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G = (Q8×D5)⋊C4order 320 = 26·5

2nd semidirect product of Q8×D5 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (Q8×D5)⋊2C4, (C4×D5).5D4, Q8.9(C4×D5), C4.163(D4×D5), C4⋊C4.148D10, Q8⋊C416D5, (C2×C8).175D10, Q8⋊Dic54C2, C20.117(C2×D4), C22.77(D4×D5), C10.Q1610C2, C20.48(C22×C4), (C2×Q8).104D10, C20.44D423C2, (C2×C40).197C22, (C2×C20).242C23, C2.2(Q16⋊D5), Dic10.19(C2×C4), (C2×Dic5).208D4, C2.4(SD16⋊D5), (C22×D5).111D4, C52(C23.38D4), C4⋊Dic5.90C22, (Q8×C10).25C22, D10.37(C22⋊C4), C10.34(C8.C22), Dic5.26(C22⋊C4), (C2×Dic10).73C22, C4.13(C2×C4×D5), (C2×Q8×D5).3C2, (C4×D5).3(C2×C4), C4⋊C47D5.1C2, (C5×Q8).18(C2×C4), C2.22(D5×C22⋊C4), (C2×C4×D5).21C22, (C5×Q8⋊C4)⋊19C2, (C2×C10).255(C2×D4), (C2×C8⋊D5).10C2, (C5×C4⋊C4).43C22, C10.62(C2×C22⋊C4), (C2×C52C8).35C22, (C2×C4).349(C22×D5), SmallGroup(320,429)

Series: Derived Chief Lower central Upper central

C1C20 — (Q8×D5)⋊C4
C1C5C10C2×C10C2×C20C2×C4×D5C2×Q8×D5 — (Q8×D5)⋊C4
C5C10C20 — (Q8×D5)⋊C4
C1C22C2×C4Q8⋊C4

Generators and relations for (Q8×D5)⋊C4
 G = < a,b,c,d,e | a4=c5=d2=e4=1, b2=a2, bab-1=eae-1=a-1, ac=ca, ad=da, bc=cb, bd=db, ebe-1=ab, dcd=c-1, ce=ec, ede-1=a2d >

Subgroups: 558 in 150 conjugacy classes, 55 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×8], C22, C22 [×4], C5, C8 [×2], C2×C4, C2×C4 [×13], Q8 [×2], Q8 [×8], C23, D5 [×2], C10 [×3], C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2) [×2], C22×C4 [×2], C2×Q8, C2×Q8 [×8], Dic5 [×2], Dic5 [×3], C20 [×2], C20 [×3], D10 [×2], D10 [×2], C2×C10, Q8⋊C4, Q8⋊C4 [×3], C42⋊C2, C2×M4(2), C22×Q8, C52C8, C40, Dic10 [×2], Dic10 [×5], C4×D5 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C5×Q8, C22×D5, C23.38D4, C8⋊D5 [×2], C2×C52C8, C4×Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, C2×Dic10, C2×Dic10, C2×C4×D5, C2×C4×D5, Q8×D5 [×4], Q8×D5 [×2], Q8×C10, C10.Q16, C20.44D4, Q8⋊Dic5, C5×Q8⋊C4, C4⋊C47D5, C2×C8⋊D5, C2×Q8×D5, (Q8×D5)⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C2×C22⋊C4, C8.C22 [×2], C4×D5 [×2], C22×D5, C23.38D4, C2×C4×D5, D4×D5 [×2], D5×C22⋊C4, SD16⋊D5, Q16⋊D5, (Q8×D5)⋊C4

