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G = C5×C4○D8order 160 = 25·5

Direct product of C5 and C4○D8

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

Aliases: C5×C4○D8, D83C10, Q163C10, C20.69D4, SD163C10, C20.47C23, C40.28C22, (C2×C8)⋊4C10, (C5×D8)⋊7C2, (C2×C40)⋊12C2, C4○D41C10, C8.6(C2×C10), (C5×Q16)⋊7C2, C4.20(C5×D4), (C5×SD16)⋊7C2, D4.2(C2×C10), C2.14(D4×C10), C10.77(C2×D4), (C2×C10).11D4, Q8.2(C2×C10), C22.1(C5×D4), C4.4(C22×C10), (C5×D4).12C22, (C5×Q8).13C22, (C2×C20).132C22, (C5×C4○D4)⋊6C2, (C2×C4).28(C2×C10), SmallGroup(160,196)

Series: Derived Chief Lower central Upper central

C1C4 — C5×C4○D8
C1C2C4C20C5×D4C5×D8 — C5×C4○D8
C1C2C4 — C5×C4○D8
C1C20C2×C20 — C5×C4○D8

Generators and relations for C5×C4○D8
 G = < a,b,c,d | a5=b4=d2=1, c4=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c3 >

Subgroups: 92 in 62 conjugacy classes, 40 normal (24 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×2], C5, C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], D4 [×2], Q8 [×2], C10, C10 [×3], C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C4○D8, C40 [×2], C2×C20, C2×C20 [×2], C5×D4 [×2], C5×D4 [×2], C5×Q8 [×2], C2×C40, C5×D8, C5×SD16 [×2], C5×Q16, C5×C4○D4 [×2], C5×C4○D8
Quotients: C1, C2 [×7], C22 [×7], C5, D4 [×2], C23, C10 [×7], C2×D4, C2×C10 [×7], C4○D8, C5×D4 [×2], C22×C10, D4×C10, C5×C4○D8

Smallest permutation representation of C5×C4○D8
On 80 points
Generators in S80
(1 23 69 49 27)(2 24 70 50 28)(3 17 71 51 29)(4 18 72 52 30)(5 19 65 53 31)(6 20 66 54 32)(7 21 67 55 25)(8 22 68 56 26)(9 77 57 35 46)(10 78 58 36 47)(11 79 59 37 48)(12 80 60 38 41)(13 73 61 39 42)(14 74 62 40 43)(15 75 63 33 44)(16 76 64 34 45)
(1 39 5 35)(2 40 6 36)(3 33 7 37)(4 34 8 38)(9 69 13 65)(10 70 14 66)(11 71 15 67)(12 72 16 68)(17 44 21 48)(18 45 22 41)(19 46 23 42)(20 47 24 43)(25 59 29 63)(26 60 30 64)(27 61 31 57)(28 62 32 58)(49 73 53 77)(50 74 54 78)(51 75 55 79)(52 76 56 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)
(2 8)(3 7)(4 6)(10 16)(11 15)(12 14)(17 21)(18 20)(22 24)(25 29)(26 28)(30 32)(33 37)(34 36)(38 40)(41 43)(44 48)(45 47)(50 56)(51 55)(52 54)(58 64)(59 63)(60 62)(66 72)(67 71)(68 70)(74 80)(75 79)(76 78)

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

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

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

C5×C4○D8 is a maximal subgroup of
D82Dic5  C20.58D8  D8⋊D10  C40.30C23  C40.31C23  D85Dic5  D84Dic5  Q16⋊D10  D815D10  D811D10  D20.47D4
C5×C4○D8 is a maximal quotient of
D8×C20  SD16×C20  Q16×C20

70 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E5A5B5C5D8A8B8C8D10A10B10C10D10E10F10G10H10I···10P20A···20H20I20J20K20L20M···20T40A···40P
order122224444455558888101010101010101010···1020···202020202020···2040···40
size112441124411112222111122224···41···122224···42···2

70 irreducible representations

dim111111111111222222
type++++++++
imageC1C2C2C2C2C2C5C10C10C10C10C10D4D4C4○D8C5×D4C5×D4C5×C4○D8
kernelC5×C4○D8C2×C40C5×D8C5×SD16C5×Q16C5×C4○D4C4○D8C2×C8D8SD16Q16C4○D4C20C2×C10C5C4C22C1
# reps1112124448481144416

Matrix representation of C5×C4○D8 in GL2(𝔽41) generated by

100
010
,
90
09
,
1229
1212
,
10
040
G:=sub<GL(2,GF(41))| [10,0,0,10],[9,0,0,9],[12,12,29,12],[1,0,0,40] >;

C5×C4○D8 in GAP, Magma, Sage, TeX

C_5\times C_4\circ D_8
% in TeX

G:=Group("C5xC4oD8");
// GroupNames label

G:=SmallGroup(160,196);
// by ID

G=gap.SmallGroup(160,196);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-2,505,374,3604,1810,88]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^4=d^2=1,c^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^3>;
// generators/relations

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