Copied to
clipboard

## G = Dic20.C4order 320 = 26·5

### 1st non-split extension by Dic20 of C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — Dic20.C4
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C8×D5 — C40.C4 — Dic20.C4
 Lower central C5 — C10 — C20 — C40 — Dic20.C4
 Upper central C1 — C2 — C4 — C8 — Q16

Generators and relations for Dic20.C4
G = < a,b,c | a40=1, b2=c4=a20, bab-1=a-1, cac-1=a3, cbc-1=a35b >

Character table of Dic20.C4

 class 1 2A 2B 4A 4B 4C 4D 5 8A 8B 8C 8D 8E 10 16A 16B 16C 16D 20A 20B 20C 40A 40B size 1 1 10 2 8 10 40 4 4 10 10 40 40 4 20 20 20 20 8 16 16 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 1 1 1 linear of order 2 ρ4 1 1 1 1 -1 1 -1 1 1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 1 1 linear of order 2 ρ5 1 1 -1 1 -1 -1 1 1 1 -1 -1 i -i 1 -i i i -i 1 -1 -1 1 1 linear of order 4 ρ6 1 1 -1 1 -1 -1 1 1 1 -1 -1 -i i 1 i -i -i i 1 -1 -1 1 1 linear of order 4 ρ7 1 1 -1 1 1 -1 -1 1 1 -1 -1 -i i 1 -i i i -i 1 1 1 1 1 linear of order 4 ρ8 1 1 -1 1 1 -1 -1 1 1 -1 -1 i -i 1 i -i -i i 1 1 1 1 1 linear of order 4 ρ9 2 2 2 2 0 2 0 2 -2 -2 -2 0 0 2 0 0 0 0 2 0 0 -2 -2 orthogonal lifted from D4 ρ10 2 2 -2 2 0 -2 0 2 -2 2 2 0 0 2 0 0 0 0 2 0 0 -2 -2 orthogonal lifted from D4 ρ11 2 2 -2 -2 0 2 0 2 0 0 0 0 0 2 √2 -√2 √2 -√2 -2 0 0 0 0 orthogonal lifted from D8 ρ12 2 2 -2 -2 0 2 0 2 0 0 0 0 0 2 -√2 √2 -√2 √2 -2 0 0 0 0 orthogonal lifted from D8 ρ13 2 2 2 -2 0 -2 0 2 0 0 0 0 0 2 -√-2 -√-2 √-2 √-2 -2 0 0 0 0 complex lifted from SD16 ρ14 2 2 2 -2 0 -2 0 2 0 0 0 0 0 2 √-2 √-2 -√-2 -√-2 -2 0 0 0 0 complex lifted from SD16 ρ15 4 4 0 4 -4 0 0 -1 4 0 0 0 0 -1 0 0 0 0 -1 1 1 -1 -1 orthogonal lifted from C2×F5 ρ16 4 4 0 4 4 0 0 -1 4 0 0 0 0 -1 0 0 0 0 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ17 4 4 0 4 0 0 0 -1 -4 0 0 0 0 -1 0 0 0 0 -1 √5 -√5 1 1 orthogonal lifted from C22⋊F5 ρ18 4 4 0 4 0 0 0 -1 -4 0 0 0 0 -1 0 0 0 0 -1 -√5 √5 1 1 orthogonal lifted from C22⋊F5 ρ19 4 -4 0 0 0 0 0 4 0 2√2 -2√2 0 0 -4 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.17D4, Schur index 2 ρ20 4 -4 0 0 0 0 0 4 0 -2√2 2√2 0 0 -4 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.17D4, Schur index 2 ρ21 8 8 0 -8 0 0 0 -2 0 0 0 0 0 -2 0 0 0 0 2 0 0 0 0 orthogonal lifted from D20⋊C4, Schur index 2 ρ22 8 -8 0 0 0 0 0 -2 0 0 0 0 0 2 0 0 0 0 0 0 0 -√10 √10 symplectic faithful, Schur index 2 ρ23 8 -8 0 0 0 0 0 -2 0 0 0 0 0 2 0 0 0 0 0 0 0 √10 -√10 symplectic faithful, Schur index 2