Smallest permutation representation of (Q8×D5)⋊C4
On 160 points
Generators in S160
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)(41 51 46 56)(42 52 47 57)(43 53 48 58)(44 54 49 59)(45 55 50 60)(61 71 66 76)(62 72 67 77)(63 73 68 78)(64 74 69 79)(65 75 70 80)(81 91 86 96)(82 92 87 97)(83 93 88 98)(84 94 89 99)(85 95 90 100)(101 111 106 116)(102 112 107 117)(103 113 108 118)(104 114 109 119)(105 115 110 120)(121 136 126 131)(122 137 127 132)(123 138 128 133)(124 139 129 134)(125 140 130 135)(141 156 146 151)(142 157 147 152)(143 158 148 153)(144 159 149 154)(145 160 150 155)
(1 86 6 81)(2 87 7 82)(3 88 8 83)(4 89 9 84)(5 90 10 85)(11 96 16 91)(12 97 17 92)(13 98 18 93)(14 99 19 94)(15 100 20 95)(21 106 26 101)(22 107 27 102)(23 108 28 103)(24 109 29 104)(25 110 30 105)(31 116 36 111)(32 117 37 112)(33 118 38 113)(34 119 39 114)(35 120 40 115)(41 126 46 121)(42 127 47 122)(43 128 48 123)(44 129 49 124)(45 130 50 125)(51 136 56 131)(52 137 57 132)(53 138 58 133)(54 139 59 134)(55 140 60 135)(61 146 66 141)(62 147 67 142)(63 148 68 143)(64 149 69 144)(65 150 70 145)(71 156 76 151)(72 157 77 152)(73 158 78 153)(74 159 79 154)(75 160 80 155)
(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)(2 9)(3 8)(4 7)(5 6)(11 20)(12 19)(13 18)(14 17)(15 16)(21 30)(22 29)(23 28)(24 27)(25 26)(31 40)(32 39)(33 38)(34 37)(35 36)(41 45)(42 44)(46 50)(47 49)(51 55)(52 54)(56 60)(57 59)(61 65)(62 64)(66 70)(67 69)(71 75)(72 74)(76 80)(77 79)(81 90)(82 89)(83 88)(84 87)(85 86)(91 100)(92 99)(93 98)(94 97)(95 96)(101 110)(102 109)(103 108)(104 107)(105 106)(111 120)(112 119)(113 118)(114 117)(115 116)(121 125)(122 124)(126 130)(127 129)(131 135)(132 134)(136 140)(137 139)(141 145)(142 144)(146 150)(147 149)(151 155)(152 154)(156 160)(157 159)
(1 61 21 41)(2 62 22 42)(3 63 23 43)(4 64 24 44)(5 65 25 45)(6 66 26 46)(7 67 27 47)(8 68 28 48)(9 69 29 49)(10 70 30 50)(11 71 31 51)(12 72 32 52)(13 73 33 53)(14 74 34 54)(15 75 35 55)(16 76 36 56)(17 77 37 57)(18 78 38 58)(19 79 39 59)(20 80 40 60)(81 151 101 131)(82 152 102 132)(83 153 103 133)(84 154 104 134)(85 155 105 135)(86 156 106 136)(87 157 107 137)(88 158 108 138)(89 159 109 139)(90 160 110 140)(91 146 111 126)(92 147 112 127)(93 148 113 128)(94 149 114 129)(95 150 115 130)(96 141 116 121)(97 142 117 122)(98 143 118 123)(99 144 119 124)(100 145 120 125)

G:=sub<Sym(160)| (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35)(41,51,46,56)(42,52,47,57)(43,53,48,58)(44,54,49,59)(45,55,50,60)(61,71,66,76)(62,72,67,77)(63,73,68,78)(64,74,69,79)(65,75,70,80)(81,91,86,96)(82,92,87,97)(83,93,88,98)(84,94,89,99)(85,95,90,100)(101,111,106,116)(102,112,107,117)(103,113,108,118)(104,114,109,119)(105,115,110,120)(121,136,126,131)(122,137,127,132)(123,138,128,133)(124,139,129,134)(125,140,130,135)(141,156,146,151)(142,157,147,152)(143,158,148,153)(144,159,149,154)(145,160,150,155), (1,86,6,81)(2,87,7,82)(3,88,8,83)(4,89,9,84)(5,90,10,85)(11,96,16,91)(12,97,17,92)(13,98,18,93)(14,99,19,94)(15,100,20,95)(21,106,26,101)(22,107,27,102)(23,108,28,103)(24,109,29,104)(25,110,30,105)(31,116,36,111)(32,117,37,112)(33,118,38,113)(34,119,39,114)(35,120,40,115)(41,126,46,121)(42,127,47,122)(43,128,48,123)(44,129,49,124)(45,130,50,125)(51,136,56,131)(52,137,57,132)(53,138,58,133)(54,139,59,134)(55,140,60,135)(61,146,66,141)(62,147,67,142)(63,148,68,143)(64,149,69,144)(65,150,70,145)(71,156,76,151)(72,157,77,152)(73,158,78,153)(74,159,79,154)(75,160,80,155), (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)(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16)(21,30)(22,29)(23,28)(24,27)(25,26)(31,40)(32,39)(33,38)(34,37)(35,36)(41,45)(42,44)(46,50)(47,49)(51,55)(52,54)(56,60)(57,59)(61,65)(62,64)(66,70)(67,69)(71,75)(72,74)(76,80)(77,79)(81,90)(82,89)(83,88)(84,87)(85,86)(91,100)(92,99)(93,98)(94,97)(95,96)(101,110)(102,109)(103,108)(104,107)(105,106)(111,120)(112,119)(113,118)(114,117)(115,116)(121,125)(122,124)(126,130)(127,129)(131,135)(132,134)(136,140)(137,139)(141,145)(142,144)(146,150)(147,149)(151,155)(152,154)(156,160)(157,159), (1,61,21,41)(2,62,22,42)(3,63,23,43)(4,64,24,44)(5,65,25,45)(6,66,26,46)(7,67,27,47)(8,68,28,48)(9,69,29,49)(10,70,30,50)(11,71,31,51)(12,72,32,52)(13,73,33,53)(14,74,34,54)(15,75,35,55)(16,76,36,56)(17,77,37,57)(18,78,38,58)(19,79,39,59)(20,80,40,60)(81,151,101,131)(82,152,102,132)(83,153,103,133)(84,154,104,134)(85,155,105,135)(86,156,106,136)(87,157,107,137)(88,158,108,138)(89,159,109,139)(90,160,110,140)(91,146,111,126)(92,147,112,127)(93,148,113,128)(94,149,114,129)(95,150,115,130)(96,141,116,121)(97,142,117,122)(98,143,118,123)(99,144,119,124)(100,145,120,125)>;

