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G = C52C824D4order 320 = 26·5

6th semidirect product of C52C8 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C52C824D4, C55(C88D4), C22⋊Q83D5, C4⋊C4.67D10, (C2×C10)⋊6SD16, C4.174(D4×D5), (C2×C20).267D4, C20.154(C2×D4), C221(Q8⋊D5), (C2×Q8).29D10, D206C439C2, C207D4.12C2, Q8⋊Dic515C2, C20.Q839C2, C10.73(C2×SD16), (C22×C10).94D4, C20.190(C4○D4), C10.101(C4○D8), C4.63(D42D5), C10.97(C4⋊D4), (C2×C20).367C23, (C22×C4).342D10, C23.41(C5⋊D4), (Q8×C10).47C22, (C2×D20).103C22, C4⋊Dic5.146C22, C2.18(Dic5⋊D4), C2.20(D4.8D10), (C22×C20).171C22, (C2×Q8⋊D5)⋊10C2, C2.10(C2×Q8⋊D5), (C5×C22⋊Q8)⋊3C2, (C22×C52C8)⋊5C2, (C2×C10).498(C2×D4), (C2×C4).107(C5⋊D4), (C5×C4⋊C4).114C22, (C2×C4).467(C22×D5), C22.173(C2×C5⋊D4), (C2×C52C8).258C22, SmallGroup(320,675)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C52C824D4
C1C5C10C20C2×C20C2×D20C207D4 — C52C824D4
C5C10C2×C20 — C52C824D4
C1C22C22×C4C22⋊Q8

Generators and relations for C52C824D4
 G = < a,b,c,d | a5=b8=c4=d2=1, bab-1=cac-1=a-1, ad=da, cbc-1=b3, bd=db, dcd=c-1 >

Subgroups: 502 in 124 conjugacy classes, 45 normal (39 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×5], C5, C8 [×3], C2×C4 [×2], C2×C4 [×5], D4 [×4], Q8 [×2], C23, C23, D5, C10 [×3], C10 [×2], C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C2×C8 [×4], SD16 [×2], C22×C4, C2×D4 [×2], C2×Q8, Dic5, C20 [×2], C20 [×3], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×2], D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, C2×SD16, C52C8 [×2], C52C8, D20 [×2], C2×Dic5, C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×4], C5×Q8 [×2], C22×D5, C22×C10, C88D4, C2×C52C8 [×2], C2×C52C8 [×2], C4⋊Dic5, D10⋊C4, Q8⋊D5 [×2], C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×D20, C2×C5⋊D4, C22×C20, Q8×C10, C20.Q8, D206C4, Q8⋊Dic5, C22×C52C8, C207D4, C2×Q8⋊D5, C5×C22⋊Q8, C52C824D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, SD16 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×SD16, C4○D8, C5⋊D4 [×2], C22×D5, C88D4, Q8⋊D5 [×2], D4×D5, D42D5, C2×C5⋊D4, Dic5⋊D4, C2×Q8⋊D5, D4.8D10, C52C824D4

