metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.180D10, C10.862+ (1+4), C10.412- (1+4), C4⋊Q8⋊18D5, C4⋊C4.127D10, (C2×Q8).90D10, D10⋊Q8⋊49C2, D10⋊3Q8⋊39C2, C42⋊2D5⋊20C2, Dic5⋊Q8⋊28C2, (C2×C20).641C23, (C4×C20).274C22, (C2×C10).279C24, C2.90(D4⋊6D10), Dic5.Q8⋊43C2, D10.13D4.5C2, C20.23D4.11C2, (C2×D20).180C22, C4⋊Dic5.256C22, (Q8×C10).146C22, C22.300(C23×D5), D10⋊C4.76C22, C5⋊6(C22.57C24), (C4×Dic5).176C22, (C2×Dic5).147C23, C10.D4.88C22, (C22×D5).124C23, C2.42(Q8.10D10), (C2×Dic10).199C22, (C5×C4⋊Q8)⋊21C2, C4⋊C4⋊D5⋊48C2, (C2×C4×D5).161C22, (C5×C4⋊C4).222C22, (C2×C4).222(C22×D5), SmallGroup(320,1407)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 678 in 196 conjugacy classes, 91 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×13], C22, C22 [×6], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×8], D4, Q8 [×3], C23 [×2], D5 [×2], C10, C10 [×2], C42, C42 [×2], C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×12], C22×C4 [×2], C2×D4, C2×Q8 [×2], C2×Q8, Dic5 [×6], C20 [×7], D10 [×6], C2×C10, C22⋊Q8 [×4], C22.D4 [×2], C4.4D4, C42.C2 [×2], C42⋊2C2 [×4], C4⋊Q8, C4⋊Q8, Dic10, C4×D5 [×2], D20, C2×Dic5 [×6], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×2], C22×D5 [×2], C22.57C24, C4×Dic5 [×2], C10.D4 [×10], C4⋊Dic5 [×2], D10⋊C4 [×10], C4×C20, C5×C4⋊C4 [×4], C2×Dic10, C2×C4×D5 [×2], C2×D20, Q8×C10 [×2], C42⋊2D5 [×2], Dic5.Q8 [×2], D10.13D4 [×2], D10⋊Q8 [×2], C4⋊C4⋊D5 [×2], Dic5⋊Q8, D10⋊3Q8 [×2], C20.23D4, C5×C4⋊Q8, C42.180D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], 2+ (1+4), 2- (1+4) [×2], C22×D5 [×7], C22.57C24, C23×D5, D4⋊6D10, Q8.10D10 [×2], C42.180D10
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, dcd-1=c9 >
►On 160 points
Generators in S160
(1 120 11 110)(2 111 12 101)(3 102 13 112)(4 113 14 103)(5 104 15 114)(6 115 16 105)(7 106 17 116)(8 117 18 107)(9 108 19 118)(10 119 20 109)(21 149 31 159)(22 160 32 150)(23 151 33 141)(24 142 34 152)(25 153 35 143)(26 144 36 154)(27 155 37 145)(28 146 38 156)(29 157 39 147)(30 148 40 158)(41 93 51 83)(42 84 52 94)(43 95 53 85)(44 86 54 96)(45 97 55 87)(46 88 56 98)(47 99 57 89)(48 90 58 100)(49 81 59 91)(50 92 60 82)(61 125 71 135)(62 136 72 126)(63 127 73 137)(64 138 74 128)(65 129 75 139)(66 140 76 130)(67 131 77 121)(68 122 78 132)(69 133 79 123)(70 124 80 134)
(1 151 97 126)(2 127 98 152)(3 153 99 128)(4 129 100 154)(5 155 81 130)(6 131 82 156)(7 157 83 132)(8 133 84 158)(9 159 85 134)(10 135 86 160)(11 141 87 136)(12 137 88 142)(13 143 89 138)(14 139 90 144)(15 145 91 140)(16 121 92 146)(17 147 93 122)(18 123 94 148)(19 149 95 124)(20 125 96 150)(21 43 70 108)(22 109 71 44)(23 45 72 110)(24 111 73 46)(25 47 74 112)(26 113 75 48)(27 49 76 114)(28 115 77 50)(29 51 78 116)(30 117 79 52)(31 53 80 118)(32 119 61 54)(33 55 62 120)(34 101 63 56)(35 57 64 102)(36 103 65 58)(37 59 66 104)(38 105 67 60)(39 41 68 106)(40 107 69 42)
(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 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 24 31 34)(22 33 32 23)(25 40 35 30)(26 29 36 39)(27 38 37 28)(41 103 51 113)(42 112 52 102)(43 101 53 111)(44 110 54 120)(45 119 55 109)(46 108 56 118)(47 117 57 107)(48 106 58 116)(49 115 59 105)(50 104 60 114)(61 72 71 62)(63 70 73 80)(64 79 74 69)(65 68 75 78)(66 77 76 67)(81 82 91 92)(83 100 93 90)(84 89 94 99)(85 98 95 88)(86 87 96 97)(121 145 131 155)(122 154 132 144)(123 143 133 153)(124 152 134 142)(125 141 135 151)(126 150 136 160)(127 159 137 149)(128 148 138 158)(129 157 139 147)(130 146 140 156)
