Copied to
clipboard

## G = C2×C20.48D4order 320 = 26·5

### Direct product of C2 and C20.48D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C2×C20.48D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C22×Dic10 — C2×C20.48D4
 Lower central C5 — C2×C10 — C2×C20.48D4
 Upper central C1 — C23 — C23×C4

Generators and relations for C2×C20.48D4
G = < a,b,c,d | a2=b20=c4=1, d2=b10, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b10c-1 >

Subgroups: 878 in 322 conjugacy classes, 143 normal (23 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×10], C22 [×12], C5, C2×C4 [×8], C2×C4 [×26], Q8 [×8], C23, C23 [×6], C23 [×4], C10 [×3], C10 [×4], C10 [×4], C22⋊C4 [×8], C4⋊C4 [×12], C22×C4 [×2], C22×C4 [×4], C22×C4 [×8], C2×Q8 [×8], C24, Dic5 [×8], C20 [×4], C20 [×2], C2×C10, C2×C10 [×10], C2×C10 [×12], C2×C22⋊C4 [×2], C2×C4⋊C4 [×3], C22⋊Q8 [×8], C23×C4, C22×Q8, Dic10 [×8], C2×Dic5 [×8], C2×Dic5 [×8], C2×C20 [×8], C2×C20 [×10], C22×C10, C22×C10 [×6], C22×C10 [×4], C2×C22⋊Q8, C10.D4 [×8], C4⋊Dic5 [×4], C23.D5 [×8], C2×Dic10 [×4], C2×Dic10 [×4], C22×Dic5 [×4], C22×C20 [×2], C22×C20 [×4], C22×C20 [×4], C23×C10, C2×C10.D4 [×2], C20.48D4 [×8], C2×C4⋊Dic5, C2×C23.D5 [×2], C22×Dic10, C23×C20, C2×C20.48D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], D5, C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, D10 [×7], C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4, Dic10 [×4], C5⋊D4 [×4], C22×D5 [×7], C2×C22⋊Q8, C2×Dic10 [×6], C4○D20 [×2], C2×C5⋊D4 [×6], C23×D5, C20.48D4 [×4], C22×Dic10, C2×C4○D20, C22×C5⋊D4, C2×C20.48D4

Smallest permutation representation of C2×C20.48D4
On 160 points
Generators in S160
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 21)(14 22)(15 23)(16 24)(17 25)(18 26)(19 27)(20 28)(41 73)(42 74)(43 75)(44 76)(45 77)(46 78)(47 79)(48 80)(49 61)(50 62)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(81 139)(82 140)(83 121)(84 122)(85 123)(86 124)(87 125)(88 126)(89 127)(90 128)(91 129)(92 130)(93 131)(94 132)(95 133)(96 134)(97 135)(98 136)(99 137)(100 138)(101 149)(102 150)(103 151)(104 152)(105 153)(106 154)(107 155)(108 156)(109 157)(110 158)(111 159)(112 160)(113 141)(114 142)(115 143)(116 144)(117 145)(118 146)(119 147)(120 148)
(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 68 159 93)(2 67 160 92)(3 66 141 91)(4 65 142 90)(5 64 143 89)(6 63 144 88)(7 62 145 87)(8 61 146 86)(9 80 147 85)(10 79 148 84)(11 78 149 83)(12 77 150 82)(13 76 151 81)(14 75 152 100)(15 74 153 99)(16 73 154 98)(17 72 155 97)(18 71 156 96)(19 70 157 95)(20 69 158 94)(21 44 103 139)(22 43 104 138)(23 42 105 137)(24 41 106 136)(25 60 107 135)(26 59 108 134)(27 58 109 133)(28 57 110 132)(29 56 111 131)(30 55 112 130)(31 54 113 129)(32 53 114 128)(33 52 115 127)(34 51 116 126)(35 50 117 125)(36 49 118 124)(37 48 119 123)(38 47 120 122)(39 46 101 121)(40 45 102 140)
(1 83 11 93)(2 82 12 92)(3 81 13 91)(4 100 14 90)(5 99 15 89)(6 98 16 88)(7 97 17 87)(8 96 18 86)(9 95 19 85)(10 94 20 84)(21 129 31 139)(22 128 32 138)(23 127 33 137)(24 126 34 136)(25 125 35 135)(26 124 36 134)(27 123 37 133)(28 122 38 132)(29 121 39 131)(30 140 40 130)(41 106 51 116)(42 105 52 115)(43 104 53 114)(44 103 54 113)(45 102 55 112)(46 101 56 111)(47 120 57 110)(48 119 58 109)(49 118 59 108)(50 117 60 107)(61 146 71 156)(62 145 72 155)(63 144 73 154)(64 143 74 153)(65 142 75 152)(66 141 76 151)(67 160 77 150)(68 159 78 149)(69 158 79 148)(70 157 80 147)

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

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

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

92 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I ··· 4P 5A 5B 10A ··· 10AD 20A ··· 20AF order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 2 2 2 2 ··· 2 20 ··· 20 2 2 2 ··· 2 2 ··· 2

92 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + - + + + - image C1 C2 C2 C2 C2 C2 C2 D4 Q8 D5 C4○D4 D10 D10 C5⋊D4 Dic10 C4○D20 kernel C2×C20.48D4 C2×C10.D4 C20.48D4 C2×C4⋊Dic5 C2×C23.D5 C22×Dic10 C23×C20 C2×C20 C22×C10 C23×C4 C2×C10 C22×C4 C24 C2×C4 C23 C22 # reps 1 2 8 1 2 1 1 4 4 2 4 12 2 16 16 16

Matrix representation of C2×C20.48D4 in GL5(𝔽41)

 40 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 40 0 0 0 0 0 40
,
 40 0 0 0 0 0 4 0 0 0 0 0 31 0 0 0 0 0 0 1 0 0 0 40 0
,
 40 0 0 0 0 0 0 1 0 0 0 40 0 0 0 0 0 0 0 32 0 0 0 32 0
,
 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 32 0 0 0 32 0

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,4,0,0,0,0,0,31,0,0,0,0,0,0,40,0,0,0,1,0],[40,0,0,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,32,0],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,32,0] >;

C2×C20.48D4 in GAP, Magma, Sage, TeX

C_2\times C_{20}._{48}D_4
% in TeX

G:=Group("C2xC20.48D4");
// GroupNames label

G:=SmallGroup(320,1456);
// by ID

G=gap.SmallGroup(320,1456);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,758,184,675,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^4=1,d^2=b^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^10*c^-1>;
// generators/relations

׿
×
𝔽