G:=Group( (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35)(41,51,46,56)(42,52,47,57)(43,53,48,58)(44,54,49,59)(45,55,50,60)(61,71,66,76)(62,72,67,77)(63,73,68,78)(64,74,69,79)(65,75,70,80)(81,91,86,96)(82,92,87,97)(83,93,88,98)(84,94,89,99)(85,95,90,100)(101,111,106,116)(102,112,107,117)(103,113,108,118)(104,114,109,119)(105,115,110,120)(121,136,126,131)(122,137,127,132)(123,138,128,133)(124,139,129,134)(125,140,130,135)(141,156,146,151)(142,157,147,152)(143,158,148,153)(144,159,149,154)(145,160,150,155), (1,86,6,81)(2,87,7,82)(3,88,8,83)(4,89,9,84)(5,90,10,85)(11,96,16,91)(12,97,17,92)(13,98,18,93)(14,99,19,94)(15,100,20,95)(21,106,26,101)(22,107,27,102)(23,108,28,103)(24,109,29,104)(25,110,30,105)(31,116,36,111)(32,117,37,112)(33,118,38,113)(34,119,39,114)(35,120,40,115)(41,126,46,121)(42,127,47,122)(43,128,48,123)(44,129,49,124)(45,130,50,125)(51,136,56,131)(52,137,57,132)(53,138,58,133)(54,139,59,134)(55,140,60,135)(61,146,66,141)(62,147,67,142)(63,148,68,143)(64,149,69,144)(65,150,70,145)(71,156,76,151)(72,157,77,152)(73,158,78,153)(74,159,79,154)(75,160,80,155), (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)(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16)(21,30)(22,29)(23,28)(24,27)(25,26)(31,40)(32,39)(33,38)(34,37)(35,36)(41,45)(42,44)(46,50)(47,49)(51,55)(52,54)(56,60)(57,59)(61,65)(62,64)(66,70)(67,69)(71,75)(72,74)(76,80)(77,79)(81,90)(82,89)(83,88)(84,87)(85,86)(91,100)(92,99)(93,98)(94,97)(95,96)(101,110)(102,109)(103,108)(104,107)(105,106)(111,120)(112,119)(113,118)(114,117)(115,116)(121,125)(122,124)(126,130)(127,129)(131,135)(132,134)(136,140)(137,139)(141,145)(142,144)(146,150)(147,149)(151,155)(152,154)(156,160)(157,159), (1,61,21,41)(2,62,22,42)(3,63,23,43)(4,64,24,44)(5,65,25,45)(6,66,26,46)(7,67,27,47)(8,68,28,48)(9,69,29,49)(10,70,30,50)(11,71,31,51)(12,72,32,52)(13,73,33,53)(14,74,34,54)(15,75,35,55)(16,76,36,56)(17,77,37,57)(18,78,38,58)(19,79,39,59)(20,80,40,60)(81,151,101,131)(82,152,102,132)(83,153,103,133)(84,154,104,134)(85,155,105,135)(86,156,106,136)(87,157,107,137)(88,158,108,138)(89,159,109,139)(90,160,110,140)(91,146,111,126)(92,147,112,127)(93,148,113,128)(94,149,114,129)(95,150,115,130)(96,141,116,121)(97,142,117,122)(98,143,118,123)(99,144,119,124)(100,145,120,125) );

