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G = C42.141D10order 320 = 26·5

141st non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.141D10, C10.902- 1+4, C4.33(D4×D5), (C4×D5).12D4, C20.62(C2×D4), C4.4D49D5, C202Q830C2, D10.80(C2×D4), (C2×D4).172D10, C42⋊D520C2, (C2×C20).80C23, (C2×Q8).136D10, C22⋊C4.35D10, Dic5.91(C2×D4), C10.89(C22×D4), Dic5⋊Q824C2, C20.17D424C2, (C2×C10).219C24, (C4×C20).185C22, C4⋊Dic5.51C22, D10.12D441C2, C23.41(C22×D5), (D4×C10).154C22, (C22×C10).49C23, (Q8×C10).126C22, C22.240(C23×D5), Dic5.14D440C2, C23.D5.54C22, C54(C23.38C23), (C4×Dic5).141C22, (C2×Dic5).114C23, (C22×D5).224C23, C2.51(D4.10D10), D10⋊C4.110C22, (C2×Dic10).183C22, C10.D4.120C22, (C22×Dic5).142C22, (C2×Q8×D5)⋊10C2, C2.62(C2×D4×D5), (C5×C4.4D4)⋊11C2, (C2×C4×D5).129C22, (C2×D42D5).11C2, (C2×C4).194(C22×D5), (C2×C5⋊D4).59C22, (C5×C22⋊C4).64C22, SmallGroup(320,1347)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.141D10
C1C5C10C2×C10C22×D5C2×C4×D5C2×Q8×D5 — C42.141D10
C5C2×C10 — C42.141D10
C1C22C4.4D4

Generators and relations for C42.141D10
 G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, dbd-1=b-1, dcd-1=c9 >

Subgroups: 926 in 270 conjugacy classes, 103 normal (29 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×12], C22, C22 [×10], C5, C2×C4, C2×C4 [×4], C2×C4 [×19], D4 [×6], Q8 [×10], C23 [×2], C23, D5 [×2], C10, C10 [×2], C10 [×2], C42, C42, C22⋊C4 [×4], C22⋊C4 [×6], C4⋊C4 [×10], C22×C4 [×5], C2×D4, C2×D4 [×2], C2×Q8, C2×Q8 [×8], C4○D4 [×4], Dic5 [×2], Dic5 [×6], C20 [×2], C20 [×4], D10 [×2], D10 [×2], C2×C10, C2×C10 [×6], C42⋊C2, C22⋊Q8 [×4], C22.D4 [×4], C4.4D4, C4.4D4, C4⋊Q8 [×2], C22×Q8, C2×C4○D4, Dic10 [×8], C4×D5 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×6], C2×Dic5 [×4], C5⋊D4 [×4], C2×C20, C2×C20 [×4], C5×D4 [×2], C5×Q8 [×2], C22×D5, C22×C10 [×2], C23.38C23, C4×Dic5, C10.D4 [×6], C4⋊Dic5 [×4], D10⋊C4 [×2], C23.D5 [×4], C4×C20, C5×C22⋊C4 [×4], C2×Dic10 [×2], C2×Dic10 [×2], C2×C4×D5, C2×C4×D5 [×2], D42D5 [×4], Q8×D5 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×2], D4×C10, Q8×C10, C202Q8, C42⋊D5, Dic5.14D4 [×4], D10.12D4 [×4], C20.17D4, Dic5⋊Q8, C5×C4.4D4, C2×D42D5, C2×Q8×D5, C42.141D10
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, 2- 1+4 [×2], C22×D5 [×7], C23.38C23, D4×D5 [×2], C23×D5, C2×D4×D5, D4.10D10 [×2], C42.141D10

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

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I···4N5A5B10A···10F10G10H10I10J20A···20L20M20N20O20P
order12222222444444444···45510···101010101020···2020202020
size1111441010224444101020···20222···288884···48888

50 irreducible representations

dim1111111111222222444
type++++++++++++++++-+-
imageC1C2C2C2C2C2C2C2C2C2D4D5D10D10D10D102- 1+4D4×D5D4.10D10
kernelC42.141D10C202Q8C42⋊D5Dic5.14D4D10.12D4C20.17D4Dic5⋊Q8C5×C4.4D4C2×D42D5C2×Q8×D5C4×D5C4.4D4C42C22⋊C4C2×D4C2×Q8C10C4C2
# reps1114411111422822248

Matrix representation of C42.141D10 in GL6(𝔽41)

100000
010000
00140340
00014034
00340270
00034027
,
0400000
100000
00303200
0091100
00003032
0000911
,
32170000
1790000
000077
00003440
00343400
007100
,
9240000
24320000
000077
00004034
00343400
001700

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

C42.141D10 in GAP, Magma, Sage, TeX

C_4^2._{141}D_{10}
% in TeX

G:=Group("C4^2.141D10");
// GroupNames label

G:=SmallGroup(320,1347);
// by ID

G=gap.SmallGroup(320,1347);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,100,675,297,192,12550]);
// 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,c*b*c^-1=a^2*b^-1,d*b*d^-1=b^-1,d*c*d^-1=c^9>;
// generators/relations

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