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G = C52C826D4order 320 = 26·5

8th semidirect product of C52C8 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C52C826D4, C57(C89D4), C22⋊C815D5, C408C414C2, C4.199(D4×D5), C10.59(C4×D4), C20.358(C2×D4), (C2×C8).164D10, (C2×C10)⋊6M4(2), D101C821C2, C23.24(C4×D5), C10.49(C8○D4), C221(C8⋊D5), C20.8Q821C2, C23.D5.14C4, D10⋊C4.20C4, C20.300(C4○D4), (C2×C40).174C22, (C2×C20).825C23, C10.D4.20C4, (C22×C4).305D10, C10.40(C2×M4(2)), C4.126(D42D5), C2.13(Dic54D4), C2.11(D20.2C4), (C22×C20).339C22, (C4×Dic5).203C22, (C2×C4).64(C4×D5), (C2×C8⋊D5)⋊14C2, (C5×C22⋊C8)⋊19C2, (C4×C5⋊D4).14C2, C2.10(C2×C8⋊D5), (C2×C5⋊D4).16C4, C22.107(C2×C4×D5), (C2×C20).328(C2×C4), (C22×C52C8)⋊17C2, (C2×C4×D5).231C22, (C2×Dic5).20(C2×C4), (C22×D5).19(C2×C4), (C2×C4).767(C22×D5), (C22×C10).111(C2×C4), (C2×C10).181(C22×C4), (C2×C52C8).310C22, SmallGroup(320,357)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C52C826D4
C1C5C10C20C2×C20C2×C4×D5C4×C5⋊D4 — C52C826D4
C5C2×C10 — C52C826D4
C1C2×C4C22⋊C8

Generators and relations for C52C826D4
 G = < a,b,c,d | a5=b8=c4=d2=1, bab-1=cac-1=a-1, ad=da, cbc-1=b5, bd=db, dcd=c-1 >

Subgroups: 398 in 124 conjugacy classes, 53 normal (47 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×5], C5, C8 [×5], C2×C4 [×2], C2×C4 [×7], D4 [×2], C23, C23, D5, C10 [×3], C10 [×2], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×4], M4(2) [×2], C22×C4, C22×C4, C2×D4, Dic5 [×3], C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×2], C8⋊C4, C22⋊C8, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C2×M4(2), C52C8 [×2], C52C8, C40 [×2], C4×D5 [×2], C2×Dic5 [×3], C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×2], C22×D5, C22×C10, C89D4, C8⋊D5 [×2], C2×C52C8 [×2], C2×C52C8 [×2], C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C40 [×2], C2×C4×D5, C2×C5⋊D4, C22×C20, C20.8Q8, C408C4, D101C8, C5×C22⋊C8, C2×C8⋊D5, C22×C52C8, C4×C5⋊D4, C52C826D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, M4(2) [×2], C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C2×M4(2), C8○D4, C4×D5 [×2], C22×D5, C89D4, C8⋊D5 [×2], C2×C4×D5, D4×D5, D42D5, Dic54D4, C2×C8⋊D5, D20.2C4, C52C826D4

Smallest permutation representation of C52C826D4
On 160 points
Generators in S160
(1 106 33 113 147)(2 148 114 34 107)(3 108 35 115 149)(4 150 116 36 109)(5 110 37 117 151)(6 152 118 38 111)(7 112 39 119 145)(8 146 120 40 105)(9 41 63 126 94)(10 95 127 64 42)(11 43 57 128 96)(12 89 121 58 44)(13 45 59 122 90)(14 91 123 60 46)(15 47 61 124 92)(16 93 125 62 48)(17 101 132 155 65)(18 66 156 133 102)(19 103 134 157 67)(20 68 158 135 104)(21 97 136 159 69)(22 70 160 129 98)(23 99 130 153 71)(24 72 154 131 100)(25 73 142 86 49)(26 50 87 143 74)(27 75 144 88 51)(28 52 81 137 76)(29 77 138 82 53)(30 54 83 139 78)(31 79 140 84 55)(32 56 85 141 80)
(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 27 48 131)(2 32 41 136)(3 29 42 133)(4 26 43 130)(5 31 44 135)(6 28 45 132)(7 25 46 129)(8 30 47 134)(9 159 107 56)(10 156 108 53)(11 153 109 50)(12 158 110 55)(13 155 111 52)(14 160 112 49)(15 157 105 54)(16 154 106 51)(17 118 137 122)(18 115 138 127)(19 120 139 124)(20 117 140 121)(21 114 141 126)(22 119 142 123)(23 116 143 128)(24 113 144 125)(33 88 93 72)(34 85 94 69)(35 82 95 66)(36 87 96 71)(37 84 89 68)(38 81 90 65)(39 86 91 70)(40 83 92 67)(57 99 150 74)(58 104 151 79)(59 101 152 76)(60 98 145 73)(61 103 146 78)(62 100 147 75)(63 97 148 80)(64 102 149 77)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 101)(10 102)(11 103)(12 104)(13 97)(14 98)(15 99)(16 100)(17 94)(18 95)(19 96)(20 89)(21 90)(22 91)(23 92)(24 93)(33 144)(34 137)(35 138)(36 139)(37 140)(38 141)(39 142)(40 143)(41 132)(42 133)(43 134)(44 135)(45 136)(46 129)(47 130)(48 131)(49 145)(50 146)(51 147)(52 148)(53 149)(54 150)(55 151)(56 152)(57 157)(58 158)(59 159)(60 160)(61 153)(62 154)(63 155)(64 156)(65 126)(66 127)(67 128)(68 121)(69 122)(70 123)(71 124)(72 125)(73 112)(74 105)(75 106)(76 107)(77 108)(78 109)(79 110)(80 111)(81 114)(82 115)(83 116)(84 117)(85 118)(86 119)(87 120)(88 113)

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

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

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

68 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D8E···8L10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order12222224444444445588888···810···101010101020···202020202040···40
size1111222011112220202022444410···102···244442···244444···4

68 irreducible representations

dim1111111111112222222222444
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C4C4C4C4D4D5C4○D4M4(2)D10D10C8○D4C4×D5C4×D5C8⋊D5D4×D5D42D5D20.2C4
kernelC52C826D4C20.8Q8C408C4D101C8C5×C22⋊C8C2×C8⋊D5C22×C52C8C4×C5⋊D4C10.D4D10⋊C4C23.D5C2×C5⋊D4C52C8C22⋊C8C20C2×C10C2×C8C22×C4C10C2×C4C23C22C4C4C2
# reps11111111222222244244416224

Matrix representation of C52C826D4 in GL4(𝔽41) generated by

1000
0100
003540
003640
,
40000
04000
002210
00419
,
04000
1000
0061
00635
,
0100
1000
0010
0001
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,35,36,0,0,40,40],[40,0,0,0,0,40,0,0,0,0,22,4,0,0,10,19],[0,1,0,0,40,0,0,0,0,0,6,6,0,0,1,35],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;

C52C826D4 in GAP, Magma, Sage, TeX

C_5\rtimes_2C_8\rtimes_{26}D_4
% in TeX

G:=Group("C5:2C8:26D4");
// GroupNames label

G:=SmallGroup(320,357);
// by ID

G=gap.SmallGroup(320,357);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,758,219,58,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^8=c^4=d^2=1,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=b^5,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