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G = C42.163D10order 320 = 26·5

163rd non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.163D10, C10.1402+ 1+4, (C4×D20)⋊16C2, C4⋊D2037C2, C4⋊C4.214D10, C422C26D5, C42⋊D58C2, D208C442C2, D10⋊D445C2, (C2×C20).96C23, (C4×C20).35C22, C22⋊C4.81D10, Dic54D437C2, D10.56(C4○D4), (C2×C10).253C24, D10.13D442C2, C2.65(D48D10), C23.59(C22×D5), Dic5.49(C4○D4), Dic5.Q839C2, (C2×D20).175C22, C22.D2030C2, C4⋊Dic5.318C22, (C22×C10).67C23, C22.274(C23×D5), D10⋊C4.46C22, (C4×Dic5).160C22, (C2×Dic5).276C23, (C22×D5).112C23, C511(C22.47C24), C10.D4.147C22, (C22×Dic5).153C22, (D5×C4⋊C4)⋊43C2, C4⋊C4⋊D543C2, C2.100(D5×C4○D4), (C5×C422C2)⋊8C2, C10.211(C2×C4○D4), (C2×C4×D5).144C22, (C2×C4).89(C22×D5), (C5×C4⋊C4).205C22, (C2×C5⋊D4).73C22, (C5×C22⋊C4).78C22, SmallGroup(320,1381)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.163D10
C1C5C10C2×C10C22×D5C2×C4×D5D5×C4⋊C4 — C42.163D10
C5C2×C10 — C42.163D10
C1C22C422C2

Generators and relations for C42.163D10
 G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=a2b, dbd-1=a2b-1, dcd-1=c9 >

Subgroups: 918 in 238 conjugacy classes, 95 normal (91 characteristic)
C1, C2 [×3], C2 [×5], C4 [×12], C22, C22 [×13], C5, C2×C4 [×6], C2×C4 [×13], D4 [×10], C23, C23 [×3], D5 [×4], C10 [×3], C10, C42, C42 [×2], C22⋊C4 [×3], C22⋊C4 [×7], C4⋊C4 [×3], C4⋊C4 [×7], C22×C4 [×6], C2×D4 [×6], Dic5 [×2], Dic5 [×4], C20 [×6], D10 [×2], D10 [×8], C2×C10, C2×C10 [×3], C2×C4⋊C4, C42⋊C2, C4×D4 [×4], C4⋊D4 [×4], C22.D4 [×2], C42.C2, C422C2, C422C2, C4×D5 [×7], D20 [×5], C2×Dic5 [×5], C2×Dic5, C5⋊D4 [×5], C2×C20 [×6], C22×D5 [×3], C22×C10, C22.47C24, C4×Dic5 [×2], C10.D4 [×5], C4⋊Dic5 [×2], D10⋊C4 [×7], C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C2×C4×D5 [×5], C2×D20 [×3], C22×Dic5, C2×C5⋊D4 [×3], C42⋊D5, C4×D20, Dic54D4 [×2], D10⋊D4 [×3], C22.D20, Dic5.Q8, D5×C4⋊C4, D208C4, D10.13D4, C4⋊D20, C4⋊C4⋊D5, C5×C422C2, C42.163D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2+ 1+4, C22×D5 [×7], C22.47C24, C23×D5, D5×C4○D4 [×2], D48D10, C42.163D10

