Copied to
clipboard

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

C1C8 — C5×SD32
C1C2C4C8C40C5×Q16 — C5×SD32
C1C2C4C8 — C5×SD32
C1C10C20C40 — C5×SD32

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

8C2
4C22
4C4
8C10
2D4
2Q8
4C20
4C2×C10
2C5×Q8
2C5×D4

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 2A2B4A4B5A5B5C5D8A8B10A10B10C10D10E10F10G10H16A16B16C16D20A20B20C20D20E20F20G20H40A···40H80A···80P
order12244555588101010101010101016161616202020202020202040···4080···80
size11828111122111188882222222288882···22···2

55 irreducible representations

dim11111111222222
type++++++
imageC1C2C2C2C5C10C10C10D4D8SD32C5×D4C5×D8C5×SD32
kernelC5×SD32C80C5×D8C5×Q16SD32C16D8Q16C20C10C5C4C2C1
# reps111144441244816

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

980
098
,
10341
200103
,
01
10
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

Export

Subgroup lattice of C5×SD32 in TeX

׿
×
𝔽