C10xC5:2C8 - GroupNames

G = C₁₀×C₅⋊₂C₈ order 400 = 2⁴·5²

Direct product of C₁₀ and C₅⋊₂C₈

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

Aliases: C₁₀×C₅⋊₂C₈, C₁₀⋊₂C₄₀, C₂₀.₆C₂₀, C₂₀.₆₆D₁₀, C₁₀².₁₀C₄, C₂₀.₁₄Dic₅, C₅⋊₄(C₂×C₄₀), (C₅×C₁₀)⋊₆C₈, C₅²⋊₁₇(C₂×C₈), (C₂×C₁₀).₆C₂₀, (C₂×C₂₀).₈C₁₀, (C₅×C₂₀).₁₇C₄, C₄.₁₄(D₅×C₁₀), (C₂×C₂₀).₂₁D₅, C₄.₃(C₅×Dic₅), C₂₀.₁₅(C₂×C₁₀), C₁₀.₁₂(C₂×C₂₀), (C₁₀×C₂₀).₁₀C₂, C₂.₁(C₁₀×Dic₅), (C₅×C₂₀).₄₄C₂², (C₂×C₁₀).₁₀Dic₅, C₁₀.₂₄(C₂×Dic₅), C₂².₂(C₅×Dic₅), (C₂×C₄).₅(C₅×D₅), (C₅×C₁₀).₆₀(C₂×C₄), SmallGroup(400,81)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₅ — C₁₀×C₅⋊₂C₈

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — C₅×C₂₀ — C₅×C₅⋊₂C₈ — C₁₀×C₅⋊₂C₈

Lower central

C₅ — C₁₀×C₅⋊₂C₈

Upper central

C₁ — C₂×C₂₀

Generators and relations for C₁₀×C₅⋊₂C₈
G = < a,b,c | a¹⁰=b⁵=c⁸=1, ab=ba, ac=ca, cbc^-1=b^-1 >

Subgroups: 100 in 60 conjugacy classes, 38 normal (26 characteristic)
C₁, C₂, C₂, C₄, C₂², C₅, C₅, C₈, C₂×C₄, C₁₀, C₁₀, C₁₀, C₂×C₈, C₂₀, C₂₀, C₂×C₁₀, C₂×C₁₀, C₅², C₅⋊₂C₈, C₄₀, C₂×C₂₀, C₂×C₂₀, C₅×C₁₀, C₅×C₁₀, C₂×C₅⋊₂C₈, C₂×C₄₀, C₅×C₂₀, C₁₀², C₅×C₅⋊₂C₈, C₁₀×C₂₀, C₁₀×C₅⋊₂C₈
Quotients: C₁, C₂, C₄, C₂², C₅, C₈, C₂×C₄, D₅, C₁₀, C₂×C₈, Dic₅, C₂₀, D₁₀, C₂×C₁₀, C₅⋊₂C₈, C₄₀, C₂×Dic₅, C₂×C₂₀, C₅×D₅, C₂×C₅⋊₂C₈, C₂×C₄₀, C₅×Dic₅, D₅×C₁₀, C₅×C₅⋊₂C₈, C₁₀×Dic₅, C₁₀×C₅⋊₂C₈

Smallest permutation representation of C₁₀×C₅⋊₂C₈
►On 80 points

Generators in S₈₀

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

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

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

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

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

160 conjugacy classes

class	1	2A	2B	2C	4A	4B	4C	4D	5A	5B	5C	5D	5E	···	5N	8A	···	8H	10A	···	10L	10M	···	10AP	20A	···	20P	20Q	···	20BD	40A	···	40AF
order	1	2	2	2	4	4	4	4	5	5	5	5	5	···	5	8	···	8	10	···	10	10	···	10	20	···	20	20	···	20	40	···	40
size	1	1	1	1	1	1	1	1	1	1	1	1	2	···	2	5	···	5	1	···	1	2	···	2	1	···	1	2	···	2	5	···	5

160 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2
type	+	+	+										+	-	+	-
image	C₁	C₂	C₂	C₄	C₄	C₅	C₈	C₁₀	C₁₀	C₂₀	C₂₀	C₄₀	D₅	Dic₅	D₁₀	Dic₅	C₅⋊₂C₈	C₅×D₅	C₅×Dic₅	D₅×C₁₀	C₅×Dic₅	C₅×C₅⋊₂C₈
kernel	C₁₀×C₅⋊₂C₈	C₅×C₅⋊₂C₈	C₁₀×C₂₀	C₅×C₂₀	C₁₀²	C₂×C₅⋊₂C₈	C₅×C₁₀	C₅⋊₂C₈	C₂×C₂₀	C₂₀	C₂×C₁₀	C₁₀	C₂×C₂₀	C₂₀	C₂₀	C₂×C₁₀	C₁₀	C₂×C₄	C₄	C₄	C₂²	C₂
# reps	1	2	1	2	2	4	8	8	4	8	8	32	2	2	2	2	8	8	8	8	8	32

Matrix representation of C₁₀×C₅⋊₂C₈ ►in GL₃(𝔽₄₁) generated by

23	0	0
0	16	0
0	0	16

1	0	0
0	10	0
0	19	37

1	0	0
0	39	36
0	19	2

G:=sub<GL(3,GF(41))| [23,0,0,0,16,0,0,0,16],[1,0,0,0,10,19,0,0,37],[1,0,0,0,39,19,0,36,2] >;

C₁₀×C₅⋊₂C₈ in GAP, Magma, Sage, TeX

C_{10}\times C_5\rtimes_2C_8

% in TeX

G:=Group("C10xC5:2C8");

// GroupNames label

G:=SmallGroup(400,81);

// by ID

G=gap.SmallGroup(400,81);

# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-5,120,69,11525]);

// Polycyclic

G:=Group<a,b,c|a^10=b^5=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;

// generators/relations

G = C10×C5⋊2C8 order 400 = 24·52

Direct product of C10 and C5⋊2C8

G = C₁₀×C₅⋊₂C₈ order 400 = 2⁴·5²

Direct product of C₁₀ and C₅⋊₂C₈