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G = C42.D10order 320 = 26·5

1st non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.1D10, C8⋊C46D5, C10.17C4≀C2, (C2×D20).1C4, (C2×C20).223D4, (C2×C4).105D20, (C4×C20).10C22, (C2×Dic10).1C4, C4.D20.6C2, C42.D51C2, C2.4(D207C4), C2.6(D204C4), C10.8(C4.D4), C2.3(C20.46D4), C53(C42.C22), C22.56(D10⋊C4), (C5×C8⋊C4)⋊15C2, (C2×C4).10(C4×D5), (C2×C20).195(C2×C4), (C2×C4).206(C5⋊D4), (C2×C10).101(C22⋊C4), SmallGroup(320,22)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C42.D10
C1C5C10C2×C10C2×C20C4×C20C4.D20 — C42.D10
C5C2×C10C2×C20 — C42.D10
C1C22C42C8⋊C4

Generators and relations for C42.D10
 G = < a,b,c,d | a4=b4=1, c10=a, d2=a-1b, ab=ba, ac=ca, dad-1=a-1b2, cbc-1=a2b, dbd-1=a2b-1, dcd-1=a2bc9 >

Subgroups: 350 in 70 conjugacy classes, 25 normal (all characteristic)
C1, C2 [×3], C2, C4 [×4], C22, C22 [×3], C5, C8 [×4], C2×C4 [×3], C2×C4, D4, Q8, C23, D5, C10 [×3], C42, C22⋊C4 [×2], C2×C8 [×2], C2×D4, C2×Q8, Dic5, C20 [×3], D10 [×3], C2×C10, C8⋊C4, C8⋊C4, C4.4D4, C52C8 [×2], C40 [×2], Dic10, D20, C2×Dic5, C2×C20 [×3], C22×D5, C42.C22, C2×C52C8, D10⋊C4 [×2], C4×C20, C2×C40, C2×Dic10, C2×D20, C42.D5, C5×C8⋊C4, C4.D20, C42.D10
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, D10, C4.D4, C4≀C2 [×2], C4×D5, D20, C5⋊D4, C42.C22, D10⋊C4, D204C4, C20.46D4, D207C4, C42.D10

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

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

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

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

59 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F5A5B8A8B8C8D8E8F8G8H10A···10F20A···20H20I···20P40A···40P
order12222444444558888888810···1020···2020···2040···40
size1111402222440224444202020202···22···24···44···4

59 irreducible representations

dim11111122222222444
type++++++++++
imageC1C2C2C2C4C4D4D5D10C4≀C2C4×D5D20C5⋊D4D204C4C4.D4C20.46D4D207C4
kernelC42.D10C42.D5C5×C8⋊C4C4.D20C2×Dic10C2×D20C2×C20C8⋊C4C42C10C2×C4C2×C4C2×C4C2C10C2C2
# reps111122222844416144

Matrix representation of C42.D10 in GL4(𝔽41) generated by

243500
71700
00320
00032
,
301300
191100
0001
00400
,
36600
34200
0044
00437
,
23500
93900
00437
0044
G:=sub<GL(4,GF(41))| [24,7,0,0,35,17,0,0,0,0,32,0,0,0,0,32],[30,19,0,0,13,11,0,0,0,0,0,40,0,0,1,0],[36,34,0,0,6,2,0,0,0,0,4,4,0,0,4,37],[2,9,0,0,35,39,0,0,0,0,4,4,0,0,37,4] >;

C42.D10 in GAP, Magma, Sage, TeX

C_4^2.D_{10}
% in TeX

G:=Group("C4^2.D10");
// GroupNames label

G:=SmallGroup(320,22);
// by ID

G=gap.SmallGroup(320,22);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,141,36,422,184,1571,570,192,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^10=a,d^2=a^-1*b,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,c*b*c^-1=a^2*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=a^2*b*c^9>;
// generators/relations

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