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G = C42.51D10order 320 = 26·5

51st non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.51D10, D4.D59C4, D4.6(C4×D5), (C4×D4).8D5, (D4×C20).9C2, C4⋊C4.246D10, C10.102(C4×D4), (C2×C20).256D4, Dic1020(C2×C4), (C4×Dic10)⋊20C2, (C2×D4).193D10, C4.39(C4○D20), C20.53(C4○D4), C10.D832C2, C10.Q1630C2, C56(SD16⋊C4), (C4×C20).89C22, C20.59(C22×C4), C42.D56C2, C10.88(C8⋊C22), (C2×C20).340C23, D4⋊Dic5.10C2, C2.3(D4.9D10), C2.4(D4.D10), (D4×C10).235C22, C4⋊Dic5.329C22, C10.108(C8.C22), (C2×Dic10).273C22, C4.24(C2×C4×D5), C52C89(C2×C4), C2.18(C4×C5⋊D4), (C5×D4).27(C2×C4), (C2×D4.D5).4C2, (C2×C10).471(C2×D4), C22.78(C2×C5⋊D4), (C2×C4).219(C5⋊D4), (C5×C4⋊C4).277C22, (C2×C52C8).95C22, (C2×C4).440(C22×D5), SmallGroup(320,645)

Series: Derived Chief Lower central Upper central

C1C20 — C42.51D10
C1C5C10C2×C10C2×C20C2×Dic10C2×D4.D5 — C42.51D10
C5C10C20 — C42.51D10
C1C22C42C4×D4

Generators and relations for C42.51D10
 G = < a,b,c,d | a4=b4=c10=1, d2=b, ab=ba, cac-1=dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=b-1c-1 >

Subgroups: 358 in 120 conjugacy classes, 51 normal (39 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×3], C2×C4 [×3], C2×C4 [×5], D4 [×2], D4, Q8 [×3], C23, C10 [×3], C10 [×2], C42, C42, C22⋊C4, C4⋊C4, C4⋊C4 [×2], C2×C8 [×2], SD16 [×4], C22×C4, C2×D4, C2×Q8, Dic5 [×3], C20 [×2], C20 [×3], C2×C10, C2×C10 [×4], C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C4×D4, C4×Q8, C2×SD16, C52C8 [×2], C52C8, Dic10 [×2], Dic10, C2×Dic5 [×2], C2×C20 [×3], C2×C20 [×3], C5×D4 [×2], C5×D4, C22×C10, SD16⋊C4, C2×C52C8 [×2], C4×Dic5, C10.D4, C4⋊Dic5, D4.D5 [×4], C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C22×C20, D4×C10, C42.D5, C10.D8, C10.Q16, D4⋊Dic5, C4×Dic10, C2×D4.D5, D4×C20, C42.51D10
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], C5⋊D4 [×2], C22×D5, SD16⋊C4, C2×C4×D5, C4○D20, C2×C5⋊D4, C4×C5⋊D4, D4.D10, D4.9D10, C42.51D10

