metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.236D10, (C4×D5)⋊3Q8, C20⋊Q8⋊34C2, C4.38(Q8×D5), D10.3(C2×Q8), C20.49(C2×Q8), C4⋊C4.204D10, C42.C2⋊15D5, (D5×C42).8C2, (C2×C20).86C23, D10⋊Q8.1C2, C20.6Q8⋊22C2, Dic5.33(C2×Q8), Dic5⋊3Q8⋊34C2, C10.41(C22×Q8), (C4×C20).192C22, (C2×C10).232C24, Dic5.18(C4○D4), C4⋊Dic5.239C22, C22.253(C23×D5), D10⋊C4.38C22, C5⋊5(C23.37C23), (C4×Dic5).147C22, (C2×Dic5).378C23, C10.D4.50C22, (C22×D5).229C23, (C2×Dic10).184C22, C2.24(C2×Q8×D5), C2.84(D5×C4○D4), (C5×C42.C2)⋊5C2, C4⋊C4⋊7D5.11C2, C10.195(C2×C4○D4), (C2×C4×D5).319C22, (C2×C4).77(C22×D5), (C5×C4⋊C4).187C22, SmallGroup(320,1360)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.236D10
G = < a,b,c,d | a4=b4=1, c10=b2, d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=dbd-1=a2b-1, dcd-1=a2c9 >
Subgroups: 686 in 222 conjugacy classes, 107 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, Q8, C23, D5, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C42, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C42.C2, C4⋊Q8, Dic10, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C23.37C23, C4×Dic5, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C4×C20, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×C4×D5, C20.6Q8, D5×C42, Dic5⋊3Q8, C20⋊Q8, C4⋊C4⋊7D5, D10⋊Q8, C5×C42.C2, C42.236D10
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, C4○D4, C24, D10, C22×Q8, C2×C4○D4, C22×D5, C23.37C23, Q8×D5, C23×D5, C2×Q8×D5, D5×C4○D4, C42.236D10
►On 160 points
Generators in S160
(1 113 55 131)(2 132 56 114)(3 115 57 133)(4 134 58 116)(5 117 59 135)(6 136 60 118)(7 119 41 137)(8 138 42 120)(9 101 43 139)(10 140 44 102)(11 103 45 121)(12 122 46 104)(13 105 47 123)(14 124 48 106)(15 107 49 125)(16 126 50 108)(17 109 51 127)(18 128 52 110)(19 111 53 129)(20 130 54 112)(21 66 154 81)(22 82 155 67)(23 68 156 83)(24 84 157 69)(25 70 158 85)(26 86 159 71)(27 72 160 87)(28 88 141 73)(29 74 142 89)(30 90 143 75)(31 76 144 91)(32 92 145 77)(33 78 146 93)(34 94 147 79)(35 80 148 95)(36 96 149 61)(37 62 150 97)(38 98 151 63)(39 64 152 99)(40 100 153 65)
(1 143 11 153)(2 21 12 31)(3 145 13 155)(4 23 14 33)(5 147 15 157)(6 25 16 35)(7 149 17 159)(8 27 18 37)(9 151 19 141)(10 29 20 39)(22 57 32 47)(24 59 34 49)(26 41 36 51)(28 43 38 53)(30 45 40 55)(42 160 52 150)(44 142 54 152)(46 144 56 154)(48 146 58 156)(50 148 60 158)(61 109 71 119)(62 138 72 128)(63 111 73 101)(64 140 74 130)(65 113 75 103)(66 122 76 132)(67 115 77 105)(68 124 78 134)(69 117 79 107)(70 126 80 136)(81 104 91 114)(82 133 92 123)(83 106 93 116)(84 135 94 125)(85 108 95 118)(86 137 96 127)(87 110 97 120)(88 139 98 129)(89 112 99 102)(90 121 100 131)
(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 10 45 54)(2 53 46 9)(3 8 47 52)(4 51 48 7)(5 6 49 50)(11 20 55 44)(12 43 56 19)(13 18 57 42)(14 41 58 17)(15 16 59 60)(21 151 144 28)(22 27 145 150)(23 149 146 26)(24 25 147 148)(29 143 152 40)(30 39 153 142)(31 141 154 38)(32 37 155 160)(33 159 156 36)(34 35 157 158)(61 78 86 83)(62 82 87 77)(63 76 88 81)(64 100 89 75)(65 74 90 99)(66 98 91 73)(67 72 92 97)(68 96 93 71)(69 70 94 95)(79 80 84 85)(101 114 129 122)(102 121 130 113)(103 112 131 140)(104 139 132 111)(105 110 133 138)(106 137 134 109)(107 108 135 136)(115 120 123 128)(116 127 124 119)(117 118 125 126)