Smallest permutation representation of C42.163D10
On 160 points
Generators in S160
(1 24 11 34)(2 107 12 117)(3 26 13 36)(4 109 14 119)(5 28 15 38)(6 111 16 101)(7 30 17 40)(8 113 18 103)(9 32 19 22)(10 115 20 105)(21 66 31 76)(23 68 33 78)(25 70 35 80)(27 72 37 62)(29 74 39 64)(41 152 51 142)(42 85 52 95)(43 154 53 144)(44 87 54 97)(45 156 55 146)(46 89 56 99)(47 158 57 148)(48 91 58 81)(49 160 59 150)(50 93 60 83)(61 108 71 118)(63 110 73 120)(65 112 75 102)(67 114 77 104)(69 116 79 106)(82 125 92 135)(84 127 94 137)(86 129 96 139)(88 131 98 121)(90 133 100 123)(122 157 132 147)(124 159 134 149)(126 141 136 151)(128 143 138 153)(130 145 140 155)
(1 125 69 59)(2 136 70 50)(3 127 71 41)(4 138 72 52)(5 129 73 43)(6 140 74 54)(7 131 75 45)(8 122 76 56)(9 133 77 47)(10 124 78 58)(11 135 79 49)(12 126 80 60)(13 137 61 51)(14 128 62 42)(15 139 63 53)(16 130 64 44)(17 121 65 55)(18 132 66 46)(19 123 67 57)(20 134 68 48)(21 99 113 157)(22 90 114 148)(23 81 115 159)(24 92 116 150)(25 83 117 141)(26 94 118 152)(27 85 119 143)(28 96 120 154)(29 87 101 145)(30 98 102 156)(31 89 103 147)(32 100 104 158)(33 91 105 149)(34 82 106 160)(35 93 107 151)(36 84 108 142)(37 95 109 153)(38 86 110 144)(39 97 111 155)(40 88 112 146)
(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 43 11 53)(2 52 12 42)(3 41 13 51)(4 50 14 60)(5 59 15 49)(6 48 16 58)(7 57 17 47)(8 46 18 56)(9 55 19 45)(10 44 20 54)(21 99 31 89)(22 88 32 98)(23 97 33 87)(24 86 34 96)(25 95 35 85)(26 84 36 94)(27 93 37 83)(28 82 38 92)(29 91 39 81)(30 100 40 90)(61 137 71 127)(62 126 72 136)(63 135 73 125)(64 124 74 134)(65 133 75 123)(66 122 76 132)(67 131 77 121)(68 140 78 130)(69 129 79 139)(70 138 80 128)(101 149 111 159)(102 158 112 148)(103 147 113 157)(104 156 114 146)(105 145 115 155)(106 154 116 144)(107 143 117 153)(108 152 118 142)(109 141 119 151)(110 150 120 160)

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

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

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

53 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I···4N4O4P5A5B10A···10F10G10H20A···20L20M···20R
order122222222444444444···4445510···10101020···2020···20
size11114101020202222444410···102020222···2884···48···8

53 irreducible representations

dim1111111111111222222444
type+++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2D5C4○D4C4○D4D10D10D102+ 1+4D5×C4○D4D48D10
kernelC42.163D10C42⋊D5C4×D20Dic54D4D10⋊D4C22.D20Dic5.Q8D5×C4⋊C4D208C4D10.13D4C4⋊D20C4⋊C4⋊D5C5×C422C2C422C2Dic5D10C42C22⋊C4C4⋊C4C10C2C2
# reps1112311111111244266184

Matrix representation of C42.163D10 in GL6(𝔽41)

100000
010000
009000
000900
00003218
000009
,
100000
010000
0032000
0021900
0000402
000001
,
34340000
710000
0040500
0016100
0000139
0000140
,
770000
40340000
009000
000900
0000402
0000401

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0,0,0,0,0,18,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,21,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,2,1],[34,7,0,0,0,0,34,1,0,0,0,0,0,0,40,16,0,0,0,0,5,1,0,0,0,0,0,0,1,1,0,0,0,0,39,40],[7,40,0,0,0,0,7,34,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,40,40,0,0,0,0,2,1] >;

C42.163D10 in GAP, Magma, Sage, TeX

C_4^2._{163}D_{10}
% in TeX

G:=Group("C4^2.163D10");
// GroupNames label

G:=SmallGroup(320,1381);
// by ID

G=gap.SmallGroup(320,1381);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,219,184,1571,297,192,12550]);
// Polycyclic

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

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