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G = C52C823D4order 320 = 26·5

5th semidirect product of C52C8 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C52C823D4, C54(C88D4), C4⋊C4.61D10, (C2×C10)⋊4SD16, C4⋊D4.6D5, C4.172(D4×D5), (C2×D4).41D10, C20.150(C2×D4), (C2×C20).264D4, C10.98(C4○D8), C10.Q1635C2, D4⋊Dic517C2, C221(D4.D5), C20.Q836C2, C10.56(C2×SD16), (C22×C10).87D4, C20.185(C4○D4), C20.48D424C2, C4.61(D42D5), C10.95(C4⋊D4), (C2×C20).360C23, (D4×C10).57C22, (C22×C4).341D10, C23.40(C5⋊D4), C4⋊Dic5.144C22, C2.16(Dic5⋊D4), C2.17(D4.8D10), (C22×C20).164C22, (C2×Dic10).107C22, (C2×D4.D5)⋊10C2, (C22×C52C8)⋊4C2, (C5×C4⋊D4).5C2, C2.10(C2×D4.D5), (C2×C10).491(C2×D4), (C2×C4).106(C5⋊D4), (C5×C4⋊C4).108C22, (C2×C4).460(C22×D5), C22.166(C2×C5⋊D4), (C2×C52C8).257C22, SmallGroup(320,668)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C52C823D4
C1C5C10C20C2×C20C2×Dic10C20.48D4 — C52C823D4
C5C10C2×C20 — C52C823D4
C1C22C22×C4C4⋊D4

Generators and relations for C52C823D4
 G = < a,b,c,d | a5=b8=c4=d2=1, bab-1=a-1, ac=ca, ad=da, cbc-1=b3, bd=db, dcd=c-1 >

Subgroups: 406 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, C10 [×3], C10 [×3], C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C2×C8 [×4], SD16 [×2], C22×C4, C2×D4, C2×D4, C2×Q8, Dic5 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×5], D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, C2×SD16, C52C8 [×2], C52C8, Dic10 [×2], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×3], C5×D4 [×4], C22×C10, C22×C10, C88D4, C2×C52C8 [×2], C2×C52C8 [×2], C10.D4, C4⋊Dic5, D4.D5 [×2], C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C22×C20, D4×C10, D4×C10, C20.Q8, C10.Q16, D4⋊Dic5, C22×C52C8, C20.48D4, C2×D4.D5, C5×C4⋊D4, C52C823D4
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, D4.D5 [×2], D4×D5, D42D5, C2×C5⋊D4, C2×D4.D5, Dic5⋊D4, D4.8D10, C52C823D4

Smallest permutation representation of C52C823D4
On 160 points
Generators in S160
(1 74 37 47 155)(2 156 48 38 75)(3 76 39 41 157)(4 158 42 40 77)(5 78 33 43 159)(6 160 44 34 79)(7 80 35 45 153)(8 154 46 36 73)(9 126 103 131 142)(10 143 132 104 127)(11 128 97 133 144)(12 137 134 98 121)(13 122 99 135 138)(14 139 136 100 123)(15 124 101 129 140)(16 141 130 102 125)(17 62 49 68 94)(18 95 69 50 63)(19 64 51 70 96)(20 89 71 52 57)(21 58 53 72 90)(22 91 65 54 59)(23 60 55 66 92)(24 93 67 56 61)(25 84 149 106 113)(26 114 107 150 85)(27 86 151 108 115)(28 116 109 152 87)(29 88 145 110 117)(30 118 111 146 81)(31 82 147 112 119)(32 120 105 148 83)
(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)
(1 113 121 50)(2 116 122 53)(3 119 123 56)(4 114 124 51)(5 117 125 54)(6 120 126 49)(7 115 127 52)(8 118 128 55)(9 62 79 32)(10 57 80 27)(11 60 73 30)(12 63 74 25)(13 58 75 28)(14 61 76 31)(15 64 77 26)(16 59 78 29)(17 34 83 142)(18 37 84 137)(19 40 85 140)(20 35 86 143)(21 38 87 138)(22 33 88 141)(23 36 81 144)(24 39 82 139)(41 147 136 93)(42 150 129 96)(43 145 130 91)(44 148 131 94)(45 151 132 89)(46 146 133 92)(47 149 134 95)(48 152 135 90)(65 159 110 102)(66 154 111 97)(67 157 112 100)(68 160 105 103)(69 155 106 98)(70 158 107 101)(71 153 108 104)(72 156 109 99)
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 49)(9 30)(10 31)(11 32)(12 25)(13 26)(14 27)(15 28)(16 29)(17 36)(18 37)(19 38)(20 39)(21 40)(22 33)(23 34)(24 35)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)(57 76)(58 77)(59 78)(60 79)(61 80)(62 73)(63 74)(64 75)(65 159)(66 160)(67 153)(68 154)(69 155)(70 156)(71 157)(72 158)(81 142)(82 143)(83 144)(84 137)(85 138)(86 139)(87 140)(88 141)(97 105)(98 106)(99 107)(100 108)(101 109)(102 110)(103 111)(104 112)(113 121)(114 122)(115 123)(116 124)(117 125)(118 126)(119 127)(120 128)(129 152)(130 145)(131 146)(132 147)(133 148)(134 149)(135 150)(136 151)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G5A5B8A···8H10A···10F10G10H10I10J10K10L10M10N20A···20H20I20J20K20L
order12222224444444558···810···10101010101010101020···2020202020
size11112282222840402210···102···2444488884···48888

50 irreducible representations

dim111111112222222222224444
type++++++++++++++++--
imageC1C2C2C2C2C2C2C2D4D4D4D5C4○D4SD16D10D10D10C4○D8C5⋊D4C5⋊D4D4×D5D42D5D4.D5D4.8D10
kernelC52C823D4C20.Q8C10.Q16D4⋊Dic5C22×C52C8C20.48D4C2×D4.D5C5×C4⋊D4C52C8C2×C20C22×C10C4⋊D4C20C2×C10C4⋊C4C22×C4C2×D4C10C2×C4C23C4C4C22C2
# reps111111112112242224442244

Matrix representation of C52C823D4 in GL6(𝔽41)

1600000
14180000
001000
000100
000010
000001
,
12310000
35290000
0001100
00151100
00003023
00002511
,
4000000
1410000
00321800
000900
00001739
00002224
,
4000000
0400000
00321800
0032900
00001739
00002124

G:=sub<GL(6,GF(41))| [16,14,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,35,0,0,0,0,31,29,0,0,0,0,0,0,0,15,0,0,0,0,11,11,0,0,0,0,0,0,30,25,0,0,0,0,23,11],[40,14,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,18,9,0,0,0,0,0,0,17,22,0,0,0,0,39,24],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,32,0,0,0,0,18,9,0,0,0,0,0,0,17,21,0,0,0,0,39,24] >;

C52C823D4 in GAP, Magma, Sage, TeX

C_5\rtimes_2C_8\rtimes_{23}D_4
% in TeX

G:=Group("C5:2C8:23D4");
// GroupNames label

G:=SmallGroup(320,668);
// by ID

G=gap.SmallGroup(320,668);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,254,219,1123,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^8=c^4=d^2=1,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=b^3,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
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