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G = D4.D55C4order 320 = 26·5

1st semidirect product of D4.D5 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.D55C4, D4.1(C4×D5), C408C415C2, C10.61(C4×D4), C4⋊C4.129D10, D4⋊C4.7D5, (C2×C8).165D10, C10.D82C2, (D4×Dic5).1C2, C22.66(D4×D5), Dic1010(C2×C4), Dic53Q81C2, (C2×D4).124D10, C2.1(D8⋊D5), C53(SD16⋊C4), C20.37(C22×C4), C20.143(C4○D4), C4.44(D42D5), C20.44D414C2, C10.24(C8⋊C22), (C2×C40).176C22, (C2×C20).197C23, (C2×Dic5).189D4, C2.1(SD16⋊D5), (D4×C10).18C22, C4⋊Dic5.57C22, (C4×Dic5).9C22, C10.24(C8.C22), C2.15(Dic54D4), (C2×Dic10).53C22, C4.2(C2×C4×D5), C52C81(C2×C4), (C5×D4).15(C2×C4), (C5×C4⋊C4).2C22, (C2×D4.D5).1C2, (C5×D4⋊C4).7C2, (C2×C10).210(C2×D4), (C2×C52C8).5C22, (C2×C4).304(C22×D5), SmallGroup(320,384)

Series: Derived Chief Lower central Upper central

C1C20 — D4.D55C4
C1C5C10C20C2×C20C4×Dic5D4×Dic5 — D4.D55C4
C5C10C20 — D4.D55C4
C1C22C2×C4D4⋊C4

Generators and relations for D4.D55C4
 G = < a,b,c,d,e | a4=b2=c5=e4=1, d2=a2, bab=dad-1=eae-1=a-1, ac=ca, bc=cb, dbd-1=ebe-1=ab, dcd-1=c-1, ce=ec, ede-1=a2d >

Subgroups: 422 in 120 conjugacy classes, 49 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×3], C2×C4, C2×C4 [×7], D4 [×2], D4, Q8 [×3], C23, C10 [×3], C10 [×2], C42 [×2], C22⋊C4, C4⋊C4, C4⋊C4 [×2], C2×C8, C2×C8, SD16 [×4], C22×C4, C2×D4, C2×Q8, Dic5 [×5], C20 [×2], C20, C2×C10, C2×C10 [×4], C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C4×D4, C4×Q8, C2×SD16, C52C8 [×2], C40, Dic10 [×2], Dic10, C2×Dic5 [×2], C2×Dic5 [×4], C2×C20, C2×C20, C5×D4 [×2], C5×D4, C22×C10, SD16⋊C4, C2×C52C8, C4×Dic5, C4×Dic5, C10.D4, C4⋊Dic5, D4.D5 [×4], C23.D5, C5×C4⋊C4, C2×C40, C2×Dic10, C22×Dic5, D4×C10, C10.D8, C408C4, C20.44D4, C5×D4⋊C4, Dic53Q8, C2×D4.D5, D4×Dic5, D4.D55C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C8⋊C22, C8.C22, C4×D5 [×2], C22×D5, SD16⋊C4, C2×C4×D5, D4×D5, D42D5, Dic54D4, D8⋊D5, SD16⋊D5, D4.D55C4

