direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×Q8⋊6D4, C10.1622+ (1+4), Q8⋊6(C5×D4), (C5×Q8)⋊24D4, (D4×C20)⋊47C2, (C4×D4)⋊18C10, C4⋊1D4⋊8C10, (Q8×C20)⋊33C2, (C4×Q8)⋊13C10, C4.44(D4×C10), C20⋊19(C4○D4), C4⋊D4⋊14C10, C20.405(C2×D4), C42.45(C2×C10), (C2×C20).677C23, (C4×C20).286C22, (C2×C10).370C24, C10.198(C22×D4), (D4×C10).220C22, C22.44(C23×C10), C23.17(C22×C10), (Q8×C10).286C22, C2.14(C5×2+ (1+4)), (C22×C10).101C23, (C22×C20).456C22, C4⋊3(C5×C4○D4), C2.22(D4×C2×C10), (C2×C4○D4)⋊9C10, (C10×C4○D4)⋊25C2, (C5×C4⋊1D4)⋊19C2, (C5×C4⋊D4)⋊41C2, C4⋊C4.72(C2×C10), C2.23(C10×C4○D4), (C2×D4).68(C2×C10), C10.242(C2×C4○D4), (C2×Q8).74(C2×C10), (C5×C4⋊C4).398C22, C22⋊C4.21(C2×C10), (C22×C4).68(C2×C10), (C2×C4).137(C22×C10), (C5×C22⋊C4).154C22, SmallGroup(320,1552)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 506 in 312 conjugacy classes, 166 normal (20 characteristic)
C1, C2 [×3], C2 [×6], C4 [×8], C4 [×5], C22, C22 [×18], C5, C2×C4, C2×C4 [×8], C2×C4 [×12], D4 [×24], Q8 [×4], C23 [×6], C10 [×3], C10 [×6], C42 [×3], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×3], C22×C4 [×6], C2×D4 [×15], C2×Q8, C4○D4 [×8], C20 [×8], C20 [×5], C2×C10, C2×C10 [×18], C4×D4 [×3], C4×Q8, C4⋊D4 [×6], C4⋊1D4 [×3], C2×C4○D4 [×2], C2×C20, C2×C20 [×8], C2×C20 [×12], C5×D4 [×24], C5×Q8 [×4], C22×C10 [×6], Q8⋊6D4, C4×C20 [×3], C5×C22⋊C4 [×6], C5×C4⋊C4, C5×C4⋊C4 [×3], C22×C20 [×6], D4×C10 [×15], Q8×C10, C5×C4○D4 [×8], D4×C20 [×3], Q8×C20, C5×C4⋊D4 [×6], C5×C4⋊1D4 [×3], C10×C4○D4 [×2], C5×Q8⋊6D4
Quotients:
C1, C2 [×15], C22 [×35], C5, D4 [×4], C23 [×15], C10 [×15], C2×D4 [×6], C4○D4 [×2], C24, C2×C10 [×35], C22×D4, C2×C4○D4, 2+ (1+4), C5×D4 [×4], C22×C10 [×15], Q8⋊6D4, D4×C10 [×6], C5×C4○D4 [×2], C23×C10, D4×C2×C10, C10×C4○D4, C5×2+ (1+4), C5×Q8⋊6D4
Generators and relations
G = < a,b,c,d,e | a5=b4=d4=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=b-1, be=eb, dcd-1=ece=b2c, ede=d-1 >
►On 160 points
Generators in S160
(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 43 35 50)(2 44 31 46)(3 45 32 47)(4 41 33 48)(5 42 34 49)(6 136 16 155)(7 137 17 151)(8 138 18 152)(9 139 19 153)(10 140 20 154)(11 143 156 150)(12 144 157 146)(13 145 158 147)(14 141 159 148)(15 142 160 149)(21 39 28 53)(22 40 29 54)(23 36 30 55)(24 37 26 51)(25 38 27 52)(56 83 75 90)(57 84 71 86)(58 85 72 87)(59 81 73 88)(60 82 74 89)(61 79 68 93)(62 80 69 94)(63 76 70 95)(64 77 66 91)(65 78 67 92)(96 123 115 130)(97 124 111 126)(98 125 112 127)(99 121 113 128)(100 122 114 129)(101 119 108 133)(102 120 109 134)(103 116 110 135)(104 117 106 131)(105 118 107 132)
(1 90 35 83)(2 86 31 84)(3 87 32 85)(4 88 33 81)(5 89 34 82)(6 96 16 115)(7 97 17 111)(8 98 18 112)(9 99 19 113)(10 100 20 114)(11 103 156 110)(12 104 157 106)(13 105 158 107)(14 101 159 108)(15 102 160 109)(21 93 28 79)(22 94 29 80)(23 95 30 76)(24 91 26 77)(25 92 27 78)(36 70 55 63)(37 66 51 64)(38 67 52 65)(39 68 53 61)(40 69 54 62)(41 73 48 59)(42 74 49 60)(43 75 50 56)(44 71 46 57)(45 72 47 58)(116 143 135 150)(117 144 131 146)(118 145 132 147)(119 141 133 148)(120 142 134 149)(121 139 128 153)(122 140 129 154)(123 136 130 155)(124 137 126 151)(125 138 127 152)
