metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.178D10, C10.402- (1+4), C10.842+ (1+4), C4⋊Q8⋊16D5, C4⋊C4.126D10, (C2×Q8).88D10, D20⋊8C4⋊44C2, D10⋊2Q8⋊46C2, D10⋊3Q8⋊38C2, C4.D20⋊27C2, C4⋊2D20.14C2, C42⋊D5⋊26C2, Dic5⋊3Q8⋊43C2, C20.140(C4○D4), C20.23D4⋊27C2, (C4×C20).218C22, (C2×C20).639C23, (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), (C2×Dic5).284C23, (C4×Dic5).174C22, (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
Subgroups: 798 in 216 conjugacy classes, 95 normal (43 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×11], C22, C22 [×9], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×9], D4 [×4], Q8 [×4], C23 [×3], D5 [×3], C10 [×3], C42, C42 [×3], C22⋊C4 [×12], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×6], C22×C4 [×3], C2×D4 [×3], C2×Q8 [×2], C2×Q8, Dic5 [×5], C20 [×2], C20 [×6], D10 [×9], C2×C10, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8 [×3], C22.D4 [×2], C4.4D4 [×3], C42⋊2C2 [×2], C4⋊Q8, Dic10 [×2], C4×D5 [×4], D20 [×4], C2×Dic5 [×3], C2×Dic5 [×2], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×2], C22×D5, C22×D5 [×2], C22.36C24, C4×Dic5, C4×Dic5 [×2], C10.D4 [×2], C10.D4 [×2], C4⋊Dic5 [×2], D10⋊C4 [×2], D10⋊C4 [×10], C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10, C2×C4×D5, C2×C4×D5 [×2], C2×D20, C2×D20 [×2], Q8×C10 [×2], C42⋊D5, C4.D20, Dic5⋊3Q8, D20⋊8C4, D10.13D4 [×2], C4⋊2D20, D10⋊2Q8, C4⋊C4⋊D5 [×2], D10⋊3Q8 [×2], C20.23D4 [×2], C5×C4⋊Q8, C42.178D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ (1+4), 2- (1+4), C22×D5 [×7], C22.36C24, Q8⋊2D5 [×2], C23×D5, D4⋊6D10, C2×Q8⋊2D5, Q8.10D10, C42.178D10
Generators and relations
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 >
►On 160 points
Generators in S160
(1 87 11 97)(2 98 12 88)(3 89 13 99)(4 100 14 90)(5 91 15 81)(6 82 16 92)(7 93 17 83)(8 84 18 94)(9 95 19 85)(10 86 20 96)(21 118 31 108)(22 109 32 119)(23 120 33 110)(24 111 34 101)(25 102 35 112)(26 113 36 103)(27 104 37 114)(28 115 38 105)(29 106 39 116)(30 117 40 107)(41 160 51 150)(42 151 52 141)(43 142 53 152)(44 153 54 143)(45 144 55 154)(46 155 56 145)(47 146 57 156)(48 157 58 147)(49 148 59 158)(50 159 60 149)(61 135 71 125)(62 126 72 136)(63 137 73 127)(64 128 74 138)(65 139 75 129)(66 130 76 140)(67 121 77 131)(68 132 78 122)(69 123 79 133)(70 134 80 124)
(1 125 112 148)(2 149 113 126)(3 127 114 150)(4 151 115 128)(5 129 116 152)(6 153 117 130)(7 131 118 154)(8 155 119 132)(9 133 120 156)(10 157 101 134)(11 135 102 158)(12 159 103 136)(13 137 104 160)(14 141 105 138)(15 139 106 142)(16 143 107 140)(17 121 108 144)(18 145 109 122)(19 123 110 146)(20 147 111 124)(21 55 83 77)(22 78 84 56)(23 57 85 79)(24 80 86 58)(25 59 87 61)(26 62 88 60)(27 41 89 63)(28 64 90 42)(29 43 91 65)(30 66 92 44)(31 45 93 67)(32 68 94 46)(33 47 95 69)(34 70 96 48)(35 49 97 71)(36 72 98 50)(37 51 99 73)(38 74 100 52)(39 53 81 75)(40 76 82 54)
(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)
