Copied to
clipboard

G = D40:16C4order 320 = 26·5

10th semidirect product of D40 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D40:16C4, Dic20:16C4, M4(2).26D10, C40:C2:8C4, C40:8C4:1C2, C8.11(C4xD5), C5:6(C8.26D4), C40.70(C2xC4), (C2xC8).69D10, C4.212(D4xD5), C10.85(C4xD4), C8.C4:4D5, C5:2C8.51D4, D20:7C4:7C2, D20.24(C2xC4), C20.371(C2xD4), D40:7C2.4C2, (C2xC40).41C22, D20.2C4:10C2, (C2xC20).310C23, C20.113(C22xC4), Dic10.25(C2xC4), C4oD20.17C22, C2.15(D20:8C4), C22.1(Q8:2D5), (C4xDic5).46C22, (C5xM4(2)).20C22, C4.46(C2xC4xD5), (C5xC8.C4):4C2, (C2xC10).1(C4oD4), (C2xC5:2C8).78C22, (C2xC4).413(C22xD5), SmallGroup(320,521)

Series: Derived Chief Lower central Upper central

C1C20 — D40:16C4
C1C5C10C20C2xC20C4oD20D40:7C2 — D40:16C4
C5C10C20 — D40:16C4
C1C4C2xC4C8.C4

Generators and relations for D40:16C4
 G = < a,b,c | a40=b2=c4=1, bab=a-1, cac-1=a29, cbc-1=a18b >

Subgroups: 390 in 104 conjugacy classes, 45 normal (27 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, C8:C4, C4wrC2, C8.C4, C8oD4, C4oD8, C5:2C8, C40, C40, Dic10, C4xD5, D20, C2xDic5, C5:D4, C2xC20, C8.26D4, C8xD5, C8:D5, C40:C2, D40, Dic20, C2xC5:2C8, C4xDic5, C2xC40, C5xM4(2), C4oD20, C40:8C4, D20:7C4, C5xC8.C4, D40:7C2, D20.2C4, D40:16C4
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, D5, C22xC4, C2xD4, C4oD4, D10, C4xD4, C4xD5, C22xD5, C8.26D4, C2xC4xD5, D4xD5, Q8:2D5, D20:8C4, D40:16C4

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

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

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

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

50 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G5A5B8A···8F8G8H8I8J10A10B10C10D20A20B20C20D20E20F40A···40H40I···40P
order122224444444558···888881010101020202020202040···4040···40
size112202011220202020224···41010101022442222444···48···8

50 irreducible representations

dim1111111112222224444
type++++++++++++
imageC1C2C2C2C2C2C4C4C4D4D5C4oD4D10D10C4xD5C8.26D4D4xD5Q8:2D5D40:16C4
kernelD40:16C4C40:8C4D20:7C4C5xC8.C4D40:7C2D20.2C4C40:C2D40Dic20C5:2C8C8.C4C2xC10C2xC8M4(2)C8C5C4C22C1
# reps1121124222222482228

Matrix representation of D40:16C4 in GL6(F41)

660000
3510000
0031010
00174001
00140100
00238241
,
660000
1350000
0004000
0040000
00302209
00717320
,
900000
13320000
0032000
000100
0025090
00132040

G:=sub<GL(6,GF(41))| [6,35,0,0,0,0,6,1,0,0,0,0,0,0,31,17,14,23,0,0,0,40,0,8,0,0,1,0,10,24,0,0,0,1,0,1],[6,1,0,0,0,0,6,35,0,0,0,0,0,0,0,40,30,7,0,0,40,0,22,17,0,0,0,0,0,32,0,0,0,0,9,0],[9,13,0,0,0,0,0,32,0,0,0,0,0,0,32,0,25,13,0,0,0,1,0,2,0,0,0,0,9,0,0,0,0,0,0,40] >;

D40:16C4 in GAP, Magma, Sage, TeX

D_{40}\rtimes_{16}C_4
% in TeX

G:=Group("D40:16C4");
// GroupNames label

G:=SmallGroup(320,521);
// by ID

G=gap.SmallGroup(320,521);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,219,58,136,1684,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^29,c*b*c^-1=a^18*b>;
// generators/relations

׿
x
:
Z
F
o
wr
Q
<