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## G = C10×SD16order 160 = 25·5

### Direct product of C10 and SD16

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C10×SD16
 Chief series C1 — C2 — C4 — C20 — C5×Q8 — C5×SD16 — C10×SD16
 Lower central C1 — C2 — C4 — C10×SD16
 Upper central C1 — C2×C10 — C2×C20 — C10×SD16

Generators and relations for C10×SD16
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, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C10, C10, C10, C2×C8, SD16, C2×D4, C2×Q8, C20, C20, C2×C10, C2×C10, C2×SD16, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×C10, C2×C40, C5×SD16, D4×C10, Q8×C10, C10×SD16
Quotients: C1, C2, C22, C5, D4, C23, C10, SD16, C2×D4, C2×C10, C2×SD16, C5×D4, C22×C10, C5×SD16, D4×C10, C10×SD16

Smallest permutation representation of C10×SD16
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)]])

70 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 5A 5B 5C 5D 8A 8B 8C 8D 10A ··· 10L 10M ··· 10T 20A ··· 20H 20I ··· 20P 40A ··· 40P order 1 2 2 2 2 2 4 4 4 4 5 5 5 5 8 8 8 8 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 4 4 2 2 4 4 1 1 1 1 2 2 2 2 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4 2 ··· 2

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C5 C10 C10 C10 C10 D4 D4 SD16 C5×D4 C5×D4 C5×SD16 kernel C10×SD16 C2×C40 C5×SD16 D4×C10 Q8×C10 C2×SD16 C2×C8 SD16 C2×D4 C2×Q8 C20 C2×C10 C10 C4 C22 C2 # reps 1 1 4 1 1 4 4 16 4 4 1 1 4 4 4 16

Matrix representation of C10×SD16 in GL4(𝔽41) generated by

 31 0 0 0 0 31 0 0 0 0 40 0 0 0 0 40
,
 11 21 0 0 2 30 0 0 0 0 26 15 0 0 26 26
,
 1 30 0 0 0 40 0 0 0 0 1 0 0 0 0 40
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] >;

C10×SD16 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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