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## G = C40⋊21(C2×C4)  order 320 = 26·5

### 11st semidirect product of C40 and C2×C4 acting via C2×C4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C40⋊21(C2×C4)
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×D20 — C2×C40⋊C2 — C40⋊21(C2×C4)
 Lower central C5 — C10 — C20 — C40⋊21(C2×C4)
 Upper central C1 — C22 — C2×C4 — C2.D8

Generators and relations for C4021(C2×C4)
G = < a,b,c | a40=b2=c4=1, bab=a19, cac-1=a31, bc=cb >

Subgroups: 502 in 120 conjugacy classes, 49 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C4×D4, C4×Q8, C2×SD16, C52C8, C40, Dic10, Dic10, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, SD16⋊C4, C40⋊C2, C2×C52C8, C4×Dic5, C4×Dic5, C10.D4, D10⋊C4, C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, C2×D20, D206C4, C10.Q16, C408C4, C5×C2.D8, Dic53Q8, D208C4, C2×C40⋊C2, C4021(C2×C4)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22×C4, C2×D4, C4○D4, D10, C4×D4, C8⋊C22, C8.C22, C4×D5, C22×D5, SD16⋊C4, C2×C4×D5, D4×D5, Q82D5, D208C4, D8⋊D5, Q16⋊D5, C4021(C2×C4)

Smallest permutation representation of C4021(C2×C4)
On 160 points
Generators in S160
(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)
(2 20)(3 39)(4 18)(5 37)(6 16)(7 35)(8 14)(9 33)(10 12)(11 31)(13 29)(15 27)(17 25)(19 23)(22 40)(24 38)(26 36)(28 34)(30 32)(41 59)(42 78)(43 57)(44 76)(45 55)(46 74)(47 53)(48 72)(49 51)(50 70)(52 68)(54 66)(56 64)(58 62)(61 79)(63 77)(65 75)(67 73)(69 71)(81 119)(82 98)(83 117)(84 96)(85 115)(86 94)(87 113)(88 92)(89 111)(91 109)(93 107)(95 105)(97 103)(99 101)(100 120)(102 118)(104 116)(106 114)(108 112)(121 139)(122 158)(123 137)(124 156)(125 135)(126 154)(127 133)(128 152)(129 131)(130 150)(132 148)(134 146)(136 144)(138 142)(141 159)(143 157)(145 155)(147 153)(149 151)
(1 80 90 140)(2 71 91 131)(3 62 92 122)(4 53 93 153)(5 44 94 144)(6 75 95 135)(7 66 96 126)(8 57 97 157)(9 48 98 148)(10 79 99 139)(11 70 100 130)(12 61 101 121)(13 52 102 152)(14 43 103 143)(15 74 104 134)(16 65 105 125)(17 56 106 156)(18 47 107 147)(19 78 108 138)(20 69 109 129)(21 60 110 160)(22 51 111 151)(23 42 112 142)(24 73 113 133)(25 64 114 124)(26 55 115 155)(27 46 116 146)(28 77 117 137)(29 68 118 128)(30 59 119 159)(31 50 120 150)(32 41 81 141)(33 72 82 132)(34 63 83 123)(35 54 84 154)(36 45 85 145)(37 76 86 136)(38 67 87 127)(39 58 88 158)(40 49 89 149)

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5A 5B 8A 8B 8C 8D 10A ··· 10F 20A 20B 20C 20D 20E ··· 20L 40A ··· 40H order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 20 20 20 20 20 ··· 20 40 ··· 40 size 1 1 1 1 20 20 2 2 4 4 4 4 10 10 10 10 20 20 2 2 4 4 20 20 2 ··· 2 4 4 4 4 8 ··· 8 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 D4 D5 C4○D4 D10 D10 C4×D5 C8⋊C22 C8.C22 Q8⋊2D5 D4×D5 D8⋊D5 Q16⋊D5 kernel C40⋊21(C2×C4) D20⋊6C4 C10.Q16 C40⋊8C4 C5×C2.D8 Dic5⋊3Q8 D20⋊8C4 C2×C40⋊C2 C40⋊C2 C2×Dic5 C2.D8 C20 C4⋊C4 C2×C8 C8 C10 C10 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 2 2 2 4 2 8 1 1 2 2 4 4

Matrix representation of C4021(C2×C4) in GL6(𝔽41)

 24 25 0 0 0 0 13 17 0 0 0 0 0 0 2 31 39 10 0 0 10 31 31 10 0 0 1 36 0 0 0 0 5 36 0 0
,
 1 0 0 0 0 0 3 40 0 0 0 0 0 0 1 0 0 0 0 0 34 40 0 0 0 0 1 0 40 0 0 0 34 40 7 1
,
 32 0 0 0 0 0 14 9 0 0 0 0 0 0 15 14 25 13 0 0 27 40 28 16 0 0 7 0 26 27 0 0 0 7 14 1

G:=sub<GL(6,GF(41))| [24,13,0,0,0,0,25,17,0,0,0,0,0,0,2,10,1,5,0,0,31,31,36,36,0,0,39,31,0,0,0,0,10,10,0,0],[1,3,0,0,0,0,0,40,0,0,0,0,0,0,1,34,1,34,0,0,0,40,0,40,0,0,0,0,40,7,0,0,0,0,0,1],[32,14,0,0,0,0,0,9,0,0,0,0,0,0,15,27,7,0,0,0,14,40,0,7,0,0,25,28,26,14,0,0,13,16,27,1] >;

C4021(C2×C4) in GAP, Magma, Sage, TeX

C_{40}\rtimes_{21}(C_2\times C_4)
% in TeX

G:=Group("C40:21(C2xC4)");
// GroupNames label

G:=SmallGroup(320,516);
// by ID

G=gap.SmallGroup(320,516);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,219,58,1684,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^2=c^4=1,b*a*b=a^19,c*a*c^-1=a^31,b*c=c*b>;
// generators/relations

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