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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(C2×F5), C40.2(C2×C4), C5⋊(M5(2)⋊C2), (C5×D8).1C4, (D5×D8).2C2, (C4×D5).23D4, C52C8.14D4, C40.C41C2, C8.F52C2, C4.4(C22⋊F5), C20.4(C22⋊C4), C2.9(D20⋊C4), (C8×D5).10C22, C10.8(D4⋊C4), SmallGroup(320,244)

Series: Derived Chief Lower central Upper central

C1C40 — D40.C4
C1C5C10C20C4×D5C8×D5C40.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 [×3], C4, C4, C22 [×5], C5, C8, C8 [×2], C2×C4, D4 [×3], C23, D5 [×2], C10, C10, C16, C2×C8, M4(2), D8, D8 [×2], C2×D4, Dic5, C20, D10, D10 [×3], C2×C10, C8.C4, M5(2), C2×D8, C52C8, C40, C5⋊C8, C4×D5, D20, C5⋊D4, C5×D4, C22×D5, M5(2)⋊C2, C5⋊C16, C8×D5, D40, D4⋊D5, C5×D8, C4.F5, D4×D5, C8.F5, C40.C4, D5×D8, D40.C4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, D8, SD16, F5, D4⋊C4, C2×F5, 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 C2×F5
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 43)(44 80)(45 79)(46 78)(47 77)(48 76)(49 75)(50 74)(51 73)(52 72)(53 71)(54 70)(55 69)(56 68)(57 67)(58 66)(59 65)(60 64)(61 63)
(1 80 11 70 21 60 31 50)(2 67 20 73 22 47 40 53)(3 54 29 76 23 74 9 56)(4 41 38 79 24 61 18 59)(5 68 7 42 25 48 27 62)(6 55 16 45 26 75 36 65)(8 69 34 51 28 49 14 71)(10 43 12 57 30 63 32 77)(13 44 39 66 33 64 19 46)(15 58 17 72 35 78 37 52)

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

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

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

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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