(1 135 23 130)(2 131 24 126)(3 132 25 127)(4 133 21 128)(5 134 22 129)(6 56 11 63)(7 57 12 64)(8 58 13 65)(9 59 14 61)(10 60 15 62)(16 75 156 70)(17 71 157 66)(18 72 158 67)(19 73 159 68)(20 74 160 69)(26 124 31 117)(27 125 32 118)(28 121 33 119)(29 122 34 120)(30 123 35 116)(36 115 43 110)(37 111 44 106)(38 112 45 107)(39 113 41 108)(40 114 42 109)(46 104 51 97)(47 105 52 98)(48 101 53 99)(49 102 54 100)(50 103 55 96)(76 155 83 150)(77 151 84 146)(78 152 85 147)(79 153 81 148)(80 154 82 149)(86 144 91 137)(87 145 92 138)(88 141 93 139)(89 142 94 140)(90 143 95 136)
(1 96)(2 97)(3 98)(4 99)(5 100)(6 90)(7 86)(8 87)(9 88)(10 89)(11 95)(12 91)(13 92)(14 93)(15 94)(16 83)(17 84)(18 85)(19 81)(20 82)(21 101)(22 102)(23 103)(24 104)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(61 141)(62 142)(63 143)(64 144)(65 145)(66 146)(67 147)(68 148)(69 149)(70 150)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)
G:=sub<Sym(160)| (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), (1,43,35,50)(2,44,31,46)(3,45,32,47)(4,41,33,48)(5,42,34,49)(6,136,16,155)(7,137,17,151)(8,138,18,152)(9,139,19,153)(10,140,20,154)(11,143,156,150)(12,144,157,146)(13,145,158,147)(14,141,159,148)(15,142,160,149)(21,39,28,53)(22,40,29,54)(23,36,30,55)(24,37,26,51)(25,38,27,52)(56,83,75,90)(57,84,71,86)(58,85,72,87)(59,81,73,88)(60,82,74,89)(61,79,68,93)(62,80,69,94)(63,76,70,95)(64,77,66,91)(65,78,67,92)(96,123,115,130)(97,124,111,126)(98,125,112,127)(99,121,113,128)(100,122,114,129)(101,119,108,133)(102,120,109,134)(103,116,110,135)(104,117,106,131)(105,118,107,132), (1,90,35,83)(2,86,31,84)(3,87,32,85)(4,88,33,81)(5,89,34,82)(6,96,16,115)(7,97,17,111)(8,98,18,112)(9,99,19,113)(10,100,20,114)(11,103,156,110)(12,104,157,106)(13,105,158,107)(14,101,159,108)(15,102,160,109)(21,93,28,79)(22,94,29,80)(23,95,30,76)(24,91,26,77)(25,92,27,78)(36,70,55,63)(37,66,51,64)(38,67,52,65)(39,68,53,61)(40,69,54,62)(41,73,48,59)(42,74,49,60)(43,75,50,56)(44,71,46,57)(45,72,47,58)(116,143,135,150)(117,144,131,146)(118,145,132,147)(119,141,133,148)(120,142,134,149)(121,139,128,153)(122,140,129,154)(123,136,130,155)(124,137,126,151)(125,138,127,152), (1,135,23,130)(2,131,24,126)(3,132,25,127)(4,133,21,128)(5,134,22,129)(6,56,11,63)(7,57,12,64)(8,58,13,65)(9,59,14,61)(10,60,15,62)(16,75,156,70)(17,71,157,66)(18,72,158,67)(19,73,159,68)(20,74,160,69)(26,124,31,117)(27,125,32,118)(28,121,33,119)(29,122,34,120)(30,123,35,116)(36,115,43,110)(37,111,44,106)(38,112,45,107)(39,113,41,108)(40,114,42,109)(46,104,51,97)(47,105,52,98)(48,101,53,99)(49,102,54,100)(50,103,55,96)(76,155,83,150)(77,151,84,146)(78,152,85,147)(79,153,81,148)(80,154,82,149)(86,144,91,137)(87,145,92,138)(88,141,93,139)(89,142,94,140)(90,143,95,136), (1,96)(2,97)(3,98)(4,99)(5,100)(6,90)(7,86)(8,87)(9,88)(10,89)(11,95)(12,91)(13,92)(14,93)(15,94)(16,83)(17,84)(18,85)(19,81)(20,82)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160)>;
