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## G = C4.89(C2×D20)  order 320 = 26·5

### 16th central extension by C4 of C2×D20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C4.89(C2×D20)
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C4×D5 — C2×C4○D20 — C4.89(C2×D20)
 Lower central C5 — C2×C10 — C4.89(C2×D20)
 Upper central C1 — C2×C4 — C2×M4(2)

Generators and relations for C4.89(C2×D20)
G = < a,b,c,d | a4=b2=1, c20=a2, d2=a, ab=ba, ac=ca, ad=da, cbc-1=a2b, bd=db, dcd-1=a-1c19 >

Subgroups: 574 in 158 conjugacy classes, 63 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C10, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C22⋊C8, C22×C8, C2×M4(2), C2×C4○D4, C52C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, (C22×C8)⋊C2, C2×C52C8, C2×C52C8, C2×C40, C5×M4(2), C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C2×C5⋊D4, C22×C20, D101C8, C22×C52C8, C10×M4(2), C2×C4○D20, C4.89(C2×D20)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C2×C22⋊C4, C8○D4, C4×D5, D20, C5⋊D4, C22×D5, (C22×C8)⋊C2, D10⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, D20.2C4, C2×D10⋊C4, C4.89(C2×D20)

Smallest permutation representation of C4.89(C2×D20)
On 160 points
Generators in S160
(1 47 21 67)(2 48 22 68)(3 49 23 69)(4 50 24 70)(5 51 25 71)(6 52 26 72)(7 53 27 73)(8 54 28 74)(9 55 29 75)(10 56 30 76)(11 57 31 77)(12 58 32 78)(13 59 33 79)(14 60 34 80)(15 61 35 41)(16 62 36 42)(17 63 37 43)(18 64 38 44)(19 65 39 45)(20 66 40 46)(81 123 101 143)(82 124 102 144)(83 125 103 145)(84 126 104 146)(85 127 105 147)(86 128 106 148)(87 129 107 149)(88 130 108 150)(89 131 109 151)(90 132 110 152)(91 133 111 153)(92 134 112 154)(93 135 113 155)(94 136 114 156)(95 137 115 157)(96 138 116 158)(97 139 117 159)(98 140 118 160)(99 141 119 121)(100 142 120 122)
(1 89)(2 110)(3 91)(4 112)(5 93)(6 114)(7 95)(8 116)(9 97)(10 118)(11 99)(12 120)(13 101)(14 82)(15 103)(16 84)(17 105)(18 86)(19 107)(20 88)(21 109)(22 90)(23 111)(24 92)(25 113)(26 94)(27 115)(28 96)(29 117)(30 98)(31 119)(32 100)(33 81)(34 102)(35 83)(36 104)(37 85)(38 106)(39 87)(40 108)(41 125)(42 146)(43 127)(44 148)(45 129)(46 150)(47 131)(48 152)(49 133)(50 154)(51 135)(52 156)(53 137)(54 158)(55 139)(56 160)(57 141)(58 122)(59 143)(60 124)(61 145)(62 126)(63 147)(64 128)(65 149)(66 130)(67 151)(68 132)(69 153)(70 134)(71 155)(72 136)(73 157)(74 138)(75 159)(76 140)(77 121)(78 142)(79 123)(80 144)
(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 46 47 20 21 66 67 40)(2 19 48 65 22 39 68 45)(3 64 49 38 23 44 69 18)(4 37 50 43 24 17 70 63)(5 42 51 16 25 62 71 36)(6 15 52 61 26 35 72 41)(7 60 53 34 27 80 73 14)(8 33 54 79 28 13 74 59)(9 78 55 12 29 58 75 32)(10 11 56 57 30 31 76 77)(81 158 123 96 101 138 143 116)(82 95 124 137 102 115 144 157)(83 136 125 114 103 156 145 94)(84 113 126 155 104 93 146 135)(85 154 127 92 105 134 147 112)(86 91 128 133 106 111 148 153)(87 132 129 110 107 152 149 90)(88 109 130 151 108 89 150 131)(97 142 139 120 117 122 159 100)(98 119 140 121 118 99 160 141)

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

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

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

68 conjugacy classes

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

68 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 D4 D5 D10 D10 C8○D4 C4×D5 D20 C5⋊D4 C4×D5 D20.2C4 kernel C4.89(C2×D20) D10⋊1C8 C22×C5⋊2C8 C10×M4(2) C2×C4○D20 C2×Dic10 C2×D20 C2×C5⋊D4 C2×C20 C2×M4(2) C2×C8 C22×C4 C10 C2×C4 C2×C4 C2×C4 C23 C2 # reps 1 4 1 1 1 2 2 4 4 2 4 2 8 4 8 8 4 8

Matrix representation of C4.89(C2×D20) in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 9 0 0 0 0 9
,
 1 0 0 0 0 1 0 0 0 0 9 2 0 0 1 32
,
 21 17 0 0 3 18 0 0 0 0 38 0 0 0 27 3
,
 6 38 0 0 26 35 0 0 0 0 38 0 0 0 0 38
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,9,1,0,0,2,32],[21,3,0,0,17,18,0,0,0,0,38,27,0,0,0,3],[6,26,0,0,38,35,0,0,0,0,38,0,0,0,0,38] >;

C4.89(C2×D20) in GAP, Magma, Sage, TeX

C_4._{89}(C_2\times D_{20})
% in TeX

G:=Group("C4.89(C2xD20)");
// GroupNames label

G:=SmallGroup(320,756);
// by ID

G=gap.SmallGroup(320,756);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,422,387,58,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=1,c^20=a^2,d^2=a,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=a^-1*c^19>;
// generators/relations

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