G:=sub<Sym(160)| (1,120,11,110)(2,111,12,101)(3,102,13,112)(4,113,14,103)(5,104,15,114)(6,115,16,105)(7,106,17,116)(8,117,18,107)(9,108,19,118)(10,119,20,109)(21,149,31,159)(22,160,32,150)(23,151,33,141)(24,142,34,152)(25,153,35,143)(26,144,36,154)(27,155,37,145)(28,146,38,156)(29,157,39,147)(30,148,40,158)(41,93,51,83)(42,84,52,94)(43,95,53,85)(44,86,54,96)(45,97,55,87)(46,88,56,98)(47,99,57,89)(48,90,58,100)(49,81,59,91)(50,92,60,82)(61,125,71,135)(62,136,72,126)(63,127,73,137)(64,138,74,128)(65,129,75,139)(66,140,76,130)(67,131,77,121)(68,122,78,132)(69,133,79,123)(70,124,80,134), (1,151,97,126)(2,127,98,152)(3,153,99,128)(4,129,100,154)(5,155,81,130)(6,131,82,156)(7,157,83,132)(8,133,84,158)(9,159,85,134)(10,135,86,160)(11,141,87,136)(12,137,88,142)(13,143,89,138)(14,139,90,144)(15,145,91,140)(16,121,92,146)(17,147,93,122)(18,123,94,148)(19,149,95,124)(20,125,96,150)(21,43,70,108)(22,109,71,44)(23,45,72,110)(24,111,73,46)(25,47,74,112)(26,113,75,48)(27,49,76,114)(28,115,77,50)(29,51,78,116)(30,117,79,52)(31,53,80,118)(32,119,61,54)(33,55,62,120)(34,101,63,56)(35,57,64,102)(36,103,65,58)(37,59,66,104)(38,105,67,60)(39,41,68,106)(40,107,69,42), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,24,31,34)(22,33,32,23)(25,40,35,30)(26,29,36,39)(27,38,37,28)(41,103,51,113)(42,112,52,102)(43,101,53,111)(44,110,54,120)(45,119,55,109)(46,108,56,118)(47,117,57,107)(48,106,58,116)(49,115,59,105)(50,104,60,114)(61,72,71,62)(63,70,73,80)(64,79,74,69)(65,68,75,78)(66,77,76,67)(81,82,91,92)(83,100,93,90)(84,89,94,99)(85,98,95,88)(86,87,96,97)(121,145,131,155)(122,154,132,144)(123,143,133,153)(124,152,134,142)(125,141,135,151)(126,150,136,160)(127,159,137,149)(128,148,138,158)(129,157,139,147)(130,146,140,156)>;
G:=Group( (1,120,11,110)(2,111,12,101)(3,102,13,112)(4,113,14,103)(5,104,15,114)(6,115,16,105)(7,106,17,116)(8,117,18,107)(9,108,19,118)(10,119,20,109)(21,149,31,159)(22,160,32,150)(23,151,33,141)(24,142,34,152)(25,153,35,143)(26,144,36,154)(27,155,37,145)(28,146,38,156)(29,157,39,147)(30,148,40,158)(41,93,51,83)(42,84,52,94)(43,95,53,85)(44,86,54,96)(45,97,55,87)(46,88,56,98)(47,99,57,89)(48,90,58,100)(49,81,59,91)(50,92,60,82)(61,125,71,135)(62,136,72,126)(63,127,73,137)(64,138,74,128)(65,129,75,139)(66,140,76,130)(67,131,77,121)(68,122,78,132)(69,133,79,123)(70,124,80,134), (1,151,97,126)(2,127,98,152)(3,153,99,128)(4,129,100,154)(5,155,81,130)(6,131,82,156)(7,157,83,132)(8,133,84,158)(9,159,85,134)(10,135,86,160)(11,141,87,136)(12,137,88,142)(13,143,89,138)(14,139,90,144)(15,145,91,140)(16,121,92,146)(17,147,93,122)(18,123,94,148)(19,149,95,124)(20,125,96,150)(21,43,70,108)(22,109,71,44)(23,45,72,110)(24,111,73,46)(25,47,74,112)(26,113,75,48)(27,49,76,114)(28,115,77,50)(29,51,78,116)(30,117,79,52)(31,53,80,118)(32,119,61,54)(33,55,62,120)(34,101,63,56)(35,57,64,102)(36,103,65,58)(37,59,66,104)(38,105,67,60)(39,41,68,106)(40,107,69,42), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,24,31,34)(22,33,32,23)(25,40,35,30)(26,29,36,39)(27,38,37,28)(41,103,51,113)(42,112,52,102)(43,101,53,111)(44,110,54,120)(45,119,55,109)(46,108,56,118)(47,117,57,107)(48,106,58,116)(49,115,59,105)(50,104,60,114)(61,72,71,62)(63,70,73,80)(64,79,74,69)(65,68,75,78)(66,77,76,67)(81,82,91,92)(83,100,93,90)(84,89,94,99)(85,98,95,88)(86,87,96,97)(121,145,131,155)(122,154,132,144)(123,143,133,153)(124,152,134,142)(125,141,135,151)(126,150,136,160)(127,159,137,149)(128,148,138,158)(129,157,139,147)(130,146,140,156) );