G=PermutationGroup([(1,16,6,11),(2,17,7,12),(3,18,8,13),(4,19,9,14),(5,20,10,15),(21,36,26,31),(22,37,27,32),(23,38,28,33),(24,39,29,34),(25,40,30,35),(41,51,46,56),(42,52,47,57),(43,53,48,58),(44,54,49,59),(45,55,50,60),(61,71,66,76),(62,72,67,77),(63,73,68,78),(64,74,69,79),(65,75,70,80),(81,91,86,96),(82,92,87,97),(83,93,88,98),(84,94,89,99),(85,95,90,100),(101,111,106,116),(102,112,107,117),(103,113,108,118),(104,114,109,119),(105,115,110,120),(121,136,126,131),(122,137,127,132),(123,138,128,133),(124,139,129,134),(125,140,130,135),(141,156,146,151),(142,157,147,152),(143,158,148,153),(144,159,149,154),(145,160,150,155)], [(1,86,6,81),(2,87,7,82),(3,88,8,83),(4,89,9,84),(5,90,10,85),(11,96,16,91),(12,97,17,92),(13,98,18,93),(14,99,19,94),(15,100,20,95),(21,106,26,101),(22,107,27,102),(23,108,28,103),(24,109,29,104),(25,110,30,105),(31,116,36,111),(32,117,37,112),(33,118,38,113),(34,119,39,114),(35,120,40,115),(41,126,46,121),(42,127,47,122),(43,128,48,123),(44,129,49,124),(45,130,50,125),(51,136,56,131),(52,137,57,132),(53,138,58,133),(54,139,59,134),(55,140,60,135),(61,146,66,141),(62,147,67,142),(63,148,68,143),(64,149,69,144),(65,150,70,145),(71,156,76,151),(72,157,77,152),(73,158,78,153),(74,159,79,154),(75,160,80,155)], [(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),(2,9),(3,8),(4,7),(5,6),(11,20),(12,19),(13,18),(14,17),(15,16),(21,30),(22,29),(23,28),(24,27),(25,26),(31,40),(32,39),(33,38),(34,37),(35,36),(41,45),(42,44),(46,50),(47,49),(51,55),(52,54),(56,60),(57,59),(61,65),(62,64),(66,70),(67,69),(71,75),(72,74),(76,80),(77,79),(81,90),(82,89),(83,88),(84,87),(85,86),(91,100),(92,99),(93,98),(94,97),(95,96),(101,110),(102,109),(103,108),(104,107),(105,106),(111,120),(112,119),(113,118),(114,117),(115,116),(121,125),(122,124),(126,130),(127,129),(131,135),(132,134),(136,140),(137,139),(141,145),(142,144),(146,150),(147,149),(151,155),(152,154),(156,160),(157,159)], [(1,61,21,41),(2,62,22,42),(3,63,23,43),(4,64,24,44),(5,65,25,45),(6,66,26,46),(7,67,27,47),(8,68,28,48),(9,69,29,49),(10,70,30,50),(11,71,31,51),(12,72,32,52),(13,73,33,53),(14,74,34,54),(15,75,35,55),(16,76,36,56),(17,77,37,57),(18,78,38,58),(19,79,39,59),(20,80,40,60),(81,151,101,131),(82,152,102,132),(83,153,103,133),(84,154,104,134),(85,155,105,135),(86,156,106,136),(87,157,107,137),(88,158,108,138),(89,159,109,139),(90,160,110,140),(91,146,111,126),(92,147,112,127),(93,148,113,128),(94,149,114,129),(95,150,115,130),(96,141,116,121),(97,142,117,122),(98,143,118,123),(99,144,119,124),(100,145,120,125)])

50 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12222244444444444455888810···102020202020···2040···40
size11111010224444101020202020224420202···244448···84···4

50 irreducible representations

dim1111111112222222244444
type+++++++++++++++-++-
imageC1C2C2C2C2C2C2C2C4D4D4D4D5D10D10D10C4×D5C8.C22D4×D5D4×D5SD16⋊D5Q16⋊D5
kernel(Q8×D5)⋊C4C10.Q16C20.44D4Q8⋊Dic5C5×Q8⋊C4C4⋊C47D5C2×C8⋊D5C2×Q8×D5Q8×D5C4×D5C2×Dic5C22×D5Q8⋊C4C4⋊C4C2×C8C2×Q8Q8C10C4C22C2C2
# reps1111111182112222822244

Matrix representation of (Q8×D5)⋊C4 in GL6(𝔽41)

100000
010000
000100
0040000
0000040
000010
,
4000000
0400000
00342700
0027700
00001434
00003427
,
7400000
8400000
001000
000100
000010
000001
,
060000
700000
0040000
0004000
000010
000001
,
3200000
0320000
000010
000001
001000
000100

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,34,27,0,0,0,0,27,7,0,0,0,0,0,0,14,34,0,0,0,0,34,27],[7,8,0,0,0,0,40,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,7,0,0,0,0,6,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

(Q8×D5)⋊C4 in GAP, Magma, Sage, TeX

(Q_8\times D_5)\rtimes C_4
% in TeX

G:=Group("(Q8xD5):C4");
// GroupNames label

G:=SmallGroup(320,429);
// by ID

G=gap.SmallGroup(320,429);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,58,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=c^5=d^2=e^4=1,b^2=a^2,b*a*b^-1=e*a*e^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,e*b*e^-1=a*b,d*c*d=c^-1,c*e=e*c,e*d*e^-1=a^2*d>;
// generators/relations

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