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

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

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

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

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

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