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## G = C40⋊20(C2×C4)  order 320 = 26·5

### 10th 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⋊20(C2×C4)
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×C4×D5 — C2×C8⋊D5 — C40⋊20(C2×C4)
 Lower central C5 — C10 — C20 — C40⋊20(C2×C4)
 Upper central C1 — C22 — C2×C4 — C2.D8

Generators and relations for C4020(C2×C4)
G = < a,b,c | a40=b2=c4=1, bab=a29, cac-1=a31, bc=cb >

Subgroups: 430 in 118 conjugacy classes, 55 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×11], C23, D5 [×2], C10 [×3], C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×3], C2×C8, C2×C8, M4(2) [×4], C22×C4 [×2], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C4.Q8 [×2], C2.D8, C2.D8, C2×C4⋊C4, C42⋊C2, C2×M4(2), C52C8 [×2], C40 [×2], C4×D5 [×4], C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, M4(2)⋊C4, C8⋊D5 [×4], C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5 [×2], D10⋊C4, C5×C4⋊C4 [×2], C2×C40, C2×C4×D5, C2×C4×D5, C10.D8, C20.Q8, C406C4, C5×C2.D8, D5×C4⋊C4, C4⋊C47D5, C2×C8⋊D5, C4020(C2×C4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, D10 [×3], C2×C4⋊C4, C8⋊C22, C8.C22, C4×D5 [×2], C22×D5, M4(2)⋊C4, C2×C4×D5, D4×D5, Q8×D5, D5×C4⋊C4, D8⋊D5, Q16⋊D5, C4020(C2×C4)

Smallest permutation representation of C4020(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 30)(3 19)(4 8)(5 37)(6 26)(7 15)(9 33)(10 22)(12 40)(13 29)(14 18)(16 36)(17 25)(20 32)(23 39)(24 28)(27 35)(34 38)(41 77)(42 66)(43 55)(45 73)(46 62)(47 51)(48 80)(49 69)(50 58)(52 76)(53 65)(56 72)(57 61)(59 79)(60 68)(63 75)(67 71)(70 78)(81 93)(83 111)(84 100)(85 89)(86 118)(87 107)(88 96)(90 114)(91 103)(94 110)(95 99)(97 117)(98 106)(101 113)(104 120)(105 109)(108 116)(115 119)(121 149)(122 138)(123 127)(124 156)(125 145)(126 134)(128 152)(129 141)(131 159)(132 148)(133 137)(135 155)(136 144)(139 151)(142 158)(143 147)(146 154)(153 157)
(1 44 82 160)(2 75 83 151)(3 66 84 142)(4 57 85 133)(5 48 86 124)(6 79 87 155)(7 70 88 146)(8 61 89 137)(9 52 90 128)(10 43 91 159)(11 74 92 150)(12 65 93 141)(13 56 94 132)(14 47 95 123)(15 78 96 154)(16 69 97 145)(17 60 98 136)(18 51 99 127)(19 42 100 158)(20 73 101 149)(21 64 102 140)(22 55 103 131)(23 46 104 122)(24 77 105 153)(25 68 106 144)(26 59 107 135)(27 50 108 126)(28 41 109 157)(29 72 110 148)(30 63 111 139)(31 54 112 130)(32 45 113 121)(33 76 114 152)(34 67 115 143)(35 58 116 134)(36 49 117 125)(37 80 118 156)(38 71 119 147)(39 62 120 138)(40 53 81 129)

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

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

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

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 10 10 2 2 4 4 4 4 10 10 20 20 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 2 4 4 4 4 4 4 type + + + + + + + + - + + + + + + - - + image C1 C2 C2 C2 C2 C2 C2 C2 C4 Q8 D4 D4 D5 D10 D10 C4×D5 C8⋊C22 C8.C22 Q8×D5 D4×D5 D8⋊D5 Q16⋊D5 kernel C40⋊20(C2×C4) C10.D8 C20.Q8 C40⋊6C4 C5×C2.D8 D5×C4⋊C4 C4⋊C4⋊7D5 C2×C8⋊D5 C8⋊D5 C4×D5 C2×Dic5 C22×D5 C2.D8 C4⋊C4 C2×C8 C8 C10 C10 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 2 1 1 2 4 2 8 1 1 2 2 4 4

Matrix representation of C4020(C2×C4) in GL6(𝔽41)

 8 10 0 0 0 0 14 33 0 0 0 0 0 0 1 36 1 36 0 0 5 36 5 36 0 0 40 5 1 36 0 0 36 5 5 36
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 34 40 0 0 0 0 0 0 1 0 0 0 0 0 34 40
,
 10 20 0 0 0 0 38 31 0 0 0 0 0 0 25 0 27 0 0 0 0 25 0 27 0 0 27 0 16 0 0 0 0 27 0 16

G:=sub<GL(6,GF(41))| [8,14,0,0,0,0,10,33,0,0,0,0,0,0,1,5,40,36,0,0,36,36,5,5,0,0,1,5,1,5,0,0,36,36,36,36],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,40],[10,38,0,0,0,0,20,31,0,0,0,0,0,0,25,0,27,0,0,0,0,25,0,27,0,0,27,0,16,0,0,0,0,27,0,16] >;

C4020(C2×C4) in GAP, Magma, Sage, TeX

C_{40}\rtimes_{20}(C_2\times C_4)
% in TeX

G:=Group("C40:20(C2xC4)");
// GroupNames label

G:=SmallGroup(320,508);
// by ID

G=gap.SmallGroup(320,508);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,120,219,58,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^2=c^4=1,b*a*b=a^29,c*a*c^-1=a^31,b*c=c*b>;
// generators/relations

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