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G = Dic20.C4order 320 = 26·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q16.2F5, Dic5.3D8, Dic20.1C4, D10.14SD16, C8.4(C2×F5), C40.4(C2×C4), C5⋊(C8.17D4), (C4×D5).27D4, C52C8.18D4, (C5×Q16).1C4, (D5×Q16).2C2, C4.8(C22⋊F5), C8.F5.1C2, C40.C4.1C2, C20.8(C22⋊C4), (C8×D5).12C22, C2.13(D20⋊C4), C10.12(D4⋊C4), SmallGroup(320,248)

Series: Derived Chief Lower central Upper central

C1C40 — Dic20.C4
C1C5C10C20C4×D5C8×D5C40.C4 — Dic20.C4
C5C10C20C40 — Dic20.C4
C1C2C4C8Q16

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 >

10C2
4C4
5C22
5C4
20C4
2D5
2Q8
5C8
5C2×C4
10Q8
20Q8
20C2×C4
20C8
4C20
4Dic5
5Q16
5C2×C8
10C2×Q8
10Q16
10M4(2)
10C16
2C5×Q8
2Dic10
4C5⋊C8
4Dic10
4C4×D5
5C2×Q16
5C8.C4
5M5(2)
2Q8×D5
2C5⋊Q16
2C5⋊C16
2C4.F5
5C8.17D4

Character table of Dic20.C4

 class 12A2B4A4B4C4D58A8B8C8D8E1016A16B16C16D20A20B20C40A40B
 size 111028104044101040404202020208161688
ρ111111111111111111111111    trivial
ρ21111-11-11111111-1-1-1-11-1-111    linear of order 2
ρ311111111111-1-11-1-1-1-111111    linear of order 2
ρ41111-11-11111-1-1111111-1-111    linear of order 2
ρ511-11-1-1111-1-1i-i1-iii-i1-1-111    linear of order 4
ρ611-11-1-1111-1-1-ii1i-i-ii1-1-111    linear of order 4
ρ711-111-1-111-1-1-ii1-iii-i11111    linear of order 4
ρ811-111-1-111-1-1i-i1i-i-ii11111    linear of order 4
ρ922220202-2-2-20020000200-2-2    orthogonal lifted from D4
ρ1022-220-202-2220020000200-2-2    orthogonal lifted from D4
ρ1122-2-202020000022-22-2-20000    orthogonal lifted from D8
ρ1222-2-20202000002-22-22-20000    orthogonal lifted from D8
ρ13222-20-202000002--2--2-2-2-20000    complex lifted from SD16
ρ14222-20-202000002-2-2--2--2-20000    complex lifted from SD16
ρ154404-400-140000-10000-111-1-1    orthogonal lifted from C2×F5
ρ164404400-140000-10000-1-1-1-1-1    orthogonal lifted from F5
ρ174404000-1-40000-10000-15-511    orthogonal lifted from C22⋊F5
ρ184404000-1-40000-10000-1-5511    orthogonal lifted from C22⋊F5
ρ194-4000004022-2200-4000000000    symplectic lifted from C8.17D4, Schur index 2
ρ204-40000040-222200-4000000000    symplectic lifted from C8.17D4, Schur index 2
ρ21880-8000-200000-2000020000    orthogonal lifted from D20⋊C4, Schur index 2
ρ228-800000-20000020000000-1010    symplectic faithful, Schur index 2
ρ238-800000-2000002000000010-10    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)

01000000
00100000
00010000
2402402402400000
0000111100
00002301100
00001752380219
00001618311219
,
2400000000
11110000
0002400000
0024000000
000021921000
00002102200
000016121841198
000018352140200
,
228021210000
212102280000
22020722000000
133434130000
00003620510
00001021391239
0000207330169
0000111129066

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

Subgroup lattice of Dic20.C4 in TeX
Character table of Dic20.C4 in TeX

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