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G = D40.C4order 320 = 26·5

1st non-split extension by D40 of C4 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D8.2F5, D40.1C4, Dic5.2D8, D10.13SD16, C8.2(C2xF5), C40.2(C2xC4), C5:(M5(2):C2), (C5xD8).1C4, (D5xD8).2C2, (C4xD5).23D4, C5:2C8.14D4, C40.C4:1C2, C8.F5:2C2, C4.4(C22:F5), C20.4(C22:C4), C2.9(D20:C4), (C8xD5).10C22, C10.8(D4:C4), SmallGroup(320,244)

Series: Derived Chief Lower central Upper central

C1C40 — D40.C4
C1C5C10C20C4xD5C8xD5C40.C4 — D40.C4
C5C10C20C40 — D40.C4
C1C2C4C8D8

Generators and relations for D40.C4
 G = < a,b,c | a40=b2=1, c4=a20, bab=a-1, cac-1=a3, cbc-1=a37b >

Subgroups: 410 in 62 conjugacy classes, 20 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2xC4, D4, C23, D5, C10, C10, C16, C2xC8, M4(2), D8, D8, C2xD4, Dic5, C20, D10, D10, C2xC10, C8.C4, M5(2), C2xD8, C5:2C8, C40, C5:C8, C4xD5, D20, C5:D4, C5xD4, C22xD5, M5(2):C2, C5:C16, C8xD5, D40, D4:D5, C5xD8, C4.F5, D4xD5, C8.F5, C40.C4, D5xD8, D40.C4
Quotients: C1, C2, C4, C22, C2xC4, D4, C22:C4, D8, SD16, F5, D4:C4, C2xF5, M5(2):C2, C22:F5, D20:C4, D40.C4

Character table of D40.C4

 class 12A2B2C2D4A4B58A8B8C8D8E10A10B10C16A16B16C16D2040A40B
 size 118104021044101040404161620202020888
ρ111111111111111111111111    trivial
ρ211-11-1111111111-1-1-1-1-1-1111    linear of order 2
ρ311111111111-1-1111-1-1-1-1111    linear of order 2
ρ411-11-1111111-1-11-1-11111111    linear of order 2
ρ5111-1-11-111-1-1-ii111-iii-i111    linear of order 4
ρ611-1-111-111-1-1-ii1-1-1i-i-ii111    linear of order 4
ρ7111-1-11-111-1-1i-i111i-i-ii111    linear of order 4
ρ811-1-111-111-1-1i-i1-1-1-iii-i111    linear of order 4
ρ922020222-2-2-20020000002-2-2    orthogonal lifted from D4
ρ10220-202-22-2220020000002-2-2    orthogonal lifted from D4
ρ11220-20-222000002002-22-2-200    orthogonal lifted from D8
ρ12220-20-22200000200-22-22-200    orthogonal lifted from D8
ρ1322020-2-2200000200-2-2--2--2-200    complex lifted from SD16
ρ1422020-2-2200000200--2--2-2-2-200    complex lifted from SD16
ρ154440040-140000-1-1-10000-1-1-1    orthogonal lifted from F5
ρ1644-40040-140000-1110000-1-1-1    orthogonal lifted from C2xF5
ρ174400040-1-40000-1-550000-111    orthogonal lifted from C22:F5
ρ184400040-1-40000-15-50000-111    orthogonal lifted from C22:F5
ρ194-4000004022-2200-4000000000    orthogonal lifted from M5(2):C2
ρ204-40000040-222200-4000000000    orthogonal lifted from M5(2):C2
ρ2188000-80-200000-2000000200    orthogonal lifted from D20:C4, Schur index 2
ρ228-800000-20000020000000-1010    orthogonal faithful, Schur index 2
ρ238-800000-2000002000000010-10    orthogonal faithful, Schur index 2