Smallest permutation representation of C52C824D4
On 160 points
Generators in S160
(1 99 125 46 145)(2 146 47 126 100)(3 101 127 48 147)(4 148 41 128 102)(5 103 121 42 149)(6 150 43 122 104)(7 97 123 44 151)(8 152 45 124 98)(9 34 63 117 96)(10 89 118 64 35)(11 36 57 119 90)(12 91 120 58 37)(13 38 59 113 92)(14 93 114 60 39)(15 40 61 115 94)(16 95 116 62 33)(17 137 106 56 65)(18 66 49 107 138)(19 139 108 50 67)(20 68 51 109 140)(21 141 110 52 69)(22 70 53 111 142)(23 143 112 54 71)(24 72 55 105 144)(25 73 156 129 87)(26 88 130 157 74)(27 75 158 131 81)(28 82 132 159 76)(29 77 160 133 83)(30 84 134 153 78)(31 79 154 135 85)(32 86 136 155 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 70 37 131)(2 65 38 134)(3 68 39 129)(4 71 40 132)(5 66 33 135)(6 69 34 130)(7 72 35 133)(8 67 36 136)(9 157 104 21)(10 160 97 24)(11 155 98 19)(12 158 99 22)(13 153 100 17)(14 156 101 20)(15 159 102 23)(16 154 103 18)(25 48 109 114)(26 43 110 117)(27 46 111 120)(28 41 112 115)(29 44 105 118)(30 47 106 113)(31 42 107 116)(32 45 108 119)(49 62 85 149)(50 57 86 152)(51 60 87 147)(52 63 88 150)(53 58 81 145)(54 61 82 148)(55 64 83 151)(56 59 84 146)(73 127 140 93)(74 122 141 96)(75 125 142 91)(76 128 143 94)(77 123 144 89)(78 126 137 92)(79 121 138 95)(80 124 139 90)
(1 70)(2 71)(3 72)(4 65)(5 66)(6 67)(7 68)(8 69)(9 86)(10 87)(11 88)(12 81)(13 82)(14 83)(15 84)(16 85)(17 148)(18 149)(19 150)(20 151)(21 152)(22 145)(23 146)(24 147)(25 89)(26 90)(27 91)(28 92)(29 93)(30 94)(31 95)(32 96)(33 135)(34 136)(35 129)(36 130)(37 131)(38 132)(39 133)(40 134)(41 137)(42 138)(43 139)(44 140)(45 141)(46 142)(47 143)(48 144)(49 103)(50 104)(51 97)(52 98)(53 99)(54 100)(55 101)(56 102)(57 157)(58 158)(59 159)(60 160)(61 153)(62 154)(63 155)(64 156)(73 118)(74 119)(75 120)(76 113)(77 114)(78 115)(79 116)(80 117)(105 127)(106 128)(107 121)(108 122)(109 123)(110 124)(111 125)(112 126)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G5A5B8A···8H10A···10F10G10H10I10J20A···20H20I···20P
order12222224444444558···810···101010101020···2020···20
size11112240222288402210···102···244444···48···8

50 irreducible representations

dim111111112222222222224444
type++++++++++++++++-+
imageC1C2C2C2C2C2C2C2D4D4D4D5C4○D4SD16D10D10D10C4○D8C5⋊D4C5⋊D4D4×D5D42D5Q8⋊D5D4.8D10
kernelC52C824D4C20.Q8D206C4Q8⋊Dic5C22×C52C8C207D4C2×Q8⋊D5C5×C22⋊Q8C52C8C2×C20C22×C10C22⋊Q8C20C2×C10C4⋊C4C22×C4C2×Q8C10C2×C4C23C4C4C22C2
# reps111111112112242224442244

Matrix representation of C52C824D4 in GL6(𝔽41)

100000
010000
001000
000100
0000040
000016
,
26150000
26260000
0030200
00211100
00002121
00001820
,
0320000
3200000
00172300
0072400
0000635
00004035
,
090000
3200000
00241800
00251700
000010
000001

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,6],[26,26,0,0,0,0,15,26,0,0,0,0,0,0,30,21,0,0,0,0,2,11,0,0,0,0,0,0,21,18,0,0,0,0,21,20],[0,32,0,0,0,0,32,0,0,0,0,0,0,0,17,7,0,0,0,0,23,24,0,0,0,0,0,0,6,40,0,0,0,0,35,35],[0,32,0,0,0,0,9,0,0,0,0,0,0,0,24,25,0,0,0,0,18,17,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C52C824D4 in GAP, Magma, Sage, TeX

C_5\rtimes_2C_8\rtimes_{24}D_4
% in TeX

G:=Group("C5:2C8:24D4");
// GroupNames label

G:=SmallGroup(320,675);
// by ID

G=gap.SmallGroup(320,675);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,219,184,1123,297,136,12550]);
// 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^3,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