(1 152 11 142)(2 141 12 151)(3 150 13 160)(4 159 14 149)(5 148 15 158)(6 157 16 147)(7 146 17 156)(8 155 18 145)(9 144 19 154)(10 153 20 143)(21 57 31 47)(22 46 32 56)(23 55 33 45)(24 44 34 54)(25 53 35 43)(26 42 36 52)(27 51 37 41)(28 60 38 50)(29 49 39 59)(30 58 40 48)(61 91 71 81)(62 100 72 90)(63 89 73 99)(64 98 74 88)(65 87 75 97)(66 96 76 86)(67 85 77 95)(68 94 78 84)(69 83 79 93)(70 92 80 82)(101 130 111 140)(102 139 112 129)(103 128 113 138)(104 137 114 127)(105 126 115 136)(106 135 116 125)(107 124 117 134)(108 133 118 123)(109 122 119 132)(110 131 120 121)
G:=sub<Sym(160)| (1,87,11,97)(2,98,12,88)(3,89,13,99)(4,100,14,90)(5,91,15,81)(6,82,16,92)(7,93,17,83)(8,84,18,94)(9,95,19,85)(10,86,20,96)(21,118,31,108)(22,109,32,119)(23,120,33,110)(24,111,34,101)(25,102,35,112)(26,113,36,103)(27,104,37,114)(28,115,38,105)(29,106,39,116)(30,117,40,107)(41,160,51,150)(42,151,52,141)(43,142,53,152)(44,153,54,143)(45,144,55,154)(46,155,56,145)(47,146,57,156)(48,157,58,147)(49,148,59,158)(50,159,60,149)(61,135,71,125)(62,126,72,136)(63,137,73,127)(64,128,74,138)(65,139,75,129)(66,130,76,140)(67,121,77,131)(68,132,78,122)(69,123,79,133)(70,134,80,124), (1,125,112,148)(2,149,113,126)(3,127,114,150)(4,151,115,128)(5,129,116,152)(6,153,117,130)(7,131,118,154)(8,155,119,132)(9,133,120,156)(10,157,101,134)(11,135,102,158)(12,159,103,136)(13,137,104,160)(14,141,105,138)(15,139,106,142)(16,143,107,140)(17,121,108,144)(18,145,109,122)(19,123,110,146)(20,147,111,124)(21,55,83,77)(22,78,84,56)(23,57,85,79)(24,80,86,58)(25,59,87,61)(26,62,88,60)(27,41,89,63)(28,64,90,42)(29,43,91,65)(30,66,92,44)(31,45,93,67)(32,68,94,46)(33,47,95,69)(34,70,96,48)(35,49,97,71)(36,72,98,50)(37,51,99,73)(38,74,100,52)(39,53,81,75)(40,76,82,54), (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), (1,152,11,142)(2,141,12,151)(3,150,13,160)(4,159,14,149)(5,148,15,158)(6,157,16,147)(7,146,17,156)(8,155,18,145)(9,144,19,154)(10,153,20,143)(21,57,31,47)(22,46,32,56)(23,55,33,45)(24,44,34,54)(25,53,35,43)(26,42,36,52)(27,51,37,41)(28,60,38,50)(29,49,39,59)(30,58,40,48)(61,91,71,81)(62,100,72,90)(63,89,73,99)(64,98,74,88)(65,87,75,97)(66,96,76,86)(67,85,77,95)(68,94,78,84)(69,83,79,93)(70,92,80,82)(101,130,111,140)(102,139,112,129)(103,128,113,138)(104,137,114,127)(105,126,115,136)(106,135,116,125)(107,124,117,134)(108,133,118,123)(109,122,119,132)(110,131,120,121)>;
G:=Group( (1,87,11,97)(2,98,12,88)(3,89,13,99)(4,100,14,90)(5,91,15,81)(6,82,16,92)(7,93,17,83)(8,84,18,94)(9,95,19,85)(10,86,20,96)(21,118,31,108)(22,109,32,119)(23,120,33,110)(24,111,34,101)(25,102,35,112)(26,113,36,103)(27,104,37,114)(28,115,38,105)(29,106,39,116)(30,117,40,107)(41,160,51,150)(42,151,52,141)(43,142,53,152)(44,153,54,143)(45,144,55,154)(46,155,56,145)(47,146,57,156)(48,157,58,147)(49,148,59,158)(50,159,60,149)(61,135,71,125)(62,126,72,136)(63,137,73,127)(64,128,74,138)(65,139,75,129)(66,130,76,140)(67,121,77,131)(68,132,78,122)(69,123,79,133)(70,134,80,124), (1,125,112,148)(2,149,113,126)(3,127,114,150)(4,151,115,128)(5,129,116,152)(6,153,117,130)(7,131,118,154)(8,155,119,132)(9,133,120,156)(10,157,101,134)(11,135,102,158)(12,159,103,136)(13,137,104,160)(14,141,105,138)(15,139,106,142)(16,143,107,140)(17,121,108,144)(18,145,109,122)(19,123,110,146)(20,147,111