metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.178D10, C10.842+ 1+4, C10.402- 1+4, C4⋊Q8⋊16D5, C4⋊C4.126D10, (C2×Q8).88D10, D20⋊8C4⋊44C2, D10⋊2Q8⋊46C2, D10⋊3Q8⋊38C2, C4⋊D20.14C2, C4.D20⋊27C2, C42⋊D5⋊26C2, Dic5⋊3Q8⋊43C2, C20.140(C4○D4), C20.23D4⋊27C2, (C2×C20).639C23, (C4×C20).218C22, (C2×C10).277C24, C4.23(Q8⋊2D5), D10.13D4⋊47C2, C2.88(D4⋊6D10), (C2×D20).179C22, C4⋊Dic5.255C22, (Q8×C10).144C22, C22.298(C23×D5), (C4×Dic5).174C22, (C2×Dic5).284C23, (C22×D5).122C23, D10⋊C4.156C22, C2.41(Q8.10D10), C5⋊11(C22.36C24), (C2×Dic10).198C22, C10.D4.169C22, (C5×C4⋊Q8)⋊19C2, C4⋊C4⋊D5⋊47C2, C10.124(C2×C4○D4), C2.32(C2×Q8⋊2D5), (C2×C4×D5).159C22, (C5×C4⋊C4).220C22, (C2×C4).602(C22×D5), SmallGroup(320,1405)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.178D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c9 >
Subgroups: 798 in 216 conjugacy classes, 95 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic5, C20, C20, D10, C2×C10, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, C4⋊Q8, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C22×D5, C22.36C24, C4×Dic5, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, D10⋊C4, D10⋊C4, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, Q8×C10, C42⋊D5, C4.D20, Dic5⋊3Q8, D20⋊8C4, D10.13D4, C4⋊D20, D10⋊2Q8, C4⋊C4⋊D5, D10⋊3Q8, C20.23D4, C5×C4⋊Q8, C42.178D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, 2- 1+4, C22×D5, C22.36C24, Q8⋊2D5, C23×D5, D4⋊6D10, C2×Q8⋊2D5, Q8.10D10, C42.178D10
►On 160 points
Generators in S160
(1 27 11 37)(2 38 12 28)(3 29 13 39)(4 40 14 30)(5 31 15 21)(6 22 16 32)(7 33 17 23)(8 24 18 34)(9 35 19 25)(10 26 20 36)(41 126 51 136)(42 137 52 127)(43 128 53 138)(44 139 54 129)(45 130 55 140)(46 121 56 131)(47 132 57 122)(48 123 58 133)(49 134 59 124)(50 125 60 135)(61 155 71 145)(62 146 72 156)(63 157 73 147)(64 148 74 158)(65 159 75 149)(66 150 76 160)(67 141 77 151)(68 152 78 142)(69 143 79 153)(70 154 80 144)(81 120 91 110)(82 111 92 101)(83 102 93 112)(84 113 94 103)(85 104 95 114)(86 115 96 105)(87 106 97 116)(88 117 98 107)(89 108 99 118)(90 119 100 109)
(1 93 79 138)(2 139 80 94)(3 95 61 140)(4 121 62 96)(5 97 63 122)(6 123 64 98)(7 99 65 124)(8 125 66 100)(9 81 67 126)(10 127 68 82)(11 83 69 128)(12 129 70 84)(13 85 71 130)(14 131 72 86)(15 87 73 132)(16 133 74 88)(17 89 75 134)(18 135 76 90)(19 91 77 136)(20 137 78 92)(21 106 147 57)(22 58 148 107)(23 108 149 59)(24 60 150 109)(25 110 151 41)(26 42 152 111)(27 112 153 43)(28 44 154 113)(29 114 155 45)(30 46 156 115)(31 116 157 47)(32 48 158 117)(33 118 159 49)(34 50 160 119)(35 120 141 51)(36 52 142 101)(37 102 143 53)(38 54 144 103)(39 104 145 55)(40 56 146 105)
(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)
