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## G = C5×SD32order 160 = 25·5

### Direct product of C5 and SD32

direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary

Aliases: C5×SD32, C806C2, D8.C10, C162C10, Q161C10, C10.16D8, C20.37D4, C40.25C22, C2.4(C5×D8), C4.2(C5×D4), C8.3(C2×C10), (C5×Q16)⋊5C2, (C5×D8).2C2, SmallGroup(160,62)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C5×SD32
 Chief series C1 — C2 — C4 — C8 — C40 — C5×Q16 — C5×SD32
 Lower central C1 — C2 — C4 — C8 — C5×SD32
 Upper central C1 — C10 — C20 — C40 — C5×SD32

Generators and relations for C5×SD32
G = < a,b,c | a5=b16=c2=1, ab=ba, ac=ca, cbc=b7 >

Smallest permutation representation of C5×SD32
On 80 points
Generators in S80
(1 71 17 37 61)(2 72 18 38 62)(3 73 19 39 63)(4 74 20 40 64)(5 75 21 41 49)(6 76 22 42 50)(7 77 23 43 51)(8 78 24 44 52)(9 79 25 45 53)(10 80 26 46 54)(11 65 27 47 55)(12 66 28 48 56)(13 67 29 33 57)(14 68 30 34 58)(15 69 31 35 59)(16 70 32 36 60)
(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)
(2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(18 24)(19 31)(20 22)(21 29)(23 27)(26 32)(28 30)(33 41)(34 48)(35 39)(36 46)(38 44)(40 42)(43 47)(49 57)(50 64)(51 55)(52 62)(54 60)(56 58)(59 63)(65 77)(66 68)(67 75)(69 73)(70 80)(72 78)(74 76)

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

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

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

C5×SD32 is a maximal subgroup of   C16⋊D10  SD32⋊D5  SD323D5

55 conjugacy classes

 class 1 2A 2B 4A 4B 5A 5B 5C 5D 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 16A 16B 16C 16D 20A 20B 20C 20D 20E 20F 20G 20H 40A ··· 40H 80A ··· 80P order 1 2 2 4 4 5 5 5 5 8 8 10 10 10 10 10 10 10 10 16 16 16 16 20 20 20 20 20 20 20 20 40 ··· 40 80 ··· 80 size 1 1 8 2 8 1 1 1 1 2 2 1 1 1 1 8 8 8 8 2 2 2 2 2 2 2 2 8 8 8 8 2 ··· 2 2 ··· 2

55 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C5 C10 C10 C10 D4 D8 SD32 C5×D4 C5×D8 C5×SD32 kernel C5×SD32 C80 C5×D8 C5×Q16 SD32 C16 D8 Q16 C20 C10 C5 C4 C2 C1 # reps 1 1 1 1 4 4 4 4 1 2 4 4 8 16

Matrix representation of C5×SD32 in GL2(𝔽241) generated by

 98 0 0 98
,
 103 41 200 103
,
 0 1 1 0
G:=sub<GL(2,GF(241))| [98,0,0,98],[103,200,41,103],[0,1,1,0] >;

C5×SD32 in GAP, Magma, Sage, TeX

C_5\times {\rm SD}_{32}
% in TeX

G:=Group("C5xSD32");
// GroupNames label

G:=SmallGroup(160,62);
// by ID

G=gap.SmallGroup(160,62);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-2,480,265,1443,729,165,3604,1810,88]);
// Polycyclic

G:=Group<a,b,c|a^5=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^7>;
// generators/relations

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