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## G = C2×C8⋊D10order 320 = 26·5

### Direct product of C2 and C8⋊D10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×C8⋊D10
 Chief series C1 — C5 — C10 — C20 — D20 — C2×D20 — C22×D20 — C2×C8⋊D10
 Lower central C5 — C10 — C20 — C2×C8⋊D10
 Upper central C1 — C22 — C22×C4 — C2×M4(2)

Generators and relations for C2×C8⋊D10
G = < a,b,c,d | a2=b8=c10=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b-1, dcd=c-1 >

Subgroups: 1438 in 298 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C10, C2×C8, M4(2), D8, SD16, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C2×M4(2), C2×D8, C2×SD16, C8⋊C22, C22×D4, C2×C4○D4, C40, Dic10, Dic10, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C2×C8⋊C22, C40⋊C2, D40, C2×C40, C5×M4(2), C2×Dic10, C2×C4×D5, C2×D20, C2×D20, C2×D20, C4○D20, C4○D20, C2×C5⋊D4, C22×C20, C23×D5, C2×C40⋊C2, C2×D40, C8⋊D10, C10×M4(2), C22×D20, C2×C4○D20, C2×C8⋊D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C8⋊C22, C22×D4, D20, C22×D5, C2×C8⋊C22, C2×D20, C23×D5, C8⋊D10, C22×D20, C2×C8⋊D10

Smallest permutation representation of C2×C8⋊D10
On 80 points
Generators in S80
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(21 42)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(29 50)(30 41)(31 36)(32 37)(33 38)(34 39)(35 40)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(61 66)(62 67)(63 68)(64 69)(65 70)
(1 25 40 52 67 41 14 77)(2 42 31 78 68 26 15 53)(3 27 32 54 69 43 16 79)(4 44 33 80 70 28 17 55)(5 29 34 56 61 45 18 71)(6 46 35 72 62 30 19 57)(7 21 36 58 63 47 20 73)(8 48 37 74 64 22 11 59)(9 23 38 60 65 49 12 75)(10 50 39 76 66 24 13 51)
(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 10)(2 9)(3 8)(4 7)(5 6)(11 32)(12 31)(13 40)(14 39)(15 38)(16 37)(17 36)(18 35)(19 34)(20 33)(21 55)(22 54)(23 53)(24 52)(25 51)(26 60)(27 59)(28 58)(29 57)(30 56)(41 76)(42 75)(43 74)(44 73)(45 72)(46 71)(47 80)(48 79)(49 78)(50 77)(61 62)(63 70)(64 69)(65 68)(66 67)

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F ··· 2K 4A 4B 4C 4D 4E 4F 5A 5B 8A 8B 8C 8D 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H 20I 20J 20K 20L 40A ··· 40P order 1 2 2 2 2 2 2 ··· 2 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 10 10 10 10 20 ··· 20 20 20 20 20 40 ··· 40 size 1 1 1 1 2 2 20 ··· 20 2 2 2 2 20 20 2 2 4 4 4 4 2 ··· 2 4 4 4 4 2 ··· 2 4 4 4 4 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D4 D5 D10 D10 D10 D20 D20 C8⋊C22 C8⋊D10 kernel C2×C8⋊D10 C2×C40⋊C2 C2×D40 C8⋊D10 C10×M4(2) C22×D20 C2×C4○D20 C2×C20 C22×C10 C2×M4(2) C2×C8 M4(2) C22×C4 C2×C4 C23 C10 C2 # reps 1 2 2 8 1 1 1 3 1 2 4 8 2 12 4 2 8

Matrix representation of C2×C8⋊D10 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 40 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 11 9 0 0 0 0 32 30 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 34 0 0 0 0 7 7 0 0 0 0 0 0 1 7 0 0 0 0 34 34
,
 40 0 0 0 0 0 0 1 0 0 0 0 0 0 40 34 0 0 0 0 0 1 0 0 0 0 0 0 30 14 0 0 0 0 9 11

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,11,32,0,0,0,0,9,30,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,7,0,0,0,0,34,7,0,0,0,0,0,0,1,34,0,0,0,0,7,34],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,34,1,0,0,0,0,0,0,30,9,0,0,0,0,14,11] >;

C2×C8⋊D10 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes D_{10}
% in TeX

G:=Group("C2xC8:D10");
// GroupNames label

G:=SmallGroup(320,1418);
// by ID

G=gap.SmallGroup(320,1418);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,675,297,80,1684,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^10=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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