(1 122 11 132)(2 131 12 121)(3 140 13 130)(4 129 14 139)(5 138 15 128)(6 127 16 137)(7 136 17 126)(8 125 18 135)(9 134 19 124)(10 123 20 133)(21 112 31 102)(22 101 32 111)(23 110 33 120)(24 119 34 109)(25 108 35 118)(26 117 36 107)(27 106 37 116)(28 115 38 105)(29 104 39 114)(30 113 40 103)(41 159 51 149)(42 148 52 158)(43 157 53 147)(44 146 54 156)(45 155 55 145)(46 144 56 154)(47 153 57 143)(48 142 58 152)(49 151 59 141)(50 160 60 150)(61 95 71 85)(62 84 72 94)(63 93 73 83)(64 82 74 92)(65 91 75 81)(66 100 76 90)(67 89 77 99)(68 98 78 88)(69 87 79 97)(70 96 80 86)
G:=sub<Sym(160)| (1,27,11,37)(2,38,12,28)(3,29,13,39)(4,40,14,30)(5,31,15,21)(6,22,16,32)(7,33,17,23)(8,24,18,34)(9,35,19,25)(10,26,20,36)(41,126,51,136)(42,137,52,127)(43,128,53,138)(44,139,54,129)(45,130,55,140)(46,121,56,131)(47,132,57,122)(48,123,58,133)(49,134,59,124)(50,125,60,135)(61,155,71,145)(62,146,72,156)(63,157,73,147)(64,148,74,158)(65,159,75,149)(66,150,76,160)(67,141,77,151)(68,152,78,142)(69,143,79,153)(70,154,80,144)(81,120,91,110)(82,111,92,101)(83,102,93,112)(84,113,94,103)(85,104,95,114)(86,115,96,105)(87,106,97,116)(88,117,98,107)(89,108,99,118)(90,119,100,109), (1,93,79,138)(2,139,80,94)(3,95,61,140)(4,121,62,96)(5,97,63,122)(6,123,64,98)(7,99,65,124)(8,125,66,100)(9,81,67,126)(10,127,68,82)(11,83,69,128)(12,129,70,84)(13,85,71,130)(14,131,72,86)(15,87,73,132)(16,133,74,88)(17,89,75,134)(18,135,76,90)(19,91,77,136)(20,137,78,92)(21,106,147,57)(22,58,148,107)(23,108,149,59)(24,60,150,109)(25,110,151,41)(26,42,152,111)(27,112,153,43)(28,44,154,113)(29,114,155,45)(30,46,156,115)(31,116,157,47)(32,48,158,117)(33,118,159,49)(34,50,160,119)(35,120,141,51)(36,52,142,101)(37,102,143,53)(38,54,144,103)(39,104,145,55)(40,56,146,105), (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), (1,122,11,132)(2,131,12,121)(3,140,13,130)(4,129,14,139)(5,138,15,128)(6,127,16,137)(7,136,17,126)(8,125,18,135)(9,134,19,124)(10,123,20,133)(21,112,31,102)(22,101,32,111)(23,110,33,120)(24,119,34,109)(25,108,35,118)(26,117,36,107)(27,106,37,116)(28,115,38,105)(29,104,39,114)(30,113,40,103)(41,159,51,149)(42,148,52,158)(43,157,53,147)(44,146,54,156)(45,155,55,145)(46,144,56,154)(47,153,57,143)(48,142,58,152)(49,151,59,141)(50,160,60,150)(61,95,71,85)(62,84,72,94)(63,93,73,83)(64,82,74,92)(65,91,75,81)(66,100,76,90)(67,89,77,99)(68,98,78,88)(69,87,79,97)(70,96,80,86)>;
G:=Group( (1,27,11,37)(2,38,12,28)(3,29,13,39)(4,40,14,30)(5,31,15,21)(6,22,16,32)(7,33,17,23)(8,24,18,34)(9,35,19,25)(10,26,20,36)(41,126,51,136)(42,137,52,127)(43,128,53,138)(44,139,54,129)(45,130,55,140)(46,121,56,131)(47,132,57,122)(48,123,58,133)(49,134,59,124)(50,125,60,135)(61,155,71,145)(62,146,72,156)(63,157,73,147)(64,148,74,158)(65,159,75,149)(66,150,76,160)(67,141,77,151)(68,152,78,142)(69,143,79,153)(70,154,80,144)(81,120,91,110)(82,111,92,101)(83,102,93,112)(84,113,94,103)(85,104,95,114)(86,115,96,105)(87,106,97,116)(88,117,98,107)(89,108,99,118)(90,119,100,109), (1,93,79,138)(2,139,80,94)(3,95,61,140)(4,121,62,96)(5,97,63,122