Copied to
clipboard

G = C10xD8order 160 = 25·5

Direct product of C10 and D8

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C10xD8, C20.41D4, C40:12C22, C20.44C23, (C2xC8):3C10, C8:2(C2xC10), C4.6(C5xD4), (C2xC40):11C2, D4:1(C2xC10), (C2xD4):4C10, (D4xC10):13C2, C10.74(C2xD4), C2.11(D4xC10), (C2xC10).52D4, (C5xD4):10C22, C4.1(C22xC10), C22.14(C5xD4), (C2xC20).129C22, (C2xC4).25(C2xC10), SmallGroup(160,193)

Series: Derived Chief Lower central Upper central

C1C4 — C10xD8
C1C2C4C20C5xD4C5xD8 — C10xD8
C1C2C4 — C10xD8
C1C2xC10C2xC20 — C10xD8

Generators and relations for C10xD8
 G = < a,b,c | a10=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 140 in 76 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C8, C2xC4, D4, D4, C23, C10, C10, C10, C2xC8, D8, C2xD4, C20, C2xC10, C2xC10, C2xD8, C40, C2xC20, C5xD4, C5xD4, C22xC10, C2xC40, C5xD8, D4xC10, C10xD8
Quotients: C1, C2, C22, C5, D4, C23, C10, D8, C2xD4, C2xC10, C2xD8, C5xD4, C22xC10, C5xD8, D4xC10, C10xD8

Smallest permutation representation of C10xD8
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 28 50 32 70 20 56 79)(2 29 41 33 61 11 57 80)(3 30 42 34 62 12 58 71)(4 21 43 35 63 13 59 72)(5 22 44 36 64 14 60 73)(6 23 45 37 65 15 51 74)(7 24 46 38 66 16 52 75)(8 25 47 39 67 17 53 76)(9 26 48 40 68 18 54 77)(10 27 49 31 69 19 55 78)
(1 79)(2 80)(3 71)(4 72)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 59)(22 60)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 69)(32 70)(33 61)(34 62)(35 63)(36 64)(37 65)(38 66)(39 67)(40 68)

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

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

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

C10xD8 is a maximal subgroup of
C10.D16  D8.Dic5  D8.D10  Dic5:D8  C40:5D4  D8:Dic5  (C2xD8).D5  C40:11D4  C40.22D4  D20:D4  C40:6D4  Dic10:D4  C40:12D4  C40.23D4  D8:13D10

70 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B5A5B5C5D8A8B8C8D10A···10L10M···10AB20A···20H40A···40P
order12222222445555888810···1010···1020···2040···40
size1111444422111122221···14···42···22···2

70 irreducible representations

dim11111111222222
type+++++++
imageC1C2C2C2C5C10C10C10D4D4D8C5xD4C5xD4C5xD8
kernelC10xD8C2xC40C5xD8D4xC10C2xD8C2xC8D8C2xD4C20C2xC10C10C4C22C2
# reps1142441681144416

Matrix representation of C10xD8 in GL3(F41) generated by

4000
0250
0025
,
4000
0029
02417
,
100
01712
01724
G:=sub<GL(3,GF(41))| [40,0,0,0,25,0,0,0,25],[40,0,0,0,0,24,0,29,17],[1,0,0,0,17,17,0,12,24] >;

C10xD8 in GAP, Magma, Sage, TeX

C_{10}\times D_8
% in TeX

G:=Group("C10xD8");
// GroupNames label

G:=SmallGroup(160,193);
// by ID

G=gap.SmallGroup(160,193);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-2,505,3604,1810,88]);
// Polycyclic

G:=Group<a,b,c|a^10=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

׿
x
:
Z
F
o
wr
Q
<