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G = C42.179D10order 320 = 26·5

179th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.179D10, C10.852+ 1+4, C4⋊Q817D5, (C4×D20)⋊52C2, C4⋊D2041C2, C4⋊C4.222D10, (C2×Q8).89D10, C20.141(C4○D4), C20.23D428C2, (C2×C10).278C24, (C4×C20).219C22, (C2×C20).640C23, C4.42(Q82D5), C2.89(D46D10), (C2×D20).283C22, C4⋊Dic5.387C22, (Q8×C10).145C22, C22.299(C23×D5), D10⋊C4.53C22, C55(C22.49C24), (C4×Dic5).175C22, (C2×Dic5).285C23, (C22×D5).123C23, (C5×C4⋊Q8)⋊20C2, C4⋊C47D544C2, C10.125(C2×C4○D4), C2.33(C2×Q82D5), (C2×C4×D5).160C22, (C5×C4⋊C4).221C22, (C2×C4).603(C22×D5), SmallGroup(320,1406)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.179D10
C1C5C10C2×C10C22×D5C2×C4×D5C4⋊C47D5 — C42.179D10
C5C2×C10 — C42.179D10
C1C22C4⋊Q8

Generators and relations for C42.179D10
 G = < a,b,c,d | a4=b4=1, c10=b2, d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, dbd-1=a2b, dcd-1=a2c9 >

Subgroups: 918 in 236 conjugacy classes, 99 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×9], C22, C22 [×12], C5, C2×C4, C2×C4 [×6], C2×C4 [×12], D4 [×8], Q8 [×2], C23 [×4], D5 [×4], C10, C10 [×2], C42, C42 [×4], C22⋊C4 [×12], C4⋊C4 [×4], C4⋊C4 [×2], C22×C4 [×4], C2×D4 [×6], C2×Q8 [×2], Dic5 [×4], C20 [×4], C20 [×5], D10 [×12], C2×C10, C42⋊C2 [×4], C4×D4 [×2], C4⋊D4 [×4], C4.4D4 [×4], C4⋊Q8, C4×D5 [×8], D20 [×8], C2×Dic5 [×4], C2×C20, C2×C20 [×6], C5×Q8 [×2], C22×D5 [×4], C22.49C24, C4×Dic5 [×4], C4⋊Dic5 [×2], D10⋊C4 [×12], C4×C20, C5×C4⋊C4 [×4], C2×C4×D5 [×4], C2×D20 [×6], Q8×C10 [×2], C4×D20 [×2], C4⋊C47D5 [×4], C4⋊D20 [×4], C20.23D4 [×4], C5×C4⋊Q8, C42.179D10
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], C22.49C24, Q82D5 [×4], C23×D5, D46D10, C2×Q82D5 [×2], C42.179D10

Smallest permutation representation of C42.179D10
On 160 points
Generators in S160
(1 127 158 26)(2 27 159 128)(3 129 160 28)(4 29 141 130)(5 131 142 30)(6 31 143 132)(7 133 144 32)(8 33 145 134)(9 135 146 34)(10 35 147 136)(11 137 148 36)(12 37 149 138)(13 139 150 38)(14 39 151 140)(15 121 152 40)(16 21 153 122)(17 123 154 22)(18 23 155 124)(19 125 156 24)(20 25 157 126)(41 115 61 99)(42 100 62 116)(43 117 63 81)(44 82 64 118)(45 119 65 83)(46 84 66 120)(47 101 67 85)(48 86 68 102)(49 103 69 87)(50 88 70 104)(51 105 71 89)(52 90 72 106)(53 107 73 91)(54 92 74 108)(55 109 75 93)(56 94 76 110)(57 111 77 95)(58 96 78 112)(59 113 79 97)(60 98 80 114)
(1 75 11 65)(2 66 12 76)(3 77 13 67)(4 68 14 78)(5 79 15 69)(6 70 16 80)(7 61 17 71)(8 72 18 62)(9 63 19 73)(10 74 20 64)(21 114 31 104)(22 105 32 115)(23 116 33 106)(24 107 34 117)(25 118 35 108)(26 109 36 119)(27 120 37 110)(28 111 38 101)(29 102 39 112)(30 113 40 103)(41 154 51 144)(42 145 52 155)(43 156 53 146)(44 147 54 157)(45 158 55 148)(46 149 56 159)(47 160 57 150)(48 151 58 141)(49 142 59 152)(50 153 60 143)(81 125 91 135)(82 136 92 126)(83 127 93 137)(84 138 94 128)(85 129 95 139)(86 140 96 130)(87 131 97 121)(88 122 98 132)(89 133 99 123)(90 124 100 134)
(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 97 148 103)(2 102 149 96)(3 95 150 101)(4 120 151 94)(5 93 152 119)(6 118 153 92)(7 91 154 117)(8 116 155 90)(9 89 156 115)(10 114 157 88)(11 87 158 113)(12 112 159 86)(13 85 160 111)(14 110 141 84)(15 83 142 109)(16 108 143 82)(17 81 144 107)(18 106 145 100)(19 99 146 105)(20 104 147 98)(21 74 132 44)(22 43 133 73)(23 72 134 42)(24 41 135 71)(25 70 136 60)(26 59 137 69)(27 68 138 58)(28 57 139 67)(29 66 140 56)(30 55 121 65)(31 64 122 54)(32 53 123 63)(33 62 124 52)(34 51 125 61)(35 80 126 50)(36 49 127 79)(37 78 128 48)(38 47 129 77)(39 76 130 46)(40 45 131 75)

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

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

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

53 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4I4J···4Q5A5B10A···10F20A···20L20M···20T
order1222222244444···44···45510···1020···2020···20
size11112020202022224···410···10222···24···48···8

53 irreducible representations

dim11111122222444
type++++++++++++
imageC1C2C2C2C2C2D5C4○D4D10D10D102+ 1+4Q82D5D46D10
kernelC42.179D10C4×D20C4⋊C47D5C4⋊D20C20.23D4C5×C4⋊Q8C4⋊Q8C20C42C4⋊C4C2×Q8C10C4C2
# reps12444128284184

Matrix representation of C42.179D10 in GL6(𝔽41)

13340000
36280000
0040000
0004000
0000400
0000040
,
2870000
5130000
001000
000100
0000320
0000209
,
610000
4350000
000600
0034700
00003237
000009
,
1380000
0400000
0034600
0033700
000090
000009

G:=sub<GL(6,GF(41))| [13,36,0,0,0,0,34,28,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[28,5,0,0,0,0,7,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,20,0,0,0,0,0,9],[6,4,0,0,0,0,1,35,0,0,0,0,0,0,0,34,0,0,0,0,6,7,0,0,0,0,0,0,32,0,0,0,0,0,37,9],[1,0,0,0,0,0,38,40,0,0,0,0,0,0,34,33,0,0,0,0,6,7,0,0,0,0,0,0,9,0,0,0,0,0,0,9] >;

C42.179D10 in GAP, Magma, Sage, TeX

C_4^2._{179}D_{10}
% in TeX

G:=Group("C4^2.179D10");
// GroupNames label

G:=SmallGroup(320,1406);
// by ID

G=gap.SmallGroup(320,1406);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,387,100,675,570,185,80,12550]);
// Polycyclic

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

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