,124)(21,55,83,77)(22,78,84,56)(23,57,85,79)(24,80,86,58)(25,59,87,61)(26,62,88,60)(27,41,89,63)(28,64,90,42)(29,43,91,65)(30,66,92,44)(31,45,93,67)(32,68,94,46)(33,47,95,69)(34,70,96,48)(35,49,97,71)(36,72,98,50)(37,51,99,73)(38,74,100,52)(39,53,81,75)(40,76,82,54), (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), (1,152,11,142)(2,141,12,151)(3,150,13,160)(4,159,14,149)(5,148,15,158)(6,157,16,147)(7,146,17,156)(8,155,18,145)(9,144,19,154)(10,153,20,143)(21,57,31,47)(22,46,32,56)(23,55,33,45)(24,44,34,54)(25,53,35,43)(26,42,36,52)(27,51,37,41)(28,60,38,50)(29,49,39,59)(30,58,40,48)(61,91,71,81)(62,100,72,90)(63,89,73,99)(64,98,74,88)(65,87,75,97)(66,96,76,86)(67,85,77,95)(68,94,78,84)(69,83,79,93)(70,92,80,82)(101,130,111,140)(102,139,112,129)(103,128,113,138)(104,137,114,127)(105,126,115,136)(106,135,116,125)(107,124,117,134)(108,133,118,123)(109,122,119,132)(110,131,120,121) );
G=PermutationGroup([(1,87,11,97),(2,98,12,88),(3,89,13,99),(4,100,14,90),(5,91,15,81),(6,82,16,92),(7,93,17,83),(8,84,18,94),(9,95,19,85),(10,86,20,96),(21,118,31,108),(22,109,32,119),(23,120,33,110),(24,111,34,101),(25,102,35,112),(26,113,36,103),(27,104,37,114),(28,115,38,105),(29,106,39,116),(30,117,40,107),(41,160,51,150),(42,151,52,141),(43,142,53,152),(44,153,54,143),(45,144,55,154),(46,155,56,145),(47,146,57,156),(48,157,58,147),(49,148,59,158),(50,159,60,149),(61,135,71,125),(62,126,72,136),(63,137,73,127),(64,128,74,138),(65,139,75,129),(66,130,76,140),(67,121,77,131),(68,132,78,122),(69,123,79,133),(70,134,80,124)], [(1,125,112,148),(2,149,113,126),(3,127,114,150),(4,151,115,128),(5,129,116,152),(6,153,117,130),(7,131,118,154),(8,155,119,132),(9,133,120,156),(10,157,101,134),(11,135,102,158),(12,159,103,136),(13,137,104,160),(14,141,105,138),(15,139,106,142),(16,143,107,140),(17,121,108,144),(18,145,109,122),(19,123,110,146),(20,147,111,124),(21,55,83,77),(22,78,84,56),(23,57,85,79),(24,80,86,58),(25,59,87,61),(26,62,88,60),(27,41,89,63),(28,64,90,42),(29,43,91,65),(30,66,92,44),(31,45,93,67),(32,68,94,46),(33,47,95,69),(34,70,96,48),(35,49,97,71),(36,72,98,50),(37,51,99,73),(38,74,100,52),(39,53,81,75),(40,76,82,54)], [(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)], [(1,152,11,142),(2,141,12,151),(3,150,13,160),(4,159,14,149),(5,148,15,158),(6,157,16,147),(7,146,17,156),(8,155,18,145),(9,144,19,154),(10,153,20,143),(21,57,31,47),(22,46,32,56),(23,55,33,45),(24,44,34,54),(25,53,35,43),(26,42,36,52),(27,51,37,41),(28,60,38,50),(29,49,39,59),(30,58,40,48),(61,91,71,81),(62,100,72,90),(63,89,73,99),(64,98,74,88),(65,87,75,97),(66,96,76,86),(67,85,77,95),(68,94,78,84),(69,83,79,93),(70,92,80,82),(101,130,111,140),(102,139,112,129),(103,128,113,138),(104,137,114,127),(105,126,115,136),(106,135,116,125),(107,124,117,134),(108,133,118,123),(109,122,119,132),(110,131,120,121)])
Matrix representation ►G ⊆ 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] >;
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⋊2D20 | 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 |
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
×
⋊
ℤ
𝔽
○
≀