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G = C42.145D10order 320 = 26·5

145th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.145D10, C10.742+ 1+4, C10.932- 1+4, C4.4D417D5, (C2×Q8).84D10, D10⋊D443C2, D103Q834C2, (C2×D4).113D10, C4.D2031C2, C22⋊C4.38D10, C20.6Q829C2, Dic5⋊D435C2, (C2×C20).633C23, (C4×C20).222C22, (C2×C10).228C24, (C2×D20).37C22, C4⋊Dic5.52C22, D10.12D447C2, C2.54(D48D10), C2.78(D46D10), C23.50(C22×D5), (D4×C10).213C22, C22.D2028C2, (C22×C10).58C23, (Q8×C10).131C22, C22.249(C23×D5), Dic5.14D443C2, C23.D5.60C22, D10⋊C4.73C22, C54(C22.56C24), (C2×Dic10).41C22, (C2×Dic5).118C23, C10.D4.84C22, (C22×D5).100C23, C2.54(D4.10D10), (C22×Dic5).147C22, (C5×C4.4D4)⋊20C2, (C2×C4×D5).132C22, (C2×C4).201(C22×D5), (C2×C5⋊D4).66C22, (C5×C22⋊C4).69C22, SmallGroup(320,1356)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.145D10
C1C5C10C2×C10C22×D5C2×C4×D5D10.12D4 — C42.145D10
C5C2×C10 — C42.145D10
C1C22C4.4D4

Generators and relations for C42.145D10
 G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c9 >

Subgroups: 854 in 220 conjugacy classes, 91 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C4 [×11], C22, C22 [×12], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×10], D4 [×6], Q8 [×2], C23 [×2], C23 [×2], D5 [×2], C10 [×3], C10 [×2], C42, C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×10], C22×C4 [×4], C2×D4, C2×D4 [×5], C2×Q8, C2×Q8, Dic5 [×6], C20 [×5], D10 [×6], C2×C10, C2×C10 [×6], C4⋊D4 [×4], C22⋊Q8 [×4], C22.D4 [×4], C4.4D4, C4.4D4, C42.C2, Dic10, C4×D5 [×2], D20, C2×Dic5 [×6], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×3], C2×C20 [×2], C5×D4, C5×Q8, C22×D5 [×2], C22×C10 [×2], C22.56C24, C10.D4 [×6], C4⋊Dic5 [×2], C4⋊Dic5 [×2], D10⋊C4 [×6], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×4], C2×Dic10, C2×C4×D5 [×2], C2×D20, C22×Dic5 [×2], C2×C5⋊D4 [×4], D4×C10, Q8×C10, C20.6Q8, C4.D20, Dic5.14D4 [×2], D10.12D4 [×2], D10⋊D4 [×2], C22.D20 [×2], Dic5⋊D4 [×2], D103Q8 [×2], C5×C4.4D4, C42.145D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], 2+ 1+4 [×2], 2- 1+4, C22×D5 [×7], C22.56C24, C23×D5, D46D10, D48D10, D4.10D10, C42.145D10

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

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4E4F···4K5A5B10A···10F10G10H10I10J20A···20L20M20N20O20P
order122222224···44···45510···101010101020···2020202020
size11114420204···420···20222···288884···48888

47 irreducible representations

dim11111111112222244444
type++++++++++++++++-+-
imageC1C2C2C2C2C2C2C2C2C2D5D10D10D10D102+ 1+42- 1+4D46D10D48D10D4.10D10
kernelC42.145D10C20.6Q8C4.D20Dic5.14D4D10.12D4D10⋊D4C22.D20Dic5⋊D4D103Q8C5×C4.4D4C4.4D4C42C22⋊C4C2×D4C2×Q8C10C10C2C2C2
# reps11122222212282221444

Matrix representation of C42.145D10 in GL8(𝔽41)

399000000
42000000
001190000
0032300000
00001210237
0000922914
0000016109
000025163238
,
4000280000
04013130000
33100000
380010000
000023361526
00004018026
000000171
0000004024
,
52230350000
283322360000
02827190000
171322170000
000020252538
0000533187
00003237334
00003203626
,
21356110000
12205190000
241222140000
392724190000
00003725730
0000333142
000013373816
000090515

G:=sub<GL(8,GF(41))| [39,4,0,0,0,0,0,0,9,2,0,0,0,0,0,0,0,0,11,32,0,0,0,0,0,0,9,30,0,0,0,0,0,0,0,0,12,9,0,25,0,0,0,0,10,22,16,16,0,0,0,0,23,9,10,32,0,0,0,0,7,14,9,38],[40,0,3,38,0,0,0,0,0,40,3,0,0,0,0,0,0,13,1,0,0,0,0,0,28,13,0,1,0,0,0,0,0,0,0,0,23,40,0,0,0,0,0,0,36,18,0,0,0,0,0,0,15,0,17,40,0,0,0,0,26,26,1,24],[5,28,0,17,0,0,0,0,22,33,28,13,0,0,0,0,30,22,27,22,0,0,0,0,35,36,19,17,0,0,0,0,0,0,0,0,20,5,32,32,0,0,0,0,25,33,37,0,0,0,0,0,25,18,3,36,0,0,0,0,38,7,34,26],[21,12,24,39,0,0,0,0,35,20,12,27,0,0,0,0,6,5,22,24,0,0,0,0,11,19,14,19,0,0,0,0,0,0,0,0,37,3,13,9,0,0,0,0,25,33,37,0,0,0,0,0,7,14,38,5,0,0,0,0,30,2,16,15] >;

C42.145D10 in GAP, Magma, Sage, TeX

C_4^2._{145}D_{10}
% in TeX

G:=Group("C4^2.145D10");
// GroupNames label

G:=SmallGroup(320,1356);
// by ID

G=gap.SmallGroup(320,1356);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,1571,570,297,136,12550]);
// Polycyclic

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

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