G=PermutationGroup([(1,120,11,110),(2,111,12,101),(3,102,13,112),(4,113,14,103),(5,104,15,114),(6,115,16,105),(7,106,17,116),(8,117,18,107),(9,108,19,118),(10,119,20,109),(21,149,31,159),(22,160,32,150),(23,151,33,141),(24,142,34,152),(25,153,35,143),(26,144,36,154),(27,155,37,145),(28,146,38,156),(29,157,39,147),(30,148,40,158),(41,93,51,83),(42,84,52,94),(43,95,53,85),(44,86,54,96),(45,97,55,87),(46,88,56,98),(47,99,57,89),(48,90,58,100),(49,81,59,91),(50,92,60,82),(61,125,71,135),(62,136,72,126),(63,127,73,137),(64,138,74,128),(65,129,75,139),(66,140,76,130),(67,131,77,121),(68,122,78,132),(69,133,79,123),(70,124,80,134)], [(1,151,97,126),(2,127,98,152),(3,153,99,128),(4,129,100,154),(5,155,81,130),(6,131,82,156),(7,157,83,132),(8,133,84,158),(9,159,85,134),(10,135,86,160),(11,141,87,136),(12,137,88,142),(13,143,89,138),(14,139,90,144),(15,145,91,140),(16,121,92,146),(17,147,93,122),(18,123,94,148),(19,149,95,124),(20,125,96,150),(21,43,70,108),(22,109,71,44),(23,45,72,110),(24,111,73,46),(25,47,74,112),(26,113,75,48),(27,49,76,114),(28,115,77,50),(29,51,78,116),(30,117,79,52),(31,53,80,118),(32,119,61,54),(33,55,62,120),(34,101,63,56),(35,57,64,102),(36,103,65,58),(37,59,66,104),(38,105,67,60),(39,41,68,106),(40,107,69,42)], [(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)], [(1,10,11,20),(2,19,12,9),(3,8,13,18),(4,17,14,7),(5,6,15,16),(21,24,31,34),(22,33,32,23),(25,40,35,30),(26,29,36,39),(27,38,37,28),(41,103,51,113),(42,112,52,102),(43,101,53,111),(44,110,54,120),(45,119,55,109),(46,108,56,118),(47,117,57,107),(48,106,58,116),(49,115,59,105),(50,104,60,114),(61,72,71,62),(63,70,73,80),(64,79,74,69),(65,68,75,78),(66,77,76,67),(81,82,91,92),(83,100,93,90),(84,89,94,99),(85,98,95,88),(86,87,96,97),(121,145,131,155),(122,154,132,144),(123,143,133,153),(124,152,134,142),(125,141,135,151),(126,150,136,160),(127,159,137,149),(128,148,138,158),(129,157,139,147),(130,146,140,156)])
Matrix representation ►G ⊆ GL10(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 35 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 34 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 32 |
| 0 | 0 | 0 | 0 | 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 32 | 0 | 0 |
| 7 | 35 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 34 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 9 |
| 0 | 0 | 0 | 0 | 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 9 | 0 | 0 |
G:=sub<GL(10,GF(41))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0],[40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[0,7,0,0,0,0,0,0,0,0,35,34,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,32,0,0],[7,8,0,0,0,0,0,0,0,0,35,34,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,9,0,0] >;
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4G | 4H | ··· | 4M | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 4 | ··· | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | D10 | D10 | D10 | 2+ (1+4) | 2- (1+4) | D4⋊6D10 | Q8.10D10 |
| kernel | C42.180D10 | C42⋊2D5 | Dic5.Q8 | D10.13D4 | D10⋊Q8 | C4⋊C4⋊D5 | Dic5⋊Q8 | D10⋊3Q8 | C20.23D4 | C5×C4⋊Q8 | C4⋊Q8 | C42 | C4⋊C4 | C2×Q8 | C10 | C10 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 8 | 4 | 1 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_4^2._{180}D_{10} % in TeX
G:=Group("C4^2.180D10"); // GroupNames label
G:=SmallGroup(320,1407);
// by ID
G=gap.SmallGroup(320,1407);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,219,100,1571,570,136,12550]);
// 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,d*c*d^-1=c^9>;
// generators/relations
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