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G = D40:13C4order 320 = 26·5

7th semidirect product of D40 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D40:13C4, Dic20:13C4, M4(2).27D10, C5:6(C8oD8), C8.16(C4xD5), C40:C2:10C4, C40.76(C2xC4), (C8xDic5):1C2, C10.86(C4xD4), C4.213(D4xD5), C8.C4:8D5, C5:2C8.56D4, D20:7C4:8C2, D20.25(C2xC4), C20.372(C2xD4), (C2xC8).252D10, D40:7C2.5C2, (C2xC40).42C22, D20.2C4:11C2, C20.114(C22xC4), (C2xC20).311C23, Dic10.26(C2xC4), C4oD20.18C22, C2.16(D20:8C4), C22.2(Q8:2D5), (C4xDic5).266C22, (C5xM4(2)).21C22, C4.47(C2xC4xD5), (C5xC8.C4):5C2, (C2xC10).2(C4oD4), (C2xC4).414(C22xD5), (C2xC5:2C8).246C22, SmallGroup(320,522)

Series: Derived Chief Lower central Upper central

C1C20 — D40:13C4
C1C5C10C20C2xC20C4oD20D40:7C2 — D40:13C4
C5C10C20 — D40:13C4
C1C4C2xC4C8.C4

Generators and relations for D40:13C4
 G = < a,b,c | a40=b2=c4=1, bab=a-1, cac-1=a9, cbc-1=a38b >

Subgroups: 390 in 106 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2xC4, C2xC4, D4, Q8, D5, C10, C10, C42, C2xC8, C2xC8, M4(2), M4(2), D8, SD16, Q16, C4oD4, Dic5, C20, D10, C2xC10, C4xC8, C4wrC2, C8.C4, C8oD4, C4oD8, C5:2C8, C40, C40, Dic10, C4xD5, D20, C2xDic5, C5:D4, C2xC20, C8oD8, C8xD5, C8:D5, C40:C2, D40, Dic20, C2xC5:2C8, C4xDic5, C2xC40, C5xM4(2), C4oD20, C8xDic5, D20:7C4, C5xC8.C4, D40:7C2, D20.2C4, D40:13C4
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, D5, C22xC4, C2xD4, C4oD4, D10, C4xD4, C4xD5, C22xD5, C8oD8, C2xC4xD5, D4xD5, Q8:2D5, D20:8C4, D40:13C4

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

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

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

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

56 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D8E8F8G8H8I8J8K8L8M8N10A10B10C10D20A20B20C20D20E20F40A···40H40I···40P
order1222244444444455888888888888881010101020202020202040···4040···40
size112202011210101010202022222244445555101022442222444···48···8

56 irreducible representations

dim1111111112222222444
type++++++++++++
imageC1C2C2C2C2C2C4C4C4D4D5C4oD4D10D10C4xD5C8oD8D4xD5Q8:2D5D40:13C4
kernelD40:13C4C8xDic5D20:7C4C5xC8.C4D40:7C2D20.2C4C40:C2D40Dic20C5:2C8C8.C4C2xC10C2xC8M4(2)C8C5C4C22C1
# reps1121124222222488228

Matrix representation of D40:13C4 in GL4(F41) generated by

7600
34000
00380
001327
,
7100
343400
00404
0001
,
343500
8700
0090
002340
G:=sub<GL(4,GF(41))| [7,34,0,0,6,0,0,0,0,0,38,13,0,0,0,27],[7,34,0,0,1,34,0,0,0,0,40,0,0,0,4,1],[34,8,0,0,35,7,0,0,0,0,9,23,0,0,0,40] >;

D40:13C4 in GAP, Magma, Sage, TeX

D_{40}\rtimes_{13}C_4
% in TeX

G:=Group("D40:13C4");
// GroupNames label

G:=SmallGroup(320,522);
// by ID

G=gap.SmallGroup(320,522);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,555,58,136,1684,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^38*b>;
// generators/relations

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