)(6,123,64,98)(7,99,65,124)(8,125,66,100)(9,81,67,126)(10,127,68,82)(11,83,69,128)(12,129,70,84)(13,85,71,130)(14,131,72,86)(15,87,73,132)(16,133,74,88)(17,89,75,134)(18,135,76,90)(19,91,77,136)(20,137,78,92)(21,106,147,57)(22,58,148,107)(23,108,149,59)(24,60,150,109)(25,110,151,41)(26,42,152,111)(27,112,153,43)(28,44,154,113)(29,114,155,45)(30,46,156,115)(31,116,157,47)(32,48,158,117)(33,118,159,49)(34,50,160,119)(35,120,141,51)(36,52,142,101)(37,102,143,53)(38,54,144,103)(39,104,145,55)(40,56,146,105), (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), (1,122,11,132)(2,131,12,121)(3,140,13,130)(4,129,14,139)(5,138,15,128)(6,127,16,137)(7,136,17,126)(8,125,18,135)(9,134,19,124)(10,123,20,133)(21,112,31,102)(22,101,32,111)(23,110,33,120)(24,119,34,109)(25,108,35,118)(26,117,36,107)(27,106,37,116)(28,115,38,105)(29,104,39,114)(30,113,40,103)(41,159,51,149)(42,148,52,158)(43,157,53,147)(44,146,54,156)(45,155,55,145)(46,144,56,154)(47,153,57,143)(48,142,58,152)(49,151,59,141)(50,160,60,150)(61,95,71,85)(62,84,72,94)(63,93,73,83)(64,82,74,92)(65,91,75,81)(66,100,76,90)(67,89,77,99)(68,98,78,88)(69,87,79,97)(70,96,80,86) );
G=PermutationGroup([[(1,27,11,37),(2,38,12,28),(3,29,13,39),(4,40,14,30),(5,31,15,21),(6,22,16,32),(7,33,17,23),(8,24,18,34),(9,35,19,25),(10,26,20,36),(41,126,51,136),(42,137,52,127),(43,128,53,138),(44,139,54,129),(45,130,55,140),(46,121,56,131),(47,132,57,122),(48,123,58,133),(49,134,59,124),(50,125,60,135),(61,155,71,145),(62,146,72,156),(63,157,73,147),(64,148,74,158),(65,159,75,149),(66,150,76,160),(67,141,77,151),(68,152,78,142),(69,143,79,153),(70,154,80,144),(81,120,91,110),(82,111,92,101),(83,102,93,112),(84,113,94,103),(85,104,95,114),(86,115,96,105),(87,106,97,116),(88,117,98,107),(89,108,99,118),(90,119,100,109)], [(1,93,79,138),(2,139,80,94),(3,95,61,140),(4,121,62,96),(5,97,63,122),(6,123,64,98),(7,99,65,124),(8,125,66,100),(9,81,67,126),(10,127,68,82),(11,83,69,128),(12,129,70,84),(13,85,71,130),(14,131,72,86),(15,87,73,132),(16,133,74,88),(17,89,75,134),(18,135,76,90),(19,91,77,136),(20,137,78,92),(21,106,147,57),(22,58,148,107),(23,108,149,59),(24,60,150,109),(25,110,151,41),(26,42,152,111),(27,112,153,43),(28,44,154,113),(29,114,155,45),(30,46,156,115),(31,116,157,47),(32,48,158,117),(33,118,159,49),(34,50,160,119),(35,120,141,51),(36,52,142,101),(37,102,143,53),(38,54,144,103),(39,104,145,55),(40,56,146,105)], [(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)], [(1,122,11,132),(2,131,12,121),(3,140,13,130),(4,129,14,139),(5,138,15,128),(6,127,16,137),(7,136,17,126),(8,125,18,135),(9,134,19,124),(10,123,20,133),(21,112,31,102),(22,101,32,111),(23,110,33,120),(24,119,34,109),(25,108,35,118),(26,117,36,107),(27,106,37,116),(28,115,38,105),(29,104,39,114),(30,113,40,103),(41,159,51,149),(42,148,52,158),(43,157,53,147),(44,146,54,156),(45,155,55,145),(46,144,56,154),(47,153,57,143),(48,142,58,152),(49,151,59,141),(50,160,60,150),(61,95,71,85),(62,84,72,94),(63,93,73,83),(64,82,74,92),(65,91,75,81),(66,100,76,90),(67,89,77,99),(68,98,78,88),(69,87,79,97),(70,96,80,86)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 20 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | 2+ 1+4 | 2- 1+4 | Q8⋊2D5 | D4⋊6D10 | Q8.10D10 |
