C10xC4.D4 - GroupNames

G = C₁₀×C₄.D₄ order 320 = 2⁶·5

Direct product of C₁₀ and C₄.D₄

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C₁₀×C₄.D₄, C₂⁴.₂C₂₀, (C₂×D₄).₆C₂₀, C₄.₄₈(D₄×C₁₀), (D₄×C₁₀).₃₁C₄, (C₂×C₂₀).₅₁₅D₄, C₂₀.₄₅₅(C₂×D₄), (C₂³×C₁₀).₂C₄, C₂³.₄(C₂×C₂₀), M₄(2)⋊₈(C₂×C₁₀), (C₂×M₄(2))⋊₈C₁₀, (C₂²×D₄).₅C₁₀, (C₁₀×M₄(2))⋊₂₆C₂, (C₂×C₂₀).₆₀₆C₂³, C₂².₈(C₂²×C₂₀), C₂₀.₁₂₅(C₂²⋊C₄), (D₄×C₁₀).₂₈₄C₂², (C₅×M₄(2))⋊₃₇C₂², (C₂²×C₂₀).₄₀₇C₂², (D₄×C₂×C₁₀).₁₇C₂, (C₂×C₄).₂₁(C₅×D₄), (C₂×C₄).₁₉(C₂×C₂₀), C₄.₁₀(C₅×C₂²⋊C₄), (C₂×C₂₀).₃₆₅(C₂×C₄), (C₂×D₄).₄₂(C₂×C₁₀), C₂.₁₄(C₁₀×C₂²⋊C₄), (C₂×C₄).₁(C₂²×C₁₀), C₁₀.₁₄₃(C₂×C₂²⋊C₄), (C₂²×C₁₀).₃₆(C₂×C₄), (C₂²×C₄).₂₆(C₂×C₁₀), C₂².₁₈(C₅×C₂²⋊C₄), (C₂×C₁₀).₂₆₂(C₂²×C₄), (C₂×C₁₀).₂₀₀(C₂²⋊C₄), SmallGroup(320,912)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₁₀×C₄.D₄

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂×C₂₀ — C₅×M₄(2) — C₅×C₄.D₄ — C₁₀×C₄.D₄

Lower central

C₁ — C₂ — C₂² — C₁₀×C₄.D₄

Upper central

C₁ — C₂×C₁₀ — C₂²×C₂₀ — C₁₀×C₄.D₄

Generators and relations for C₁₀×C₄.D₄
G = < a,b,c,d | a¹⁰=b⁴=1, c⁴=b², d²=b, ab=ba, ac=ca, ad=da, cbc^-1=b^-1, bd=db, dcd^-1=b^-1c³ >

Subgroups: 370 in 186 conjugacy classes, 82 normal (22 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₁₀, C₁₀, C₁₀, C₂×C₈, M₄(2), M₄(2), C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂₀, C₂×C₁₀, C₂×C₁₀, C₄.D₄, C₂×M₄(2), C₂²×D₄, C₄₀, C₂×C₂₀, C₂×C₂₀, C₅×D₄, C₂²×C₁₀, C₂²×C₁₀, C₂²×C₁₀, C₂×C₄.D₄, C₂×C₄₀, C₅×M₄(2), C₅×M₄(2), C₂²×C₂₀, D₄×C₁₀, D₄×C₁₀, C₂³×C₁₀, C₅×C₄.D₄, C₁₀×M₄(2), D₄×C₂×C₁₀, C₁₀×C₄.D₄
Quotients: C₁, C₂, C₄, C₂², C₅, C₂×C₄, D₄, C₂³, C₁₀, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂₀, C₂×C₁₀, C₄.D₄, C₂×C₂²⋊C₄, C₂×C₂₀, C₅×D₄, C₂²×C₁₀, C₂×C₄.D₄, C₅×C₂²⋊C₄, C₂²×C₂₀, D₄×C₁₀, C₅×C₄.D₄, C₁₀×C₂²⋊C₄, C₁₀×C₄.D₄

Smallest permutation representation of C₁₀×C₄.D₄
►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 56 70 50)(2 57 61 41)(3 58 62 42)(4 59 63 43)(5 60 64 44)(6 51 65 45)(7 52 66 46)(8 53 67 47)(9 54 68 48)(10 55 69 49)(11 80 29 33)(12 71 30 34)(13 72 21 35)(14 73 22 36)(15 74 23 37)(16 75 24 38)(17 76 25 39)(18 77 26 40)(19 78 27 31)(20 79 28 32)

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

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

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

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

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

110 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	4A	4B	4C	4D	5A	5B	5C	5D	8A	···	8H	10A	···	10L	10M	···	10T	10U	···	10AJ	20A	···	20P	40A	···	40AF
order	1	2	2	2	2	2	2	2	2	2	4	4	4	4	5	5	5	5	8	···	8	10	···	10	10	···	10	10	···	10	20	···	20	40	···	40
size	1	1	1	1	2	2	4	4	4	4	2	2	2	2	1	1	1	1	4	···	4	1	···	1	2	···	2	4	···	4	2	···	2	4	···	4

110 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	4	4
type	+	+	+	+									+		+
image	C₁	C₂	C₂	C₂	C₄	C₄	C₅	C₁₀	C₁₀	C₁₀	C₂₀	C₂₀	D₄	C₅×D₄	C₄.D₄	C₅×C₄.D₄
kernel	C₁₀×C₄.D₄	C₅×C₄.D₄	C₁₀×M₄(2)	D₄×C₂×C₁₀	D₄×C₁₀	C₂³×C₁₀	C₂×C₄.D₄	C₄.D₄	C₂×M₄(2)	C₂²×D₄	C₂×D₄	C₂⁴	C₂×C₂₀	C₂×C₄	C₁₀	C₂
# reps	1	4	2	1	4	4	4	16	8	4	16	16	4	16	2	8

Matrix representation of C₁₀×C₄.D₄ ►in GL₆(𝔽₄₁)

40	0	0	0	0	0
0	40	0	0	0	0
0	0	10	0	0	0
0	0	0	10	0	0
0	0	0	0	10	0
0	0	0	0	0	10

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	1	0	0
0	0	40	0	0	0
0	0	0	0	0	1
0	0	0	0	40	0

0	1	0	0	0	0
40	0	0	0	0	0
0	0	0	0	0	1
0	0	0	0	1	0
0	0	40	0	0	0
0	0	0	1	0	0

1	0	0	0	0	0
0	40	0	0	0	0
0	0	0	0	40	0
0	0	0	0	0	40
0	0	0	40	0	0
0	0	1	0	0	0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,40,0,0,0,0,0,0,40,0,0] >;

C₁₀×C₄.D₄ in GAP, Magma, Sage, TeX

C_{10}\times C_4.D_4

% in TeX

G:=Group("C10xC4.D4");

// GroupNames label

G:=SmallGroup(320,912);

// by ID

G=gap.SmallGroup(320,912);

# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,7004,5052,124]);

// Polycyclic

G:=Group<a,b,c,d|a^10=b^4=1,c^4=b^2,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^3>;

// generators/relations

G = C10×C4.D4 order 320 = 26·5

Direct product of C10 and C4.D4

G = C₁₀×C₄.D₄ order 320 = 2⁶·5

Direct product of C₁₀ and C₄.D₄