G:=sub<Sym(160)| (1,113,55,131)(2,132,56,114)(3,115,57,133)(4,134,58,116)(5,117,59,135)(6,136,60,118)(7,119,41,137)(8,138,42,120)(9,101,43,139)(10,140,44,102)(11,103,45,121)(12,122,46,104)(13,105,47,123)(14,124,48,106)(15,107,49,125)(16,126,50,108)(17,109,51,127)(18,128,52,110)(19,111,53,129)(20,130,54,112)(21,66,154,81)(22,82,155,67)(23,68,156,83)(24,84,157,69)(25,70,158,85)(26,86,159,71)(27,72,160,87)(28,88,141,73)(29,74,142,89)(30,90,143,75)(31,76,144,91)(32,92,145,77)(33,78,146,93)(34,94,147,79)(35,80,148,95)(36,96,149,61)(37,62,150,97)(38,98,151,63)(39,64,152,99)(40,100,153,65), (1,143,11,153)(2,21,12,31)(3,145,13,155)(4,23,14,33)(5,147,15,157)(6,25,16,35)(7,149,17,159)(8,27,18,37)(9,151,19,141)(10,29,20,39)(22,57,32,47)(24,59,34,49)(26,41,36,51)(28,43,38,53)(30,45,40,55)(42,160,52,150)(44,142,54,152)(46,144,56,154)(48,146,58,156)(50,148,60,158)(61,109,71,119)(62,138,72,128)(63,111,73,101)(64,140,74,130)(65,113,75,103)(66,122,76,132)(67,115,77,105)(68,124,78,134)(69,117,79,107)(70,126,80,136)(81,104,91,114)(82,133,92,123)(83,106,93,116)(84,135,94,125)(85,108,95,118)(86,137,96,127)(87,110,97,120)(88,139,98,129)(89,112,99,102)(90,121,100,131), (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), (1,10,45,54)(2,53,46,9)(3,8,47,52)(4,51,48,7)(5,6,49,50)(11,20,55,44)(12,43,56,19)(13,18,57,42)(14,41,58,17)(15,16,59,60)(21,151,144,28)(22,27,145,150)(23,149,146,26)(24,25,147,148)(29,143,152,40)(30,39,153,142)(31,141,154,38)(32,37,155,160)(33,159,156,36)(34,35,157,158)(61,78,86,83)(62,82,87,77)(63,76,88,81)(64,100,89,75)(65,74,90,99)(66,98,91,73)(67,72,92,97)(68,96,93,71)(69,70,94,95)(79,80,84,85)(101,114,129,122)(102,121,130,113)(103,112,131,140)(104,139,132,111)(105,110,133,138)(106,137,134,109)(107,108,135,136)(115,120,123,128)(116,127,124,119)(117,118,125,126)>;
G:=Group( (1,113,55,131)(2,132,56,114)(3,115,57,133)(4,134,58,116)(5,117,59,135)(6,136,60,118)(7,119,41,137)(8,138,42,120)(9,101,43,139)(10,140,44,102)(11,103,45,121)(12,122,46,104)(13,105,47,123)(14,124,48,106)(15,107,49,125)(16,126,50,108)(17,109,51,127)(18,128,52,110)(19,111,53,129)(20,130,54,112)(21,66,154,81)(22,82,155,67)(23,68,156,83)(24,84,157,69)(25,70,158,85)(26,86,159,71)(27,72,160,87)(28,88,141,73)(29,74,142,89)(30,90,143,75)(31,76,144,91)(32,92,145,77)(33,78,146,93)(34,94,147,79)(35,80,148,95)(36,96,149,61)(37,62,150,97)(38,98,151,63)(39,64,152,99)(40,100,153,65), (1,143,11,153)(2,21,12,31)(3,145,13,155)(4,23,14,33)(5,147,15,157)(6,25,16,35)(7,149,17,159)(8,27,18,37)(9,151,19,141)(10,29,20,39)(22,57,32,47)(24,59,34,49)(26,41,36,51)(28,43,38,53)(30,45,40,55)(42,160,52,150)(44,142,54,152)(46,144,56,154)(48,146,58,156)(50,148,60,158)(61,109,71,119)(62,138,72,128)(63,111,73,101)(64,140,74,130)(65,113,75,103)(66,122,76,132)(67,115,77,105)(68,124,78,134)(69,117,79,107)(70,126,80,136)(81,104,91,114)(82,133,92,123)(83,106,93,116)(84,135,94,125)(85,108,95,118)(86,137,96,127)(87,110,97,120)(88,139,98,129)(89,112,99,102)(90,121,100,131), (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), (1,10,45,54)(2,53,46,9)(3,8,47,52)(4,51,48,7)(5,6,49,50)(11,20,55,44)(12,43,56,19)(13,18,57,42)(14,41,58,17)(15,16,59,60)(21,151,144,28)(22,27,145,150)(23,149,146,26)(24,25,147,148)(29,143,152,40)(30,39,153,142)(31,141,154,38)(32,37,155,160)(33,159,156,36)(34,35,157,158)(61,78,86,83)(62,82,87,77)(63,76,88,81)(64,100,89,75)(65,74,90,99)(66,98,91,73)(67,72,92,97)(68,96,93,71)(69,70,94,95)(79,80,84,85)(101,114,129,122)(102,121,130,113)(103,112,131,140)(104,139,132,111)(105,110,133,138)(106,137,134,109)(107,108,135,136)(115,120,123,128)(116,127,124,119)(117,118,125,126) );
