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G = C4×Dic5order 80 = 24·5

Direct product of C4 and Dic5

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

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

C1C5 — C4×Dic5
C1C5C10C2×C10C2×Dic5 — C4×Dic5
C5 — C4×Dic5
C1C2×C4

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

5C4
5C4
5C4
5C4
5C2×C4
5C2×C4
5C42

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

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

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

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

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 2A2B2C4A4B4C4D4E···4L5A5B10A···10F20A···20H
order122244444···45510···1020···20
size111111115···5222···22···2

32 irreducible representations

dim111112222
type++++-+
imageC1C2C2C4C4D5Dic5D10C4×D5
kernelC4×Dic5C2×Dic5C2×C20Dic5C20C2×C4C4C22C2
# reps121842428

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

9000
0900
0010
0001
,
1000
04000
0001
004034
,
1000
0900
002621
00315
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

Export

Subgroup lattice of C4×Dic5 in TeX

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