Copied to
clipboard

## 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, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C24, C20, C20, C20, D10, C2×C10, C2×C10, D4⋊C4, C2×C4⋊C4, C22×C8, C22×D4, C52C8, D20, D20, C2×C20, C2×C20, C2×C20, C22×D5, C22×C10, C2×D4⋊C4, C2×C52C8, C2×C52C8, C5×C4⋊C4, C5×C4⋊C4, C2×D20, C2×D20, C22×C20, C22×C20, C23×D5, D206C4, C22×C52C8, C10×C4⋊C4, C22×D20, C2×D206C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, D8, SD16, C22×C4, C2×D4, D10, D4⋊C4, C2×C22⋊C4, C2×D8, C2×SD16, C4×D5, D20, C5⋊D4, C22×D5, C2×D4⋊C4, D10⋊C4, D4⋊D5, Q8⋊D5, C2×C4×D5, C2×D20, C2×C5⋊D4, D206C4, C2×D10⋊C4, C2×D4⋊D5, C2×Q8⋊D5, C2×D206C4

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

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

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

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

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