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G = C42.116D10order 320 = 26·5

116th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.116D10, C10.1052+ 1+4, (C4×D4)⋊24D5, (D4×C20)⋊26C2, (C4×D20)⋊33C2, C20⋊D410C2, C4⋊D2016C2, C207D412C2, C202D410C2, C4⋊C4.287D10, D10⋊D411C2, (C2×D4).223D10, C4.46(C4○D20), C42⋊D515C2, C4.Dic1016C2, D10.12D49C2, C20.113(C4○D4), (C2×C10).106C24, (C4×C20).160C22, (C2×C20).164C23, C22⋊C4.118D10, (C22×C4).214D10, C2.24(D46D10), C2.18(D48D10), (D4×C10).265C22, (C2×D20).268C22, C23.23D104C2, C4⋊Dic5.364C22, (C22×C20).83C22, (C4×Dic5).86C22, (C2×Dic5).47C23, C10.D4.7C22, (C22×D5).40C23, C22.131(C23×D5), C23.103(C22×D5), C23.D5.16C22, D10⋊C4.88C22, (C22×C10).176C23, C52(C22.34C24), C10.48(C2×C4○D4), C2.55(C2×C4○D20), (C2×C4×D5).254C22, (C5×C4⋊C4).334C22, (C2×C4).581(C22×D5), (C2×C5⋊D4).19C22, (C5×C22⋊C4).129C22, SmallGroup(320,1234)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.116D10
C1C5C10C2×C10C22×D5C2×C4×D5D10.12D4 — C42.116D10
C5C2×C10 — C42.116D10
C1C22C4×D4

Generators and relations for C42.116D10
 G = < a,b,c,d | a4=b4=c10=1, d2=a2, ab=ba, cac-1=dad-1=a-1b2, bc=cb, dbd-1=a2b-1, dcd-1=a2c-1 >

Subgroups: 958 in 240 conjugacy classes, 95 normal (43 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×9], C22, C22 [×15], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×11], D4 [×12], C23 [×2], C23 [×3], D5 [×3], C10 [×3], C10 [×2], C42, C42, C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×7], C22×C4 [×2], C22×C4 [×3], C2×D4, C2×D4 [×9], Dic5 [×5], C20 [×2], C20 [×4], D10 [×9], C2×C10, C2×C10 [×6], C42⋊C2, C4×D4, C4×D4, C4⋊D4 [×6], C22.D4 [×4], C42.C2, C41D4, C4×D5 [×4], D20 [×4], C2×Dic5 [×3], C2×Dic5 [×2], C5⋊D4 [×6], C2×C20 [×3], C2×C20 [×2], C2×C20 [×2], C5×D4 [×2], C22×D5, C22×D5 [×2], C22×C10 [×2], C22.34C24, C4×Dic5, C10.D4 [×2], C10.D4 [×2], C4⋊Dic5, C4⋊Dic5 [×2], D10⋊C4 [×2], D10⋊C4 [×4], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×C4×D5, C2×C4×D5 [×2], C2×D20, C2×D20 [×2], C2×C5⋊D4 [×6], C22×C20 [×2], D4×C10, C42⋊D5, C4×D20, D10.12D4 [×2], D10⋊D4 [×2], C4.Dic10, C4⋊D20, C23.23D10 [×2], C207D4 [×2], C202D4, C20⋊D4, D4×C20, C42.116D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ 1+4 [×2], C22×D5 [×7], C22.34C24, C4○D20 [×2], C23×D5, C2×C4○D20, D46D10, D48D10, C42.116D10

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

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A···4F4G4H4I···4M5A5B10A···10F10G···10N20A···20H20I···20X
order1222222224···4444···45510···1010···1020···2020···20
size1111442020202···24420···20222···24···42···24···4

62 irreducible representations

dim11111111111122222222444
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D5C4○D4D10D10D10D10D10C4○D202+ 1+4D46D10D48D10
kernelC42.116D10C42⋊D5C4×D20D10.12D4D10⋊D4C4.Dic10C4⋊D20C23.23D10C207D4C202D4C20⋊D4D4×C20C4×D4C20C42C22⋊C4C4⋊C4C22×C4C2×D4C4C10C2C2
# reps111221122111242424216244

Matrix representation of C42.116D10 in GL6(𝔽41)

3200000
0320000
0030090
0003009
00320110
00032011
,
3200000
0320000
00244000
0011700
00002440
0000117
,
17390000
21240000
0000407
0000347
0040700
0034700
,
17390000
22240000
0014113323
00272788
008182730
0033331414

G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,30,0,32,0,0,0,0,30,0,32,0,0,9,0,11,0,0,0,0,9,0,11],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,24,1,0,0,0,0,40,17,0,0,0,0,0,0,24,1,0,0,0,0,40,17],[17,21,0,0,0,0,39,24,0,0,0,0,0,0,0,0,40,34,0,0,0,0,7,7,0,0,40,34,0,0,0,0,7,7,0,0],[17,22,0,0,0,0,39,24,0,0,0,0,0,0,14,27,8,33,0,0,11,27,18,33,0,0,33,8,27,14,0,0,23,8,30,14] >;

C42.116D10 in GAP, Magma, Sage, TeX

C_4^2._{116}D_{10}
% in TeX

G:=Group("C4^2.116D10");
// GroupNames label

G:=SmallGroup(320,1234);
// by ID

G=gap.SmallGroup(320,1234);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,675,192,12550]);
// Polycyclic

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

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