direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×D4⋊6D4, C10.1172- (1+4), D4⋊6(C5×D4), (C5×D4)⋊24D4, C4⋊Q8⋊13C10, (C4×D4)⋊16C10, (D4×C20)⋊45C2, C4.41(D4×C10), C20⋊16(C4○D4), C4⋊D4⋊12C10, C20.402(C2×D4), C22⋊Q8⋊12C10, C22.6(D4×C10), C42.42(C2×C10), (C4×C20).283C22, (C2×C10).367C24, (C2×C20).675C23, C22.D4⋊9C10, C10.195(C22×D4), C2.9(C5×2- (1+4)), (D4×C10).321C22, C23.41(C22×C10), (C22×C10).99C23, C22.41(C23×C10), (Q8×C10).273C22, (C22×C20).453C22, C4⋊2(C5×C4○D4), (C5×C4⋊Q8)⋊34C2, (C10×C4⋊C4)⋊47C2, (C2×C4⋊C4)⋊20C10, C2.19(D4×C2×C10), (C2×C4○D4)⋊7C10, (C10×C4○D4)⋊23C2, C4⋊C4.31(C2×C10), (C5×C4⋊D4)⋊39C2, C2.21(C10×C4○D4), (C5×C22⋊Q8)⋊39C2, (C2×D4).66(C2×C10), C10.240(C2×C4○D4), (C2×C10).183(C2×D4), C22⋊C4.5(C2×C10), (C2×Q8).60(C2×C10), (C5×C4⋊C4).395C22, (C2×C4).33(C22×C10), (C22×C4).65(C2×C10), (C5×C22.D4)⋊28C2, (C5×C22⋊C4).87C22, SmallGroup(320,1549)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 426 in 292 conjugacy classes, 166 normal (34 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×9], C22, C22 [×4], C22 [×10], C5, C2×C4 [×3], C2×C4 [×8], C2×C4 [×16], D4 [×4], D4 [×10], Q8 [×4], C23 [×4], C10 [×3], C10 [×6], C42, C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×8], C22×C4 [×8], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×8], C20 [×4], C20 [×9], C2×C10, C2×C10 [×4], C2×C10 [×10], C2×C4⋊C4 [×2], C4×D4 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×4], C4⋊Q8, C2×C4○D4 [×2], C2×C20 [×3], C2×C20 [×8], C2×C20 [×16], C5×D4 [×4], C5×D4 [×10], C5×Q8 [×4], C22×C10 [×4], D4⋊6D4, C4×C20, C5×C22⋊C4 [×8], C5×C4⋊C4 [×2], C5×C4⋊C4 [×8], C22×C20 [×8], D4×C10 [×2], D4×C10 [×4], Q8×C10 [×2], C5×C4○D4 [×8], C10×C4⋊C4 [×2], D4×C20 [×2], C5×C4⋊D4 [×2], C5×C22⋊Q8 [×2], C5×C22.D4 [×4], C5×C4⋊Q8, C10×C4○D4 [×2], C5×D4⋊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], D4⋊6D4, D4×C10 [×6], C5×C4○D4 [×2], C23×C10, D4×C2×C10, C10×C4○D4, C5×2- (1+4), C5×D4⋊6D4
Generators and relations
G = < a,b,c,d,e | a5=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, 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)
(1 135 35 116)(2 131 31 117)(3 132 32 118)(4 133 33 119)(5 134 34 120)(6 70 16 63)(7 66 17 64)(8 67 18 65)(9 68 19 61)(10 69 20 62)(11 75 156 56)(12 71 157 57)(13 72 158 58)(14 73 159 59)(15 74 160 60)(21 128 28 121)(22 129 29 122)(23 130 30 123)(24 126 26 124)(25 127 27 125)(36 115 55 96)(37 111 51 97)(38 112 52 98)(39 113 53 99)(40 114 54 100)(41 108 48 101)(42 109 49 102)(43 110 50 103)(44 106 46 104)(45 107 47 105)(76 155 95 136)(77 151 91 137)(78 152 92 138)(79 153 93 139)(80 154 94 140)(81 148 88 141)(82 149 89 142)(83 150 90 143)(84 146 86 144)(85 147 87 145)
