,149)(26,150)(27,141)(28,142)(29,143)(30,144)(31,118)(32,119)(33,120)(34,111)(35,112)(36,113)(37,114)(38,115)(39,116)(40,117)(41,106)(42,107)(43,108)(44,109)(45,110)(46,101)(47,102)(48,103)(49,104)(50,105)(51,79)(52,80)(53,71)(54,72)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78)(61,86)(62,87)(63,88)(64,89)(65,90)(66,81)(67,82)(68,83)(69,84)(70,85)(131,159)(132,160)(133,151)(134,152)(135,153)(136,154)(137,155)(138,156)(139,157)(140,158), (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), (1,19,41,31,55,27,61,153)(2,20,42,32,56,28,62,154)(3,11,43,33,57,29,63,155)(4,12,44,34,58,30,64,156)(5,13,45,35,59,21,65,157)(6,14,46,36,60,22,66,158)(7,15,47,37,51,23,67,159)(8,16,48,38,52,24,68,160)(9,17,49,39,53,25,69,151)(10,18,50,40,54,26,70,152)(71,149,84,133,98,129,104,116)(72,150,85,134,99,130,105,117)(73,141,86,135,100,121,106,118)(74,142,87,136,91,122,107,119)(75,143,88,137,92,123,108,120)(76,144,89,138,93,124,109,111)(77,145,90,139,94,125,110,112)(78,146,81,140,95,126,101,113)(79,147,82,131,96,127,102,114)(80,148,83,132,97,128,103,115), (1,6)(2,7)(3,8)(4,9)(5,10)(11,38)(12,39)(13,40)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,37)(21,152)(22,153)(23,154)(24,155)(25,156)(26,157)(27,158)(28,159)(29,160)(30,151)(41,66)(42,67)(43,68)(44,69)(45,70)(46,61)(47,62)(48,63)(49,64)(50,65)(51,56)(52,57)(53,58)(54,59)(55,60)(71,76)(72,77)(73,78)(74,79)(75,80)(81,106)(82,107)(83,108)(84,109)(85,110)(86,101)(87,102)(88,103)(89,104)(90,105)(91,96)(92,97)(93,98)(94,99)(95,100)(111,129)(112,130)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)(120,128)(131,142)(132,143)(133,144)(134,145)(135,146)(136,147)(137,148)(138,149)(139,150)(140,141) );
G=PermutationGroup([(1,100),(2,91),(3,92),(4,93),(5,94),(6,95),(7,96),(8,97),(9,98),(10,99),(11,123),(12,124),(13,125),(14,126),(15,127),(16,128),(17,129),(18,130),(19,121),(20,122),(21,145),(22,146),(23,147),(24,148),(25,149),(26,150),(27,141),(28,142),(29,143),(30,144),(31,118),(32,119),(33,120),(34,111),(35,112),(36,113),(37,114),(38,115),(39,116),(40,117),(41,106),(42,107),(43,108),(44,109),(45,110),(46,101),(47,102),(48,103),(49,104),(50,105),(51,79),(52,80),(53,71),(54,72),(55,73),(56,74),(57,75),(58,76),(59,77),(60,78),(61,86),(62,87),(63,88),(64,89),(65,90),(66,81),(67,82),(68,83),(69,84),(70,85),(131,159),(132,160),(133,151),(134,152),(135,153),(136,154),(137,155),(138,156),(139,157),(140,158)], [(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)], [(1,19,41,31,55,27,61,153),(2,20,42,32,56,28,62,154),(3,11,43,33,57,29,63,155),(4,12,44,34,58,30,64,156),(5,13,45,35,59,21,65,157),(6,14,46,36,60,22,66,158),(7,15,47,37,51,23,67,159),(8,16,48,38,52,24,68,160),(9,17,49,39,53,25,69,151),(10,18,50,40,54,26,70,152),(71,149,84,133,98,129,104,116),(72,150,85,134,99,130,105,117),(73,141,86,135,100,121,106,118),(74,142,87,136,91,122,107,119),(75,143,88,137,92,123,108,120),(76,144,89,138,93,124,109,111),(77,145,90,139,94,125,110,112),(78,146,81,140,95,126,101,113),(79,147,82,131,96,127,102,114),(80,148,83,132,97,128,103,115)], [(1,6),(2,7),(3,8),(4,9),(5,10),(11,38),(12,39),(13,40),(14,31),(15,32),(16,33),(17,34),(18,35),(19,36),(20,37),(21,152),(22,153),(23,154),(24,155),(25,156),(26,157),(27,158),(28,159),(29,160),(30,151),(41,66),(42,67),(43,68),(44,69),(45,70),(46,61),(47,62),(48,63),(49,64),(50,65),(51,56),(52,57),(53,58),(54,59),(55,60),(71,76),(72,77),(73,78),(74,79),(75,80),(81,106),(82,107),(83,108),(84,109),(85,110),(86,101),(87,102),(88,103),(89,104),(90,105),(91,96),(92,97),(93,98),(94,99),(95,100),(111,129),(112,130),(113,121),(114,122),(115,123),(116,124),(117,125),(118,126),(119,127),(120,128),(131,142),(132,143),(133,144),(134,145),(135,146),(136,147),(137,148),(138,149),(139,150),(140,141)])
Matrix representation ►G ⊆ 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] >;
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 |
In GAP, Magma, Sage, TeX
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
×
⋊
ℤ
𝔽
○
≀