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G = C42.64D10order 320 = 26·5

64th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.64D10, C52C87D4, C52(C83D4), C4.11(D4×D5), C204D49C2, C20.25(C2×D4), (C2×C20).82D4, C4.4D42D5, (C2×D4).48D10, (C2×Q8).38D10, C2.8(C20⋊D4), C42.D59C2, C10.17(C41D4), (C2×C20).376C23, (C4×C20).107C22, (D4×C10).64C22, (Q8×C10).56C22, C2.19(D4⋊D10), C10.120(C8⋊C22), (C2×D20).105C22, (C2×D4⋊D5)⋊12C2, (C2×Q8⋊D5)⋊13C2, (C5×C4.4D4)⋊2C2, (C2×C10).507(C2×D4), (C2×C4).62(C5⋊D4), (C2×C4).476(C22×D5), C22.182(C2×C5⋊D4), (C2×C52C8).122C22, SmallGroup(320,685)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C42.64D10
C1C5C10C20C2×C20C2×D20C204D4 — C42.64D10
C5C10C2×C20 — C42.64D10
C1C22C42C4.4D4

Generators and relations for C42.64D10
 G = < a,b,c,d | a4=b4=c10=1, d2=cbc-1=b-1, ab=ba, cac-1=a-1b2, dad-1=ab2, bd=db, dcd-1=b-1c-1 >

Subgroups: 734 in 144 conjugacy classes, 43 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×2], C4 [×3], C22, C22 [×9], C5, C8 [×4], C2×C4, C2×C4 [×2], C2×C4, D4 [×10], Q8 [×2], C23 [×3], D5 [×2], C10, C10 [×2], C10, C42, C22⋊C4 [×2], C2×C8 [×2], D8 [×4], SD16 [×4], C2×D4, C2×D4 [×4], C2×Q8, C20 [×2], C20 [×3], D10 [×6], C2×C10, C2×C10 [×3], C8⋊C4, C4.4D4, C41D4, C2×D8 [×2], C2×SD16 [×2], C52C8 [×4], D20 [×8], C2×C20, C2×C20 [×2], C2×C20, C5×D4 [×2], C5×Q8 [×2], C22×D5 [×2], C22×C10, C83D4, C2×C52C8 [×2], D4⋊D5 [×4], Q8⋊D5 [×4], C4×C20, C5×C22⋊C4 [×2], C2×D20 [×2], C2×D20 [×2], D4×C10, Q8×C10, C42.D5, C204D4, C2×D4⋊D5 [×2], C2×Q8⋊D5 [×2], C5×C4.4D4, C42.64D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C41D4, C8⋊C22 [×2], C5⋊D4 [×2], C22×D5, C83D4, D4×D5 [×2], C2×C5⋊D4, C20⋊D4, D4⋊D10 [×2], C42.64D10

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

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E5A5B8A8B8C8D10A···10F10G10H10I10J20A···20L20M20N20O20P
order12222224444455888810···101010101020···2020202020
size1111840402244822202020202···288884···48888

44 irreducible representations

dim1111112222222444
type+++++++++++++++
imageC1C2C2C2C2C2D4D4D5D10D10D10C5⋊D4C8⋊C22D4×D5D4⋊D10
kernelC42.64D10C42.D5C204D4C2×D4⋊D5C2×Q8⋊D5C5×C4.4D4C52C8C2×C20C4.4D4C42C2×D4C2×Q8C2×C4C10C4C2
# reps1112214222228248

Matrix representation of C42.64D10 in GL8(𝔽41)

400000000
040000000
00450000
0013370000
00002503636
0000025365
000055160
0000536016
,
10000000
01000000
00100000
00010000
00000100
000040000
000000040
00000010
,
3434000000
71000000
004000000
001810000
00000010
00000001
00001000
00000100
,
3434000000
17000000
004000000
000400000
00000001
00000010
00001000
000004000

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,4,13,0,0,0,0,0,0,5,37,0,0,0,0,0,0,0,0,25,0,5,5,0,0,0,0,0,25,5,36,0,0,0,0,36,36,16,0,0,0,0,0,36,5,0,16],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0],[34,7,0,0,0,0,0,0,34,1,0,0,0,0,0,0,0,0,40,18,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[34,1,0,0,0,0,0,0,34,7,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0] >;

C42.64D10 in GAP, Magma, Sage, TeX

C_4^2._{64}D_{10}
% in TeX

G:=Group("C4^2.64D10");
// GroupNames label

G:=SmallGroup(320,685);
// by ID

G=gap.SmallGroup(320,685);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,555,1123,297,136,12550]);
// Polycyclic

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

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