G:=Group( (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), (1,43,35,50)(2,44,31,46)(3,45,32,47)(4,41,33,48)(5,42,34,49)(6,136,16,155)(7,137,17,151)(8,138,18,152)(9,139,19,153)(10,140,20,154)(11,143,156,150)(12,144,157,146)(13,145,158,147)(14,141,159,148)(15,142,160,149)(21,39,28,53)(22,40,29,54)(23,36,30,55)(24,37,26,51)(25,38,27,52)(56,83,75,90)(57,84,71,86)(58,85,72,87)(59,81,73,88)(60,82,74,89)(61,79,68,93)(62,80,69,94)(63,76,70,95)(64,77,66,91)(65,78,67,92)(96,123,115,130)(97,124,111,126)(98,125,112,127)(99,121,113,128)(100,122,114,129)(101,119,108,133)(102,120,109,134)(103,116,110,135)(104,117,106,131)(105,118,107,132), (1,90,35,83)(2,86,31,84)(3,87,32,85)(4,88,33,81)(5,89,34,82)(6,96,16,115)(7,97,17,111)(8,98,18,112)(9,99,19,113)(10,100,20,114)(11,103,156,110)(12,104,157,106)(13,105,158,107)(14,101,159,108)(15,102,160,109)(21,93,28,79)(22,94,29,80)(23,95,30,76)(24,91,26,77)(25,92,27,78)(36,70,55,63)(37,66,51,64)(38,67,52,65)(39,68,53,61)(40,69,54,62)(41,73,48,59)(42,74,49,60)(43,75,50,56)(44,71,46,57)(45,72,47,58)(116,143,135,150)(117,144,131,146)(118,145,132,147)(119,141,133,148)(120,142,134,149)(121,139,128,153)(122,140,129,154)(123,136,130,155)(124,137,126,151)(125,138,127,152), (1,135,23,130)(2,131,24,126)(3,132,25,127)(4,133,21,128)(5,134,22,129)(6,56,11,63)(7,57,12,64)(8,58,13,65)(9,59,14,61)(10,60,15,62)(16,75,156,70)(17,71,157,66)(18,72,158,67)(19,73,159,68)(20,74,160,69)(26,124,31,117)(27,125,32,118)(28,121,33,119)(29,122,34,120)(30,123,35,116)(36,115,43,110)(37,111,44,106)(38,112,45,107)(39,113,41,108)(40,114,42,109)(46,104,51,97)(47,105,52,98)(48,101,53,99)(49,102,54,100)(50,103,55,96)(76,155,83,150)(77,151,84,146)(78,152,85,147)(79,153,81,148)(80,154,82,149)(86,144,91,137)(87,145,92,138)(88,141,93,139)(89,142,94,140)(90,143,95,136), (1,96)(2,97)(3,98)(4,99)(5,100)(6,90)(7,86)(8,87)(9,88)(10,89)(11,95)(12,91)(13,92)(14,93)(15,94)(16,83)(17,84)(18,85)(19,81)(20,82)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160) );
G=PermutationGroup([(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)], [(1,43,35,50),(2,44,31,46),(3,45,32,47),(4,41,33,48),(5,42,34,49),(6,136,16,155),(7,137,17,151),(8,138,18,152),(9,139,19,153),(10,140,20,154),(11,143,156,150),(12,144,157,146),(13,145,158,147),(14,141,159,148),(15,142,160,149),(21,39,28,53),(22,40,29,54),(23,36,30,55),(24,37,26,51),(25,38,27,52),(56,83,75,90),(57,84,71,86),(58,85,72,87),(59,81,73,88),(60,82,74,89),(61,79,68,93),(62,80,69,94),(63,76,70,95),(64,77,66,91),(65,78,67,92),(96,123,115,130),(97,124,111,126),(98,125,112,127),(99,121,113,128),(100,122,114,129),(101,119,108,133),(102,120,109,134),(103,116,110,135),(104,117,106,131),(105,118,107,132)], [(1,90,35,83),(2,86,31,84),(3,87,32,85),(4,88,33,81),(5,89,34,82),(6,96,16,115),(7,97,17,111),(8,98,18,112),(9,99,19,113),(10,100,20,114),(11,103,156,110),(12,104,157,106),(13,105,158,107),(14,101,159,108),(15,102,160,109),(21,93,28,79),(22,94,29,80),(23,95,30,76),(24,91,26,77),(25,92,27,78),(36,70,55,63),(37,66,51,64),(38,67,52,65),(39,68,53,61),(40,69,54,62),(41,73