Smallest permutation representation of D40.C4
On 80 points
Generators in S80
(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)
(1 10)(2 9)(3 8)(4 7)(5 6)(11 40)(12 39)(13 38)(14 37)(15 36)(16 35)(17 34)(18 33)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(25 26)(41 77)(42 76)(43 75)(44 74)(45 73)(46 72)(47 71)(48 70)(49 69)(50 68)(51 67)(52 66)(53 65)(54 64)(55 63)(56 62)(57 61)(58 60)(78 80)
(1 57 11 47 21 77 31 67)(2 44 20 50 22 64 40 70)(3 71 29 53 23 51 9 73)(4 58 38 56 24 78 18 76)(5 45 7 59 25 65 27 79)(6 72 16 62 26 52 36 42)(8 46 34 68 28 66 14 48)(10 60 12 74 30 80 32 54)(13 61 39 43 33 41 19 63)(15 75 17 49 35 55 37 69)

G:=sub<Sym(80)| (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(41,77)(42,76)(43,75)(44,74)(45,73)(46,72)(47,71)(48,70)(49,69)(50,68)(51,67)(52,66)(53,65)(54,64)(55,63)(56,62)(57,61)(58,60)(78,80), (1,57,11,47,21,77,31,67)(2,44,20,50,22,64,40,70)(3,71,29,53,23,51,9,73)(4,58,38,56,24,78,18,76)(5,45,7,59,25,65,27,79)(6,72,16,62,26,52,36,42)(8,46,34,68,28,66,14,48)(10,60,12,74,30,80,32,54)(13,61,39,43,33,41,19,63)(15,75,17,49,35,55,37,69)>;

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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(41,77)(42,76)(43,75)(44,74)(45,73)(46,72)(47,71)(48,70)(49,69)(50,68)(51,67)(52,66)(53,65)(54,64)(55,63)(56,62)(57,61)(58,60)(78,80), (1,57,11,47,21,77,31,67)(2,44,20,50,22,64,40,70)(3,71,29,53,23,51,9,73)(4,58,38,56,24,78,18,76)(5,45,7,59,25,65,27,79)(6,72,16,62,26,52,36,42)(8,46,34,68,28,66,14,48)(10,60,12,74,30,80,32,54)(13,61,39,43,33,41,19,63)(15,75,17,49,35,55,37,69) );

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)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,40),(12,39),(13,38),(14,37),(15,36),(16,35),(17,34),(18,33),(19,32),(20,31),(21,30),(22,29),(23,28),(24,27),(25,26),(41,77),(42,76),(43,75),(44,74),(45,73),(46,72),(47,71),(48,70),(49,69),(50,68),(51,67),(52,66),(53,65),(54,64),(55,63),(56,62),(57,61),(58,60),(78,80)], [(1,57,11,47,21,77,31,67),(2,44,20,50,22,64,40,70),(3,71,29,53,23,51,9,73),(4,58,38,56,24,78,18,76),(5,45,7,59,25,65,27,79),(6,72,16,62,26,52,36,42),(8,46,34,68,28,66,14,48),(10,60,12,74,30,80,32,54),(13,61,39,43,33,41,19,63),(15,75,17,49,35,55,37,69)]])

Matrix representation of D40.C4 in GL8(F241)

24052000000
18952000000
01412401900000
100141511900000
000007100
00001122200
00000021971
0000001120
,
10000000
52240000000
00100000
14101902400000
0000017000
0000112000
0000001125
0000000240
,
222236166310000
222016600000
14371950000
14301900000
00000010
00000001
000024011600
000054100

G:=sub<GL(8,GF(241))| [240,189,0,100,0,0,0,0,52,52,141,141,0,0,0,0,0,0,240,51,0,0,0,0,0,0,190,190,0,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,71,22,0,0,0,0,0,0,0,0,219,112,0,0,0,0,0,0,71,0],[1,52,0,141,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,190,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,170,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,125,240],[222,222,143,143,0,0,0,0,236,0,7,0,0,0,0,0,166,166,19,19,0,0,0,0,31,0,5,0,0,0,0,0,0,0,0,0,0,0,240,54,0,0,0,0,0,0,116,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;

D40.C4 in GAP, Magma, Sage, TeX

D_{40}.C_4
% in TeX

G:=Group("D40.C4");
// GroupNames label

G:=SmallGroup(320,244);
// by ID

G=gap.SmallGroup(320,244);
# by ID

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

G:=Group<a,b,c|a^40=b^2=1,c^4=a^20,b*a*b=a^-1,c*a*c^-1=a^3,c*b*c^-1=a^37*b>;
// generators/relations

Export

Character table of D40.C4 in TeX

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