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## G = C5×C8○D4order 160 = 25·5

### Direct product of C5 and C8○D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C5×C8○D4
 Chief series C1 — C2 — C4 — C20 — C40 — C2×C40 — C5×C8○D4
 Lower central C1 — C2 — C5×C8○D4
 Upper central C1 — C40 — C5×C8○D4

Generators and relations for C5×C8○D4
G = < a,b,c,d | a5=b8=d2=1, c2=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c >

Subgroups: 68 in 62 conjugacy classes, 56 normal (14 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, D4, Q8, C10, C10, C2×C8, M4(2), C4○D4, C20, C20, C2×C10, C8○D4, C40, C40, C2×C20, C5×D4, C5×Q8, C2×C40, C5×M4(2), C5×C4○D4, C5×C8○D4
Quotients: C1, C2, C4, C22, C5, C2×C4, C23, C10, C22×C4, C20, C2×C10, C8○D4, C2×C20, C22×C10, C22×C20, C5×C8○D4

Smallest permutation representation of C5×C8○D4
On 80 points
Generators in S80
(1 63 71 23 31)(2 64 72 24 32)(3 57 65 17 25)(4 58 66 18 26)(5 59 67 19 27)(6 60 68 20 28)(7 61 69 21 29)(8 62 70 22 30)(9 74 50 34 46)(10 75 51 35 47)(11 76 52 36 48)(12 77 53 37 41)(13 78 54 38 42)(14 79 55 39 43)(15 80 56 40 44)(16 73 49 33 45)
(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 35 5 39)(2 36 6 40)(3 37 7 33)(4 38 8 34)(9 66 13 70)(10 67 14 71)(11 68 15 72)(12 69 16 65)(17 77 21 73)(18 78 22 74)(19 79 23 75)(20 80 24 76)(25 53 29 49)(26 54 30 50)(27 55 31 51)(28 56 32 52)(41 61 45 57)(42 62 46 58)(43 63 47 59)(44 64 48 60)
(1 39)(2 40)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 66)(10 67)(11 68)(12 69)(13 70)(14 71)(15 72)(16 65)(17 73)(18 74)(19 75)(20 76)(21 77)(22 78)(23 79)(24 80)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(41 61)(42 62)(43 63)(44 64)(45 57)(46 58)(47 59)(48 60)

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

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

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

C5×C8○D4 is a maximal subgroup of
C40.92D4  C40.70C23  D4.3D20  D4.4D20  D4.5D20  C40.93D4  C40.50D4  C20.72C24  D4.11D20  D4.12D20  D4.13D20
C5×C8○D4 is a maximal quotient of
D4×C40  Q8×C40

100 conjugacy classes

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

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C2 C4 C4 C5 C10 C10 C10 C20 C20 C8○D4 C5×C8○D4 kernel C5×C8○D4 C2×C40 C5×M4(2) C5×C4○D4 C5×D4 C5×Q8 C8○D4 C2×C8 M4(2) C4○D4 D4 Q8 C5 C1 # reps 1 3 3 1 6 2 4 12 12 4 24 8 4 16

Matrix representation of C5×C8○D4 in GL2(𝔽41) generated by

 10 0 0 10
,
 3 0 0 3
,
 1 39 1 40
,
 1 39 0 40
G:=sub<GL(2,GF(41))| [10,0,0,10],[3,0,0,3],[1,1,39,40],[1,0,39,40] >;

C5×C8○D4 in GAP, Magma, Sage, TeX

C_5\times C_8\circ D_4
% in TeX

G:=Group("C5xC8oD4");
// GroupNames label

G:=SmallGroup(160,192);
// by ID

G=gap.SmallGroup(160,192);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-2,240,764,88]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^8=d^2=1,c^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c>;
// generators/relations

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