(1 110)(2 106)(3 107)(4 108)(5 109)(6 76)(7 77)(8 78)(9 79)(10 80)(11 83)(12 84)(13 85)(14 81)(15 82)(16 95)(17 91)(18 92)(19 93)(20 94)(21 113)(22 114)(23 115)(24 111)(25 112)(26 97)(27 98)(28 99)(29 100)(30 96)(31 104)(32 105)(33 101)(34 102)(35 103)(36 130)(37 126)(38 127)(39 128)(40 129)(41 133)(42 134)(43 135)(44 131)(45 132)(46 117)(47 118)(48 119)(49 120)(50 116)(51 124)(52 125)(53 121)(54 122)(55 123)(56 150)(57 146)(58 147)(59 148)(60 149)(61 153)(62 154)(63 155)(64 151)(65 152)(66 137)(67 138)(68 139)(69 140)(70 136)(71 144)(72 145)(73 141)(74 142)(75 143)(86 157)(87 158)(88 159)(89 160)(90 156)
(1 6 30 156)(2 7 26 157)(3 8 27 158)(4 9 28 159)(5 10 29 160)(11 35 16 23)(12 31 17 24)(13 32 18 25)(14 33 19 21)(15 34 20 22)(36 143 50 155)(37 144 46 151)(38 145 47 152)(39 141 48 153)(40 142 49 154)(41 139 53 148)(42 140 54 149)(43 136 55 150)(44 137 51 146)(45 138 52 147)(56 135 70 123)(57 131 66 124)(58 132 67 125)(59 133 68 121)(60 134 69 122)(61 128 73 119)(62 129 74 120)(63 130 75 116)(64 126 71 117)(65 127 72 118)(76 96 90 110)(77 97 86 106)(78 98 87 107)(79 99 88 108)(80 100 89 109)(81 101 93 113)(82 102 94 114)(83 103 95 115)(84 104 91 111)(85 105 92 112)
(1 30)(2 26)(3 27)(4 28)(5 29)(21 33)(22 34)(23 35)(24 31)(25 32)(36 43)(37 44)(38 45)(39 41)(40 42)(46 51)(47 52)(48 53)(49 54)(50 55)(76 95)(77 91)(78 92)(79 93)(80 94)(81 88)(82 89)(83 90)(84 86)(85 87)(96 103)(97 104)(98 105)(99 101)(100 102)(106 111)(107 112)(108 113)(109 114)(110 115)(116 130)(117 126)(118 127)(119 128)(120 129)(121 133)(122 134)(123 135)(124 131)(125 132)(136 155)(137 151)(138 152)(139 153)(140 154)(141 148)(142 149)(143 150)(144 146)(145 147)
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), (1,135,35,116)(2,131,31,117)(3,132,32,118)(4,133,33,119)(5,134,34,120)(6,70,16,63)(7,66,17,64)(8,67,18,65)(9,68,19,61)(10,69,20,62)(11,75,156,56)(12,71,157,57)(13,72,158,58)(14,73,159,59)(15,74,160,60)(21,128,28,121)(22,129,29,122)(23,130,30,123)(24,126,26,124)(25,127,27,125)(36,115,55,96)(37,111,51,97)(38,112,52,98)(39,113,53,99)(40,114,54,100)(41,108,48,101)(42,109,49,102)(43,110,50,103)(44,106,46,104)(45,107,47,105)(76,155,95,136)(77,151,91,137)(78,152,92,138)(79,153,93,139)(80,154,94,140)(81,148,88,141)(82,149,89,142)(83,150,90,143)(84,146,86,144)(85,147,87,145), (1,110)(2,106)(3,107)(4,108)(5,109)(6,76)(7,77)(8,78)(9,79)(10,80)(11,83)(12,84)(13,85)(14,81)(15,82)(16,95)(17,91)(18,92)(19,93)(20,94)(21,113)(22,114)(23,115)(24,111)(25,112)(26,97)(27,98)(28,99)(29,100)(30,96)(31,104)(32,105)(33,101)(34,102)(35,103)(36,130)(37,126)(38,127)(39,128)(40,