Smallest permutation representation of C42.51D10
On 160 points
Generators in S160
(1 63 23 93)(2 69 24 99)(3 65 25 95)(4 61 21 91)(5 67 22 97)(6 66 26 96)(7 62 27 92)(8 68 28 98)(9 64 29 94)(10 70 30 100)(11 87 31 71)(12 83 32 77)(13 89 33 73)(14 85 34 79)(15 81 35 75)(16 78 38 84)(17 74 39 90)(18 80 40 86)(19 76 36 82)(20 72 37 88)(41 117 153 137)(42 106 154 121)(43 119 155 139)(44 108 156 123)(45 111 157 131)(46 110 158 125)(47 113 159 133)(48 102 160 127)(49 115 151 135)(50 104 152 129)(51 132 141 112)(52 126 142 101)(53 134 143 114)(54 128 144 103)(55 136 145 116)(56 130 146 105)(57 138 147 118)(58 122 148 107)(59 140 149 120)(60 124 150 109)
(1 80 8 75)(2 76 9 71)(3 72 10 77)(4 78 6 73)(5 74 7 79)(11 69 36 64)(12 65 37 70)(13 61 38 66)(14 67 39 62)(15 63 40 68)(16 96 33 91)(17 92 34 97)(18 98 35 93)(19 94 31 99)(20 100 32 95)(21 84 26 89)(22 90 27 85)(23 86 28 81)(24 82 29 87)(25 88 30 83)(41 46 146 141)(42 142 147 47)(43 48 148 143)(44 144 149 49)(45 50 150 145)(51 153 158 56)(52 57 159 154)(53 155 160 58)(54 59 151 156)(55 157 152 60)(101 118 113 106)(102 107 114 119)(103 120 115 108)(104 109 116 111)(105 112 117 110)(121 126 138 133)(122 134 139 127)(123 128 140 135)(124 136 131 129)(125 130 132 137)
(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)
(1 150 80 145 8 45 75 50)(2 144 76 149 9 49 71 44)(3 148 72 143 10 43 77 48)(4 142 78 147 6 47 73 42)(5 146 74 141 7 41 79 46)(11 120 69 115 36 108 64 103)(12 114 65 119 37 102 70 107)(13 118 61 113 38 106 66 101)(14 112 67 117 39 110 62 105)(15 116 63 111 40 104 68 109)(16 121 96 126 33 138 91 133)(17 125 92 130 34 132 97 137)(18 129 98 124 35 136 93 131)(19 123 94 128 31 140 99 135)(20 127 100 122 32 134 95 139)(21 52 84 57 26 159 89 154)(22 56 90 51 27 153 85 158)(23 60 86 55 28 157 81 152)(24 54 82 59 29 151 87 156)(25 58 88 53 30 155 83 160)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E4A···4F4G4H4I4J4K4L5A5B8A8B8C8D10A···10F10G···10N20A···20H20I···20X
order1222224···444444455888810···1010···1020···2020···20
size1111442···2442020202022202020202···24···42···24···4

62 irreducible representations

dim1111111112222222224444
type++++++++++++++--
imageC1C2C2C2C2C2C2C2C4D4D5C4○D4D10D10D10C5⋊D4C4×D5C4○D20C8⋊C22C8.C22D4.D10D4.9D10
kernelC42.51D10C42.D5C10.D8C10.Q16D4⋊Dic5C4×Dic10C2×D4.D5D4×C20D4.D5C2×C20C4×D4C20C42C4⋊C4C2×D4C2×C4D4C4C10C10C2C2
# reps1111111182222228881144

Matrix representation of C42.51D10 in GL6(𝔽41)

900000
090000
002313917
00518536
00194010
0019241740
,
100000
010000
0010017
00013936
00221400
00240040
,
0340000
6350000
000700
0035600
00353740
00043731
,
29150000
26120000
00002622
000075
002218538
00216936

G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,23,5,19,19,0,0,1,18,40,24,0,0,39,5,1,17,0,0,17,36,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,22,24,0,0,0,1,1,0,0,0,0,39,40,0,0,0,17,36,0,40],[0,6,0,0,0,0,34,35,0,0,0,0,0,0,0,35,35,0,0,0,7,6,37,4,0,0,0,0,4,37,0,0,0,0,0,31],[29,26,0,0,0,0,15,12,0,0,0,0,0,0,0,0,22,2,0,0,0,0,18,16,0,0,26,7,5,9,0,0,22,5,38,36] >;

C42.51D10 in GAP, Magma, Sage, TeX

C_4^2._{51}D_{10}
% in TeX

G:=Group("C4^2.51D10");
// GroupNames label

G:=SmallGroup(320,645);
// by ID

G=gap.SmallGroup(320,645);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,387,58,1684,851,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=b,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^-1>;
// generators/relations

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