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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C5×C4○D8
 Chief series C1 — C2 — C4 — C20 — C5×D4 — C5×D8 — C5×C4○D8
 Lower central C1 — C2 — C4 — C5×C4○D8
 Upper central C1 — C20 — C2×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, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C10, C10, C2×C8, D8, SD16, Q16, C4○D4, C20, C20, C2×C10, C2×C10, C4○D8, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C2×C40, C5×D8, C5×SD16, C5×Q16, C5×C4○D4, C5×C4○D8
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C2×C10, C4○D8, C5×D4, 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 78 58 36 47)(10 79 59 37 48)(11 80 60 38 41)(12 73 61 39 42)(13 74 62 40 43)(14 75 63 33 44)(15 76 64 34 45)(16 77 57 35 46)
(1 39 5 35)(2 40 6 36)(3 33 7 37)(4 34 8 38)(9 70 13 66)(10 71 14 67)(11 72 15 68)(12 65 16 69)(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)(9 15)(10 14)(11 13)(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)(9,78,58,36,47)(10,79,59,37,48)(11,80,60,38,41)(12,73,61,39,42)(13,74,62,40,43)(14,75,63,33,44)(15,76,64,34,45)(16,77,57,35,46), (1,39,5,35)(2,40,6,36)(3,33,7,37)(4,34,8,38)(9,70,13,66)(10,71,14,67)(11,72,15,68)(12,65,16,69)(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)(9,15)(10,14)(11,13)(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)(9,78,58,36,47)(10,79,59,37,48)(11,80,60,38,41)(12,73,61,39,42)(13,74,62,40,43)(14,75,63,33,44)(15,76,64,34,45)(16,77,57,35,46), (1,39,5,35)(2,40,6,36)(3,33,7,37)(4,34,8,38)(9,70,13,66)(10,71,14,67)(11,72,15,68)(12,65,16,69)(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)(9,15)(10,14)(11,13)(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),(9,78,58,36,47),(10,79,59,37,48),(11,80,60,38,41),(12,73,61,39,42),(13,74,62,40,43),(14,75,63,33,44),(15,76,64,34,45),(16,77,57,35,46)], [(1,39,5,35),(2,40,6,36),(3,33,7,37),(4,34,8,38),(9,70,13,66),(10,71,14,67),(11,72,15,68),(12,65,16,69),(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),(9,15),(10,14),(11,13),(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 2A 2B 2C 2D 4A 4B 4C 4D 4E 5A 5B 5C 5D 8A 8B 8C 8D 10A 10B 10C 10D 10E 10F 10G 10H 10I ··· 10P 20A ··· 20H 20I 20J 20K 20L 20M ··· 20T 40A ··· 40P order 1 2 2 2 2 4 4 4 4 4 5 5 5 5 8 8 8 8 10 10 10 10 10 10 10 10 10 ··· 10 20 ··· 20 20 20 20 20 20 ··· 20 40 ··· 40 size 1 1 2 4 4 1 1 2 4 4 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 4 ··· 4 1 ··· 1 2 2 2 2 4 ··· 4 2 ··· 2

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 D4 D4 C4○D8 C5×D4 C5×D4 C5×C4○D8 kernel C5×C4○D8 C2×C40 C5×D8 C5×SD16 C5×Q16 C5×C4○D4 C4○D8 C2×C8 D8 SD16 Q16 C4○D4 C20 C2×C10 C5 C4 C22 C1 # reps 1 1 1 2 1 2 4 4 4 8 4 8 1 1 4 4 4 16

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

 10 0 0 10
,
 9 0 0 9
,
 12 29 12 12
,
 1 0 0 40
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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