129)(41,133)(42,134)(43,135)(44,131)(45,132)(46,117)(47,118)(48,119)(49,120)(50,116)(51,124)(52,125)(53,121)(54,122)(55,123)(56,150)(57,146)(58,147)(59,148)(60,149)(61,153)(62,154)(63,155)(64,151)(65,152)(66,137)(67,138)(68,139)(69,140)(70,136)(71,144)(72,145)(73,141)(74,142)(75,143)(86,157)(87,158)(88,159)(89,160)(90,156), (1,6,30,156)(2,7,26,157)(3,8,27,158)(4,9,28,159)(5,10,29,160)(11,35,16,23)(12,31,17,24)(13,32,18,25)(14,33,19,21)(15,34,20,22)(36,143,50,155)(37,144,46,151)(38,145,47,152)(39,141,48,153)(40,142,49,154)(41,139,53,148)(42,140,54,149)(43,136,55,150)(44,137,51,146)(45,138,52,147)(56,135,70,123)(57,131,66,124)(58,132,67,125)(59,133,68,121)(60,134,69,122)(61,128,73,119)(62,129,74,120)(63,130,75,116)(64,126,71,117)(65,127,72,118)(76,96,90,110)(77,97,86,106)(78,98,87,107)(79,99,88,108)(80,100,89,109)(81,101,93,113)(82,102,94,114)(83,103,95,115)(84,104,91,111)(85,105,92,112), (1,30)(2,26)(3,27)(4,28)(5,29)(21,33)(22,34)(23,35)(24,31)(25,32)(36,43)(37,44)(38,45)(39,41)(40,42)(46,51)(47,52)(48,53)(49,54)(50,55)(76,95)(77,91)(78,92)(79,93)(80,94)(81,88)(82,89)(83,90)(84,86)(85,87)(96,103)(97,104)(98,105)(99,101)(100,102)(106,111)(107,112)(108,113)(109,114)(110,115)(116,130)(117,126)(118,127)(119,128)(120,129)(121,133)(122,134)(123,135)(124,131)(125,132)(136,155)(137,151)(138,152)(139,153)(140,154)(141,148)(142,149)(143,150)(144,146)(145,147)>;
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), (1,135,35,116)(2,131,31,117)(3,132,32,118)(4,133,33,119)(5,134,34,120)(6,70,16,63)(7,66,17,64)(8,67,18,65)(9,68,19,61)(10,69,20,62)(11,75,156,56)(12,71,157,57)(13,72,158,58)(14,73,159,59)(15,74,160,60)(21,128,28,121)(22,129,29,122)(23,130,30,123)(24,126,26,124)(25,127,27,125)(36,115,55,96)(37,111,51,97)(38,112,52,98)(39,113,53,99)(40,114,54,100)(41,108,48,101)(42,109,49,102)(43,110,50,103)(44,106,46,104)(45,107,47,105)(76,155,95,136)(77,151,91,137)(78,152,92,138)(79,153,93,139)(80,154,94,140)(81,148,88,141)(82,149,89,142)(83,150,90,143)(84,146,86,144)(85,147,87,145), (1,110)(2,106)(3,107)(4,108)(5,109)(6,76)(7,77)(8,78)(9,79)(10,80)(11,83)(12,84)(13,85)(14,81)(15,82)(16,95)(17,91)(18,92)(19,93)(20,94)(21,113)(22,114)(23,115)(24,111)(25,112)(26,97)(27,98)(28,99)(29,100)(30,96)(31,104)(32,105)(33,101)(34,102)(35,103)(36,130)(37,126)(38,127)(39,128)(40,129)(41,133)(42,134)(43,135)(44,131)(45,132)(46,117)(47,118)(48,119)(49,120)(50,116)(51,124)(52,125)(53,121)(54,122)(55,123)(56,150)(57,146)(58,147)(59,148)(60,149)(61,153)(62,154)(63,155)(64,151)(65,152)(66,137)(67,138)(68,139)(69,140)(70,136)(71,144)(72,145)(73,141)(74,142)(75,143)(86,157)(87,158)(88,159)(89,160)(90