Smallest permutation representation of D4.D55C4
On 160 points
Generators in S160
(1 21 6 26)(2 22 7 27)(3 23 8 28)(4 24 9 29)(5 25 10 30)(11 31 16 36)(12 32 17 37)(13 33 18 38)(14 34 19 39)(15 35 20 40)(41 66 46 61)(42 67 47 62)(43 68 48 63)(44 69 49 64)(45 70 50 65)(51 76 56 71)(52 77 57 72)(53 78 58 73)(54 79 59 74)(55 80 60 75)(81 101 86 106)(82 102 87 107)(83 103 88 108)(84 104 89 109)(85 105 90 110)(91 111 96 116)(92 112 97 117)(93 113 98 118)(94 114 99 119)(95 115 100 120)(121 146 126 141)(122 147 127 142)(123 148 128 143)(124 149 129 144)(125 150 130 145)(131 156 136 151)(132 157 137 152)(133 158 138 153)(134 159 139 154)(135 160 140 155)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 21)(7 22)(8 23)(9 24)(10 25)(11 36)(12 37)(13 38)(14 39)(15 40)(16 31)(17 32)(18 33)(19 34)(20 35)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(81 106)(82 107)(83 108)(84 109)(85 110)(86 101)(87 102)(88 103)(89 104)(90 105)(91 116)(92 117)(93 118)(94 119)(95 120)(96 111)(97 112)(98 113)(99 114)(100 115)(121 126)(122 127)(123 128)(124 129)(125 130)(131 136)(132 137)(133 138)(134 139)(135 140)
(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)
(1 138 6 133)(2 137 7 132)(3 136 8 131)(4 140 9 135)(5 139 10 134)(11 128 16 123)(12 127 17 122)(13 126 18 121)(14 130 19 125)(15 129 20 124)(21 158 26 153)(22 157 27 152)(23 156 28 151)(24 160 29 155)(25 159 30 154)(31 148 36 143)(32 147 37 142)(33 146 38 141)(34 150 39 145)(35 149 40 144)(41 88 46 83)(42 87 47 82)(43 86 48 81)(44 90 49 85)(45 89 50 84)(51 98 56 93)(52 97 57 92)(53 96 58 91)(54 100 59 95)(55 99 60 94)(61 108 66 103)(62 107 67 102)(63 106 68 101)(64 110 69 105)(65 109 70 104)(71 118 76 113)(72 117 77 112)(73 116 78 111)(74 120 79 115)(75 119 80 114)
(1 51 11 41)(2 52 12 42)(3 53 13 43)(4 54 14 44)(5 55 15 45)(6 56 16 46)(7 57 17 47)(8 58 18 48)(9 59 19 49)(10 60 20 50)(21 71 31 61)(22 72 32 62)(23 73 33 63)(24 74 34 64)(25 75 35 65)(26 76 36 66)(27 77 37 67)(28 78 38 68)(29 79 39 69)(30 80 40 70)(81 136 91 126)(82 137 92 127)(83 138 93 128)(84 139 94 129)(85 140 95 130)(86 131 96 121)(87 132 97 122)(88 133 98 123)(89 134 99 124)(90 135 100 125)(101 156 111 146)(102 157 112 147)(103 158 113 148)(104 159 114 149)(105 160 115 150)(106 151 116 141)(107 152 117 142)(108 153 118 143)(109 154 119 144)(110 155 120 145)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L5A5B8A8B8C8D10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222244444444444455888810···1010101010202020202020202040···40
size11114422441010101020202020224420202···28888444488884···4

50 irreducible representations

dim1111111112222222444444
type++++++++++++++--+-
imageC1C2C2C2C2C2C2C2C4D4D5C4○D4D10D10D10C4×D5C8⋊C22C8.C22D42D5D4×D5D8⋊D5SD16⋊D5
kernelD4.D55C4C10.D8C408C4C20.44D4C5×D4⋊C4Dic53Q8C2×D4.D5D4×Dic5D4.D5C2×Dic5D4⋊C4C20C4⋊C4C2×C8C2×D4D4C10C10C4C22C2C2
# reps1111111182222228112244

Matrix representation of D4.D55C4 in GL6(𝔽41)

100000
010000
0040200
0040100
0000139
0000140
,
100000
010000
0040200
000100
0000400
0000401
,
6400000
100000
001000
000100
000010
000001
,
36380000
850000
00932260
002532026
00260329
00026169
,
3200000
0320000
000010
000001
0040000
0004000

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,40,0,0,0,0,2,1,0,0,0,0,0,0,1,1,0,0,0,0,39,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,2,1,0,0,0,0,0,0,40,40,0,0,0,0,0,1],[6,1,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[36,8,0,0,0,0,38,5,0,0,0,0,0,0,9,25,26,0,0,0,32,32,0,26,0,0,26,0,32,16,0,0,0,26,9,9],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0] >;

D4.D55C4 in GAP, Magma, Sage, TeX

D_4.D_5\rtimes_5C_4
% in TeX

G:=Group("D4.D5:5C4");
// GroupNames label

G:=SmallGroup(320,384);
// by ID

G=gap.SmallGroup(320,384);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,758,135,100,570,297,136,12550]);
// Polycyclic

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

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