,48,59),(42,74,49,60),(43,75,50,56),(44,71,46,57),(45,72,47,58),(116,143,135,150),(117,144,131,146),(118,145,132,147),(119,141,133,148),(120,142,134,149),(121,139,128,153),(122,140,129,154),(123,136,130,155),(124,137,126,151),(125,138,127,152)], [(1,135,23,130),(2,131,24,126),(3,132,25,127),(4,133,21,128),(5,134,22,129),(6,56,11,63),(7,57,12,64),(8,58,13,65),(9,59,14,61),(10,60,15,62),(16,75,156,70),(17,71,157,66),(18,72,158,67),(19,73,159,68),(20,74,160,69),(26,124,31,117),(27,125,32,118),(28,121,33,119),(29,122,34,120),(30,123,35,116),(36,115,43,110),(37,111,44,106),(38,112,45,107),(39,113,41,108),(40,114,42,109),(46,104,51,97),(47,105,52,98),(48,101,53,99),(49,102,54,100),(50,103,55,96),(76,155,83,150),(77,151,84,146),(78,152,85,147),(79,153,81,148),(80,154,82,149),(86,144,91,137),(87,145,92,138),(88,141,93,139),(89,142,94,140),(90,143,95,136)], [(1,96),(2,97),(3,98),(4,99),(5,100),(6,90),(7,86),(8,87),(9,88),(10,89),(11,95),(12,91),(13,92),(14,93),(15,94),(16,83),(17,84),(18,85),(19,81),(20,82),(21,101),(22,102),(23,103),(24,104),(25,105),(26,106),(27,107),(28,108),(29,109),(30,110),(31,111),(32,112),(33,113),(34,114),(35,115),(36,116),(37,117),(38,118),(39,119),(40,120),(41,121),(42,122),(43,123),(44,124),(45,125),(46,126),(47,127),(48,128),(49,129),(50,130),(51,131),(52,132),(53,133),(54,134),(55,135),(56,136),(57,137),(58,138),(59,139),(60,140),(61,141),(62,142),(63,143),(64,144),(65,145),(66,146),(67,147),(68,148),(69,149),(70,150),(71,151),(72,152),(73,153),(74,154),(75,155),(76,156),(77,157),(78,158),(79,159),(80,160)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 0 | 9 | 0 |
| 5 | 39 | 0 | 0 |
| 13 | 36 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 32 |
| 36 | 2 | 0 | 0 |
| 29 | 5 | 0 | 0 |
| 0 | 0 | 0 | 32 |
| 0 | 0 | 9 | 0 |
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,18,0,0,0,0,18],[40,0,0,0,0,40,0,0,0,0,0,40,0,0,1,0],[40,0,0,0,0,40,0,0,0,0,0,9,0,0,9,0],[5,13,0,0,39,36,0,0,0,0,9,0,0,0,0,32],[36,29,0,0,2,5,0,0,0,0,0,9,0,0,32,0] >;
125 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | ··· | 4L | 4M | 4N | 4O | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 10M | ··· | 10AJ | 20A | ··· | 20AV | 20AW | ··· | 20BH |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
125 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | D4 | C4○D4 | C5×D4 | C5×C4○D4 | 2+ (1+4) | C5×2+ (1+4) |
| kernel | C5×Q8⋊6D4 | D4×C20 | Q8×C20 | C5×C4⋊D4 | C5×C4⋊1D4 | C10×C4○D4 | Q8⋊6D4 | C4×D4 | C4×Q8 | C4⋊D4 | C4⋊1D4 | C2×C4○D4 | C5×Q8 | C20 | Q8 | C4 | C10 | C2 |
| # reps | 1 | 3 | 1 | 6 | 3 | 2 | 4 | 12 | 4 | 24 | 12 | 8 | 4 | 4 | 16 | 16 | 1 | 4 |
In GAP, Magma, Sage, TeX
C_5\times Q_8\rtimes_6D_4
% in TeX
G:=Group("C5xQ8:6D4"); // GroupNames label
G:=SmallGroup(320,1552);
// by ID
G=gap.SmallGroup(320,1552);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,568,3446,436,1242,304]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=d^4=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations
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