,156), (1,6,30,156)(2,7,26,157)(3,8,27,158)(4,9,28,159)(5,10,29,160)(11,35,16,23)(12,31,17,24)(13,32,18,25)(14,33,19,21)(15,34,20,22)(36,143,50,155)(37,144,46,151)(38,145,47,152)(39,141,48,153)(40,142,49,154)(41,139,53,148)(42,140,54,149)(43,136,55,150)(44,137,51,146)(45,138,52,147)(56,135,70,123)(57,131,66,124)(58,132,67,125)(59,133,68,121)(60,134,69,122)(61,128,73,119)(62,129,74,120)(63,130,75,116)(64,126,71,117)(65,127,72,118)(76,96,90,110)(77,97,86,106)(78,98,87,107)(79,99,88,108)(80,100,89,109)(81,101,93,113)(82,102,94,114)(83,103,95,115)(84,104,91,111)(85,105,92,112), (1,30)(2,26)(3,27)(4,28)(5,29)(21,33)(22,34)(23,35)(24,31)(25,32)(36,43)(37,44)(38,45)(39,41)(40,42)(46,51)(47,52)(48,53)(49,54)(50,55)(76,95)(77,91)(78,92)(79,93)(80,94)(81,88)(82,89)(83,90)(84,86)(85,87)(96,103)(97,104)(98,105)(99,101)(100,102)(106,111)(107,112)(108,113)(109,114)(110,115)(116,130)(117,126)(118,127)(119,128)(120,129)(121,133)(122,134)(123,135)(124,131)(125,132)(136,155)(137,151)(138,152)(139,153)(140,154)(141,148)(142,149)(143,150)(144,146)(145,147) );
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)], [(1,135,35,116),(2,131,31,117),(3,132,32,118),(4,133,33,119),(5,134,34,120),(6,70,16,63),(7,66,17,64),(8,67,18,65),(9,68,19,61),(10,69,20,62),(11,75,156,56),(12,71,157,57),(13,72,158,58),(14,73,159,59),(15,74,160,60),(21,128,28,121),(22,129,29,122),(23,130,30,123),(24,126,26,124),(25,127,27,125),(36,115,55,96),(37,111,51,97),(38,112,52,98),(39,113,53,99),(40,114,54,100),(41,108,48,101),(42,109,49,102),(43,110,50,103),(44,106,46,104),(45,107,47,105),(76,155,95,136),(77,151,91,137),(78,152,92,138),(79,153,93,139),(80,154,94,140),(81,148,88,141),(82,149,89,142),(83,150,90,143),(84,146,86,144),(85,147,87,145)], [(1,110),(2,106),(3,107),(4,108),(5,109),(6,76),(7,77),(8,78),(9,79),(10,80),(11,83),(12,84),(13,85),(14,81),(15,82),(16,95),(17,91),(18,92),(19,93),(20,94),(21,113),(22,114),(23,115),(24,111),(25,112),(26,97),(27,98),(28,99),(29,100),(30,96),(31,104),(32,105),(33,101),(34,102),(35,103),(36,130),(37,126),(38,127),(39,128),(40,129),(41,133),(42,134),(43,135),(44,131),(45,132),(46,117),(47,118),(48,119),(49,120),(50,116),(51,124),(52,125),(53,121),(54,122),(55,123),(56,150),(57,146),(58,147),(59,148),(60,149),(61,153),(62,154),(63,155),(64,151),(65,152),(66,137),(67,138),(68,139),(69,140),(70,136),(71,144),(72,145),(73,141),(74,142),(75,143),(86,157),(87,158),(88,159),(89,160),(90,156)], [(1,6,30,156),(2,7,26,157),(3,8,27,158),(4,9,28,159),(5,10,29,160),(11,35,16,23),(12,31,17,24),(13,32,18,25),(14,33,19,21),(15,34,20,22),(36,143,50,155),(37,144,46,151),(38,145,47,152),(39