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## G = C2×C4.Dic5order 160 = 25·5

### Direct product of C2 and C4.Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×C4.Dic5
 Chief series C1 — C5 — C10 — C20 — C5⋊2C8 — C2×C5⋊2C8 — C2×C4.Dic5
 Lower central C5 — C10 — C2×C4.Dic5
 Upper central C1 — C2×C4 — C22×C4

Generators and relations for C2×C4.Dic5
G = < a,b,c,d | a2=b4=1, c10=b2, d2=c5, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c9 >

Subgroups: 120 in 68 conjugacy classes, 49 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C10, C2×C8, M4(2), C22×C4, C20, C20, C2×C10, C2×C10, C2×C10, C2×M4(2), C52C8, C2×C20, C2×C20, C22×C10, C2×C52C8, C4.Dic5, C22×C20, C2×C4.Dic5
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, M4(2), C22×C4, Dic5, D10, C2×M4(2), C2×Dic5, C22×D5, C4.Dic5, C22×Dic5, C2×C4.Dic5

Smallest permutation representation of C2×C4.Dic5
On 80 points
Generators in S80
(1 36)(2 37)(3 38)(4 39)(5 40)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(41 78)(42 79)(43 80)(44 61)(45 62)(46 63)(47 64)(48 65)(49 66)(50 67)(51 68)(52 69)(53 70)(54 71)(55 72)(56 73)(57 74)(58 75)(59 76)(60 77)
(1 6 11 16)(2 7 12 17)(3 8 13 18)(4 9 14 19)(5 10 15 20)(21 26 31 36)(22 27 32 37)(23 28 33 38)(24 29 34 39)(25 30 35 40)(41 56 51 46)(42 57 52 47)(43 58 53 48)(44 59 54 49)(45 60 55 50)(61 76 71 66)(62 77 72 67)(63 78 73 68)(64 79 74 69)(65 80 75 70)
(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 50 6 55 11 60 16 45)(2 59 7 44 12 49 17 54)(3 48 8 53 13 58 18 43)(4 57 9 42 14 47 19 52)(5 46 10 51 15 56 20 41)(21 72 26 77 31 62 36 67)(22 61 27 66 32 71 37 76)(23 70 28 75 33 80 38 65)(24 79 29 64 34 69 39 74)(25 68 30 73 35 78 40 63)

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

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

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

52 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 5A 5B 8A ··· 8H 10A ··· 10N 20A ··· 20P order 1 2 2 2 2 2 4 4 4 4 4 4 5 5 8 ··· 8 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 1 1 1 1 2 2 2 2 10 ··· 10 2 ··· 2 2 ··· 2

52 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + - + - image C1 C2 C2 C2 C4 C4 D5 M4(2) Dic5 D10 Dic5 C4.Dic5 kernel C2×C4.Dic5 C2×C5⋊2C8 C4.Dic5 C22×C20 C2×C20 C22×C10 C22×C4 C10 C2×C4 C2×C4 C23 C2 # reps 1 2 4 1 6 2 2 4 6 6 2 16

Matrix representation of C2×C4.Dic5 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 40 0 0 0 0 40 0 0 0 0 9 33 0 0 0 32
,
 7 1 0 0 40 0 0 0 0 0 5 15 0 0 0 8
,
 9 0 0 0 19 32 0 0 0 0 33 19 0 0 23 8
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,9,0,0,0,33,32],[7,40,0,0,1,0,0,0,0,0,5,0,0,0,15,8],[9,19,0,0,0,32,0,0,0,0,33,23,0,0,19,8] >;

C2×C4.Dic5 in GAP, Magma, Sage, TeX

C_2\times C_4.{\rm Dic}_5
% in TeX

G:=Group("C2xC4.Dic5");
// GroupNames label

G:=SmallGroup(160,142);
// by ID

G=gap.SmallGroup(160,142);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,362,69,4613]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=1,c^10=b^2,d^2=c^5,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^9>;
// generators/relations

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