G=PermutationGroup([[(1,113,55,131),(2,132,56,114),(3,115,57,133),(4,134,58,116),(5,117,59,135),(6,136,60,118),(7,119,41,137),(8,138,42,120),(9,101,43,139),(10,140,44,102),(11,103,45,121),(12,122,46,104),(13,105,47,123),(14,124,48,106),(15,107,49,125),(16,126,50,108),(17,109,51,127),(18,128,52,110),(19,111,53,129),(20,130,54,112),(21,66,154,81),(22,82,155,67),(23,68,156,83),(24,84,157,69),(25,70,158,85),(26,86,159,71),(27,72,160,87),(28,88,141,73),(29,74,142,89),(30,90,143,75),(31,76,144,91),(32,92,145,77),(33,78,146,93),(34,94,147,79),(35,80,148,95),(36,96,149,61),(37,62,150,97),(38,98,151,63),(39,64,152,99),(40,100,153,65)], [(1,143,11,153),(2,21,12,31),(3,145,13,155),(4,23,14,33),(5,147,15,157),(6,25,16,35),(7,149,17,159),(8,27,18,37),(9,151,19,141),(10,29,20,39),(22,57,32,47),(24,59,34,49),(26,41,36,51),(28,43,38,53),(30,45,40,55),(42,160,52,150),(44,142,54,152),(46,144,56,154),(48,146,58,156),(50,148,60,158),(61,109,71,119),(62,138,72,128),(63,111,73,101),(64,140,74,130),(65,113,75,103),(66,122,76,132),(67,115,77,105),(68,124,78,134),(69,117,79,107),(70,126,80,136),(81,104,91,114),(82,133,92,123),(83,106,93,116),(84,135,94,125),(85,108,95,118),(86,137,96,127),(87,110,97,120),(88,139,98,129),(89,112,99,102),(90,121,100,131)], [(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)], [(1,10,45,54),(2,53,46,9),(3,8,47,52),(4,51,48,7),(5,6,49,50),(11,20,55,44),(12,43,56,19),(13,18,57,42),(14,41,58,17),(15,16,59,60),(21,151,144,28),(22,27,145,150),(23,149,146,26),(24,25,147,148),(29,143,152,40),(30,39,153,142),(31,141,154,38),(32,37,155,160),(33,159,156,36),(34,35,157,158),(61,78,86,83),(62,82,87,77),(63,76,88,81),(64,100,89,75),(65,74,90,99),(66,98,91,73),(67,72,92,97),(68,96,93,71),(69,70,94,95),(79,80,84,85),(101,114,129,122),(102,121,130,113),(103,112,131,140),(104,139,132,111),(105,110,133,138),(106,137,134,109),(107,108,135,136),(115,120,123,128),(116,127,124,119),(117,118,125,126)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 4S | 4T | 4U | 4V | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | C4○D4 | D10 | D10 | Q8×D5 | D5×C4○D4 |
| kernel | C42.236D10 | C20.6Q8 | D5×C42 | Dic5⋊3Q8 | C20⋊Q8 | C4⋊C4⋊7D5 | D10⋊Q8 | C5×C42.C2 | C4×D5 | C42.C2 | Dic5 | C42 | C4⋊C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 1 | 4 | 2 | 8 | 2 | 12 | 4 | 8 |
Matrix representation of C42.236D10 ►in GL6(𝔽41)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 1 | 32 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 6 | 0 | 0 |
| 0 | 0 | 34 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 18 |
| 0 | 0 | 0 | 0 | 9 | 1 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 6 | 0 | 0 |
| 0 | 0 | 33 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 23 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,1,0,0,0,0,0,32],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,34,0,0,0,0,6,7,0,0,0,0,0,0,40,9,0,0,0,0,18,1],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,34,33,0,0,0,0,6,7,0,0,0,0,0,0,1,0,0,0,0,0,23,40] >;
C42.236D10 in GAP, Magma, Sage, TeX
C_4^2._{236}D_{10} % in TeX
G:=Group("C4^2.236D10"); // GroupNames label
G:=SmallGroup(320,1360);
// by ID
G=gap.SmallGroup(320,1360);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,100,1123,570,409,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=b^2,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^2*b^-1,d*c*d^-1=a^2*c^9>;
// generators/relations