,141,48,153),(40,142,49,154),(41,139,53,148),(42,140,54,149),(43,136,55,150),(44,137,51,146),(45,138,52,147),(56,135,70,123),(57,131,66,124),(58,132,67,125),(59,133,68,121),(60,134,69,122),(61,128,73,119),(62,129,74,120),(63,130,75,116),(64,126,71,117),(65,127,72,118),(76,96,90,110),(77,97,86,106),(78,98,87,107),(79,99,88,108),(80,100,89,109),(81,101,93,113),(82,102,94,114),(83,103,95,115),(84,104,91,111),(85,105,92,112)], [(1,30),(2,26),(3,27),(4,28),(5,29),(21,33),(22,34),(23,35),(24,31),(25,32),(36,43),(37,44),(38,45),(39,41),(40,42),(46,51),(47,52),(48,53),(49,54),(50,55),(76,95),(77,91),(78,92),(79,93),(80,94),(81,88),(82,89),(83,90),(84,86),(85,87),(96,103),(97,104),(98,105),(99,101),(100,102),(106,111),(107,112),(108,113),(109,114),(110,115),(116,130),(117,126),(118,127),(119,128),(120,129),(121,133),(122,134),(123,135),(124,131),(125,132),(136,155),(137,151),(138,152),(139,153),(140,154),(141,148),(142,149),(143,150),(144,146),(145,147)])
Matrix representation ►G ⊆ GL5(𝔽41)
| 37 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 6 | 2 |
| 0 | 0 | 0 | 2 | 35 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 6 | 1 |
G:=sub<GL(5,GF(41))| [37,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,9,0,0,0,0,0,32,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,0,32,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,6,2,0,0,0,2,35],[40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,40,6,0,0,0,0,1] >;
125 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4H | 4I | ··· | 4O | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 10M | ··· | 10AB | 10AC | ··· | 10AJ | 20A | ··· | 20AF | 20AG | ··· | 20BH |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
125 irreducible representations
| dim | 1 | 1 | 1 | 1 | 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 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | C10 | D4 | C4○D4 | C5×D4 | C5×C4○D4 | 2- (1+4) | C5×2- (1+4) |
| kernel | C5×D4⋊6D4 | C10×C4⋊C4 | D4×C20 | C5×C4⋊D4 | C5×C22⋊Q8 | C5×C22.D4 | C5×C4⋊Q8 | C10×C4○D4 | D4⋊6D4 | C2×C4⋊C4 | C4×D4 | C4⋊D4 | C22⋊Q8 | C22.D4 | C4⋊Q8 | C2×C4○D4 | C5×D4 | C20 | D4 | C4 | C10 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 2 | 4 | 8 | 8 | 8 | 8 | 16 | 4 | 8 | 4 | 4 | 16 | 16 | 1 | 4 |
In GAP, Magma, Sage, TeX
C_5\times D_4\rtimes_6D_4
% in TeX
G:=Group("C5xD4:6D4"); // GroupNames label
G:=SmallGroup(320,1549);
// by ID
G=gap.SmallGroup(320,1549);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,560,1149,3446,436,1242]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations
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