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G = C10xSD16order 160 = 25·5

Direct product of C10 and SD16

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

Aliases: C10xSD16, C20.42D4, C40:13C22, C20.45C23, (C2xC8):5C10, C8:3(C2xC10), C4.7(C5xD4), (C2xC40):13C2, (C2xQ8):3C10, Q8:1(C2xC10), (Q8xC10):10C2, D4.1(C2xC10), (C2xD4).6C10, C10.75(C2xD4), (C2xC10).53D4, C2.12(D4xC10), (C5xQ8):9C22, (D4xC10).13C2, C4.2(C22xC10), C22.15(C5xD4), (C5xD4).11C22, (C2xC20).130C22, (C2xC4).26(C2xC10), SmallGroup(160,194)

Series: Derived Chief Lower central Upper central

C1C4 — C10xSD16
C1C2C4C20C5xQ8C5xSD16 — C10xSD16
C1C2C4 — C10xSD16
C1C2xC10C2xC20 — C10xSD16

Generators and relations for C10xSD16
 G = < a,b,c | a10=b8=c2=1, ab=ba, ac=ca, cbc=b3 >

Subgroups: 108 in 68 conjugacy classes, 44 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C10, C10, C10, C2xC8, SD16, C2xD4, C2xQ8, C20, C20, C2xC10, C2xC10, C2xSD16, C40, C2xC20, C2xC20, C5xD4, C5xD4, C5xQ8, C5xQ8, C22xC10, C2xC40, C5xSD16, D4xC10, Q8xC10, C10xSD16
Quotients: C1, C2, C22, C5, D4, C23, C10, SD16, C2xD4, C2xC10, C2xSD16, C5xD4, C22xC10, C5xSD16, D4xC10, C10xSD16

Smallest permutation representation of C10xSD16
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 31 66 11 60 28 50 79)(2 32 67 12 51 29 41 80)(3 33 68 13 52 30 42 71)(4 34 69 14 53 21 43 72)(5 35 70 15 54 22 44 73)(6 36 61 16 55 23 45 74)(7 37 62 17 56 24 46 75)(8 38 63 18 57 25 47 76)(9 39 64 19 58 26 48 77)(10 40 65 20 59 27 49 78)
(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(21 72)(22 73)(23 74)(24 75)(25 76)(26 77)(27 78)(28 79)(29 80)(30 71)(41 67)(42 68)(43 69)(44 70)(45 61)(46 62)(47 63)(48 64)(49 65)(50 66)

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

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

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

C10xSD16 is a maximal subgroup of
Dic5:3SD16  Dic5:5SD16  SD16:Dic5  (C5xD4).D4  (C5xQ8).D4  C40.31D4  C40.43D4  D10:6SD16  D10:8SD16  C40:14D4  D20:7D4  Dic10.16D4  C40:8D4  C40:15D4  C40:9D4  C40.44D4  D20.29D4

70 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D5A5B5C5D8A8B8C8D10A···10L10M···10T20A···20H20I···20P40A···40P
order12222244445555888810···1010···1020···2020···2040···40
size1111442244111122221···14···42···24···42···2

70 irreducible representations

dim1111111111222222
type+++++++
imageC1C2C2C2C2C5C10C10C10C10D4D4SD16C5xD4C5xD4C5xSD16
kernelC10xSD16C2xC40C5xSD16D4xC10Q8xC10C2xSD16C2xC8SD16C2xD4C2xQ8C20C2xC10C10C4C22C2
# reps114114416441144416

Matrix representation of C10xSD16 in GL4(F41) generated by

31000
03100
00400
00040
,
112100
23000
002615
002626
,
13000
04000
0010
00040
G:=sub<GL(4,GF(41))| [31,0,0,0,0,31,0,0,0,0,40,0,0,0,0,40],[11,2,0,0,21,30,0,0,0,0,26,26,0,0,15,26],[1,0,0,0,30,40,0,0,0,0,1,0,0,0,0,40] >;

C10xSD16 in GAP, Magma, Sage, TeX

C_{10}\times {\rm SD}_{16}
% in TeX

G:=Group("C10xSD16");
// GroupNames label

G:=SmallGroup(160,194);
// by ID

G=gap.SmallGroup(160,194);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-2,480,505,3604,1810,88]);
// 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^3>;
// generators/relations

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