Smallest permutation representation of Dic20.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)
(1 70 21 50)(2 69 22 49)(3 68 23 48)(4 67 24 47)(5 66 25 46)(6 65 26 45)(7 64 27 44)(8 63 28 43)(9 62 29 42)(10 61 30 41)(11 60 31 80)(12 59 32 79)(13 58 33 78)(14 57 34 77)(15 56 35 76)(16 55 36 75)(17 54 37 74)(18 53 38 73)(19 52 39 72)(20 51 40 71)(81 160 101 140)(82 159 102 139)(83 158 103 138)(84 157 104 137)(85 156 105 136)(86 155 106 135)(87 154 107 134)(88 153 108 133)(89 152 109 132)(90 151 110 131)(91 150 111 130)(92 149 112 129)(93 148 113 128)(94 147 114 127)(95 146 115 126)(96 145 116 125)(97 144 117 124)(98 143 118 123)(99 142 119 122)(100 141 120 121)
(1 109 31 119 21 89 11 99)(2 96 40 82 22 116 20 102)(3 83 9 85 23 103 29 105)(4 110 18 88 24 90 38 108)(5 97 27 91 25 117 7 111)(6 84 36 94 26 104 16 114)(8 98 14 100 28 118 34 120)(10 112 32 106 30 92 12 86)(13 113 19 115 33 93 39 95)(15 87 37 81 35 107 17 101)(41 134 79 160 61 154 59 140)(42 121 48 123 62 141 68 143)(43 148 57 126 63 128 77 146)(44 135 66 129 64 155 46 149)(45 122 75 132 65 142 55 152)(47 136 53 138 67 156 73 158)(49 150 71 144 69 130 51 124)(50 137 80 147 70 157 60 127)(52 151 58 153 72 131 78 133)(54 125 76 159 74 145 56 139)

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

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

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

Matrix representation of Dic20.C4 in GL8(𝔽241)

 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 240 240 240 240 0 0 0 0 0 0 0 0 11 11 0 0 0 0 0 0 230 11 0 0 0 0 0 0 175 238 0 219 0 0 0 0 161 83 11 219
,
 240 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 0 219 210 0 0 0 0 0 0 210 22 0 0 0 0 0 0 161 218 41 198 0 0 0 0 183 52 140 200
,
 228 0 21 21 0 0 0 0 21 21 0 228 0 0 0 0 220 207 220 0 0 0 0 0 13 34 34 13 0 0 0 0 0 0 0 0 36 205 1 0 0 0 0 0 102 139 1 239 0 0 0 0 207 33 0 169 0 0 0 0 111 129 0 66

G:=sub<GL(8,GF(241))| [0,0,0,240,0,0,0,0,1,0,0,240,0,0,0,0,0,1,0,240,0,0,0,0,0,0,1,240,0,0,0,0,0,0,0,0,11,230,175,161,0,0,0,0,11,11,238,83,0,0,0,0,0,0,0,11,0,0,0,0,0,0,219,219],[240,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,240,0,0,0,0,0,1,240,0,0,0,0,0,0,0,0,0,219,210,161,183,0,0,0,0,210,22,218,52,0,0,0,0,0,0,41,140,0,0,0,0,0,0,198,200],[228,21,220,13,0,0,0,0,0,21,207,34,0,0,0,0,21,0,220,34,0,0,0,0,21,228,0,13,0,0,0,0,0,0,0,0,36,102,207,111,0,0,0,0,205,139,33,129,0,0,0,0,1,1,0,0,0,0,0,0,0,239,169,66] >;

Dic20.C4 in GAP, Magma, Sage, TeX

{\rm Dic}_{20}.C_4
% in TeX

G:=Group("Dic20.C4");
// GroupNames label

G:=SmallGroup(320,248);
// by ID

G=gap.SmallGroup(320,248);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,387,184,675,794,80,1684,851,102,6278,3156]);
// Polycyclic

G:=Group<a,b,c|a^40=1,b^2=c^4=a^20,b*a*b^-1=a^-1,c*a*c^-1=a^3,c*b*c^-1=a^35*b>;
// generators/relations

Export

׿
×
𝔽