| kernel | C42.178D10 | C42⋊D5 | C4.D20 | Dic5⋊3Q8 | D20⋊8C4 | D10.13D4 | C4⋊D20 | D10⋊2Q8 | C4⋊C4⋊D5 | D10⋊3Q8 | C20.23D4 | C5×C4⋊Q8 | C4⋊Q8 | C20 | C42 | C4⋊C4 | C2×Q8 | C10 | C10 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 2 | 4 | 2 | 8 | 4 | 1 | 1 | 4 | 4 | 4 |
Matrix representation of C42.178D10 ►in GL8(𝔽41)
| 0 | 0 | 40 | 1 | 0 | 0 | 0 | 0 |
| 7 | 1 | 39 | 34 | 0 | 0 | 0 | 0 |
| 38 | 23 | 40 | 0 | 0 | 0 | 0 | 0 |
| 37 | 23 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 1 | 0 | 38 |
| 0 | 0 | 0 | 0 | 5 | 18 | 3 | 38 |
| 0 | 0 | 0 | 0 | 28 | 13 | 24 | 40 |
| 0 | 0 | 0 | 0 | 28 | 0 | 1 | 17 |
| 0 | 0 | 40 | 1 | 0 | 0 | 0 | 0 |
| 7 | 1 | 39 | 34 | 0 | 0 | 0 | 0 |
| 38 | 23 | 40 | 0 | 0 | 0 | 0 | 0 |
| 37 | 23 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 17 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 24 |
| 15 | 12 | 0 | 2 | 0 | 0 | 0 | 0 |
| 31 | 39 | 39 | 29 | 0 | 0 | 0 | 0 |
| 24 | 0 | 12 | 29 | 0 | 0 | 0 | 0 |
| 30 | 24 | 12 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 9 | 9 |
| 0 | 0 | 0 | 0 | 19 | 22 | 19 | 18 |
| 0 | 0 | 0 | 0 | 31 | 29 | 31 | 29 |
| 0 | 0 | 0 | 0 | 3 | 19 | 12 | 29 |
| 33 | 34 | 8 | 14 | 0 | 0 | 0 | 0 |
| 40 | 35 | 7 | 6 | 0 | 0 | 0 | 0 |
| 8 | 6 | 7 | 7 | 0 | 0 | 0 | 0 |
| 2 | 13 | 7 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 24 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 17 |
G:=sub<GL(8,GF(41))| [0,7,38,37,0,0,0,0,0,1,23,23,0,0,0,0,40,39,40,40,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,0,23,5,28,28,0,0,0,0,1,18,13,0,0,0,0,0,0,3,24,1,0,0,0,0,38,38,40,17],[0,7,38,37,0,0,0,0,0,1,23,23,0,0,0,0,40,39,40,40,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,0,23,5,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,17,40,0,0,0,0,0,0,1,24],[15,31,24,30,0,0,0,0,12,39,0,24,0,0,0,0,0,39,12,12,0,0,0,0,2,29,29,16,0,0,0,0,0,0,0,0,0,19,31,3,0,0,0,0,12,22,29,19,0,0,0,0,9,19,31,12,0,0,0,0,9,18,29,29],[33,40,8,2,0,0,0,0,34,35,6,13,0,0,0,0,8,7,7,7,0,0,0,0,14,6,7,7,0,0,0,0,0,0,0,0,21,3,0,0,0,0,0,0,3,20,0,0,0,0,0,0,0,0,24,3,0,0,0,0,0,0,40,17] >;
C42.178D10 in GAP, Magma, Sage, TeX
C_4^2._{178}D_{10} % in TeX
G:=Group("C4^2.178D10"); // GroupNames label
G:=SmallGroup(320,1405);
// by ID
G=gap.SmallGroup(320,1405);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,219,100,675,570,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=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=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^9>;
// generators/relations