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## G = C4×Dic5order 80 = 24·5

### Direct product of C4 and Dic5

Aliases: C4×Dic5, C204C4, C52C42, C22.3D10, (C2×C4).6D5, C2.2(C4×D5), (C2×C20).7C2, C10.10(C2×C4), C2.2(C2×Dic5), (C2×C10).3C22, (C2×Dic5).6C2, SmallGroup(80,11)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C4×Dic5
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C4×Dic5
 Lower central C5 — C4×Dic5
 Upper central C1 — C2×C4

Generators and relations for C4×Dic5
G = < a,b,c | a4=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >

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

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

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

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

C4×Dic5 is a maximal subgroup of
C20.8Q8  C408C4  D207C4  D42Dic5  C20⋊C8  C10.C42  Dic5⋊C8  D5×C42  C42⋊D5  C23.11D10  C23.D10  Dic54D4  Dic5.5D4  Dic53Q8  C20⋊Q8  Dic5.Q8  C4.Dic10  C4⋊C47D5  D208C4  C4⋊C4⋊D5  C23.21D10  C20.17D4  C20⋊D4  Dic5⋊Q8  C20.23D4
C4×Dic5 is a maximal quotient of
C42.D5  C408C4  C10.10C42

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 type + + + + - + image C1 C2 C2 C4 C4 D5 Dic5 D10 C4×D5 kernel C4×Dic5 C2×Dic5 C2×C20 Dic5 C20 C2×C4 C4 C22 C2 # reps 1 2 1 8 4 2 4 2 8

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

 9 0 0 0 0 9 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 40 0 0 0 0 0 1 0 0 40 34
,
 1 0 0 0 0 9 0 0 0 0 26 21 0 0 3 15
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,40,0,0,0,0,0,40,0,0,1,34],[1,0,0,0,0,9,0,0,0,0,26,3,0,0,21,15] >;

C4×Dic5 in GAP, Magma, Sage, TeX

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

G:=Group("C4xDic5");
// GroupNames label

G:=SmallGroup(80,11);
// by ID

G=gap.SmallGroup(80,11);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-5,20,46,1604]);
// Polycyclic

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

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