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## G = C2×D20⋊6C4order 320 = 26·5

### Direct product of C2 and D20⋊6C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×D20⋊6C4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×D20 — C22×D20 — C2×D20⋊6C4
 Lower central C5 — C10 — C20 — C2×D20⋊6C4
 Upper central C1 — C23 — C22×C4 — C2×C4⋊C4

Generators and relations for C2×D206C4
G = < a,b,c,d | a2=b20=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b11, dcd-1=b15c >

Subgroups: 990 in 202 conjugacy classes, 79 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×6], C22 [×16], C5, C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], D4 [×10], C23, C23 [×10], D5 [×4], C10 [×3], C10 [×4], C4⋊C4 [×2], C4⋊C4, C2×C8 [×4], C22×C4, C22×C4, C2×D4 [×9], C24, C20 [×2], C20 [×2], C20 [×2], D10 [×16], C2×C10, C2×C10 [×6], D4⋊C4 [×4], C2×C4⋊C4, C22×C8, C22×D4, C52C8 [×2], D20 [×4], D20 [×6], C2×C20 [×2], C2×C20 [×4], C2×C20 [×4], C22×D5 [×10], C22×C10, C2×D4⋊C4, C2×C52C8 [×2], C2×C52C8 [×2], C5×C4⋊C4 [×2], C5×C4⋊C4, C2×D20 [×6], C2×D20 [×3], C22×C20, C22×C20, C23×D5, D206C4 [×4], C22×C52C8, C10×C4⋊C4, C22×D20, C2×D206C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], D10 [×3], D4⋊C4 [×4], C2×C22⋊C4, C2×D8, C2×SD16, C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C2×D4⋊C4, D10⋊C4 [×4], D4⋊D5 [×2], Q8⋊D5 [×2], C2×C4×D5, C2×D20, C2×C5⋊D4, D206C4 [×4], C2×D10⋊C4, C2×D4⋊D5, C2×Q8⋊D5, C2×D206C4

Smallest permutation representation of C2×D206C4
On 160 points
Generators in S160
(1 91)(2 92)(3 93)(4 94)(5 95)(6 96)(7 97)(8 98)(9 99)(10 100)(11 81)(12 82)(13 83)(14 84)(15 85)(16 86)(17 87)(18 88)(19 89)(20 90)(21 111)(22 112)(23 113)(24 114)(25 115)(26 116)(27 117)(28 118)(29 119)(30 120)(31 101)(32 102)(33 103)(34 104)(35 105)(36 106)(37 107)(38 108)(39 109)(40 110)(41 126)(42 127)(43 128)(44 129)(45 130)(46 131)(47 132)(48 133)(49 134)(50 135)(51 136)(52 137)(53 138)(54 139)(55 140)(56 121)(57 122)(58 123)(59 124)(60 125)(61 151)(62 152)(63 153)(64 154)(65 155)(66 156)(67 157)(68 158)(69 159)(70 160)(71 141)(72 142)(73 143)(74 144)(75 145)(76 146)(77 147)(78 148)(79 149)(80 150)
(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)
(1 115)(2 114)(3 113)(4 112)(5 111)(6 110)(7 109)(8 108)(9 107)(10 106)(11 105)(12 104)(13 103)(14 102)(15 101)(16 120)(17 119)(18 118)(19 117)(20 116)(21 95)(22 94)(23 93)(24 92)(25 91)(26 90)(27 89)(28 88)(29 87)(30 86)(31 85)(32 84)(33 83)(34 82)(35 81)(36 100)(37 99)(38 98)(39 97)(40 96)(41 160)(42 159)(43 158)(44 157)(45 156)(46 155)(47 154)(48 153)(49 152)(50 151)(51 150)(52 149)(53 148)(54 147)(55 146)(56 145)(57 144)(58 143)(59 142)(60 141)(61 135)(62 134)(63 133)(64 132)(65 131)(66 130)(67 129)(68 128)(69 127)(70 126)(71 125)(72 124)(73 123)(74 122)(75 121)(76 140)(77 139)(78 138)(79 137)(80 136)
(1 141 36 131)(2 152 37 122)(3 143 38 133)(4 154 39 124)(5 145 40 135)(6 156 21 126)(7 147 22 137)(8 158 23 128)(9 149 24 139)(10 160 25 130)(11 151 26 121)(12 142 27 132)(13 153 28 123)(14 144 29 134)(15 155 30 125)(16 146 31 136)(17 157 32 127)(18 148 33 138)(19 159 34 129)(20 150 35 140)(41 96 66 111)(42 87 67 102)(43 98 68 113)(44 89 69 104)(45 100 70 115)(46 91 71 106)(47 82 72 117)(48 93 73 108)(49 84 74 119)(50 95 75 110)(51 86 76 101)(52 97 77 112)(53 88 78 103)(54 99 79 114)(55 90 80 105)(56 81 61 116)(57 92 62 107)(58 83 63 118)(59 94 64 109)(60 85 65 120)

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

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

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

68 conjugacy classes

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

68 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 D4 D4 D5 D8 SD16 D10 D10 C4×D5 D20 C5⋊D4 C5⋊D4 D4⋊D5 Q8⋊D5 kernel C2×D20⋊6C4 D20⋊6C4 C22×C5⋊2C8 C10×C4⋊C4 C22×D20 C2×D20 C2×C20 C22×C10 C2×C4⋊C4 C2×C10 C2×C10 C4⋊C4 C22×C4 C2×C4 C2×C4 C2×C4 C23 C22 C22 # reps 1 4 1 1 1 8 3 1 2 4 4 4 2 8 8 4 4 4 4

Matrix representation of C2×D206C4 in GL5(𝔽41)

 40 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 39 0 0 0 1 40 0 0 0 0 0 0 1 0 0 0 40 35
,
 40 0 0 0 0 0 1 39 0 0 0 0 40 0 0 0 0 0 35 40 0 0 0 35 6
,
 32 0 0 0 0 0 30 11 0 0 0 15 11 0 0 0 0 0 39 13 0 0 0 28 2

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,1,0,0,0,39,40,0,0,0,0,0,0,40,0,0,0,1,35],[40,0,0,0,0,0,1,0,0,0,0,39,40,0,0,0,0,0,35,35,0,0,0,40,6],[32,0,0,0,0,0,30,15,0,0,0,11,11,0,0,0,0,0,39,28,0,0,0,13,2] >;

C2×D206C4 in GAP, Magma, Sage, TeX

C_2\times D_{20}\rtimes_6C_4
% in TeX

G:=Group("C2xD20:6C4");
// GroupNames label

G:=SmallGroup(320,592);
// by ID

G=gap.SmallGroup(320,592);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,422,58,1684,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^11,d*c*d^-1=b^15*c>;
// generators/relations

׿
×
𝔽