C10xC4wrC2 - GroupNames

G = C₁₀xC₄wrC₂ order 320 = 2⁶·5

Direct product of C₁₀ and C₄wrC₂

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

Aliases: C₁₀xC₄wrC₂, C₄oD₄:₃C₂₀, (C₂xD₄):₉C₂₀, D₄:₅(C₂xC₂₀), (C₂xQ₈):₇C₂₀, Q₈:₅(C₂xC₂₀), (D₄xC₁₀):₃₃C₄, (C₂xC₄²):₆C₁₀, (Q₈xC₁₀):₂₇C₄, C₄.₇₂(D₄xC₁₀), (C₄xC₂₀):₅₆C₂², C₄²:₁₅(C₂xC₁₀), (C₂xC₂₀).₅₁₉D₄, C₂₀.₄₇₇(C₂xD₄), C₄.₇(C₂²xC₂₀), C₂³.₃₉(C₅xD₄), M₄(2):₉(C₂xC₁₀), C₂².₁₂(D₄xC₁₀), (C₁₀xM₄(2)):₃₀C₂, (C₂xM₄(2)):₁₂C₁₀, C₂₀.₂₁₁(C₂²xC₄), (C₂xC₂₀).₈₉₆C₂³, (C₂²xC₁₀).₁₆₁D₄, C₂₀.₁₆₄(C₂²:C₄), (C₅xM₄(2)):₃₈C₂², (C₂²xC₂₀).₅₈₄C₂², (C₂xC₄xC₂₀):₁₉C₂, (C₅xC₄oD₄):₁₅C₄, (C₅xD₄):₃₅(C₂xC₄), (C₅xQ₈):₃₂(C₂xC₄), (C₂xC₄).₇₀(C₅xD₄), (C₂xC₄).₅₀(C₂xC₂₀), C₄oD₄.₆(C₂xC₁₀), (C₂xC₄oD₄).₆C₁₀, C₄.₃₃(C₅xC₂²:C₄), (C₂xC₂₀).₄₄₄(C₂xC₄), (C₁₀xC₄oD₄).₂₀C₂, (C₂xC₁₀).₄₀₇(C₂xD₄), C₂.₂₃(C₁₀xC₂²:C₄), C₂².₆(C₅xC₂²:C₄), C₁₀.₁₅₂(C₂xC₂²:C₄), (C₂xC₄).₇₁(C₂²xC₁₀), (C₅xC₄oD₄).₅₁C₂², (C₂²xC₄).₁₁₃(C₂xC₁₀), (C₂xC₁₀).₂₀₄(C₂²:C₄), SmallGroup(320,921)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₄ — C₁₀xC₄wrC₂

Chief series

C₁ — C₂ — C₄ — C₂xC₄ — C₂xC₂₀ — C₅xM₄(2) — C₅xC₄wrC₂ — C₁₀xC₄wrC₂

Lower central

C₁ — C₂ — C₄ — C₁₀xC₄wrC₂

Upper central

C₁ — C₂xC₂₀ — C₂²xC₂₀ — C₁₀xC₄wrC₂

Generators and relations for C₁₀xC₄wrC₂
G = < a,b,c,d | a¹⁰=b⁴=c²=d⁴=1, ab=ba, ac=ca, ad=da, cbc=b^-1, bd=db, dcd^-1=b^-1c >

Subgroups: 274 in 170 conjugacy classes, 82 normal (46 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₁₀, C₁₀, C₁₀, C₄², C₄², C₂xC₈, M₄(2), M₄(2), C₂²xC₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂xQ₈, C₄oD₄, C₄oD₄, C₂₀, C₂₀, C₂xC₁₀, C₂xC₁₀, C₄wrC₂, C₂xC₄², C₂xM₄(2), C₂xC₄oD₄, C₄₀, C₂xC₂₀, C₂xC₂₀, C₅xD₄, C₅xD₄, C₅xQ₈, C₅xQ₈, C₂²xC₁₀, C₂²xC₁₀, C₂xC₄wrC₂, C₄xC₂₀, C₄xC₂₀, C₂xC₄₀, C₅xM₄(2), C₅xM₄(2), C₂²xC₂₀, C₂²xC₂₀, D₄xC₁₀, D₄xC₁₀, Q₈xC₁₀, C₅xC₄oD₄, C₅xC₄oD₄, C₅xC₄wrC₂, C₂xC₄xC₂₀, C₁₀xM₄(2), C₁₀xC₄oD₄, C₁₀xC₄wrC₂
Quotients: C₁, C₂, C₄, C₂², C₅, C₂xC₄, D₄, C₂³, C₁₀, C₂²:C₄, C₂²xC₄, C₂xD₄, C₂₀, C₂xC₁₀, C₄wrC₂, C₂xC₂²:C₄, C₂xC₂₀, C₅xD₄, C₂²xC₁₀, C₂xC₄wrC₂, C₅xC₂²:C₄, C₂²xC₂₀, D₄xC₁₀, C₅xC₄wrC₂, C₁₀xC₂²:C₄, C₁₀xC₄wrC₂

Smallest permutation representation of C₁₀xC₄wrC₂
►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 60 64 50)(2 51 65 41)(3 52 66 42)(4 53 67 43)(5 54 68 44)(6 55 69 45)(7 56 70 46)(8 57 61 47)(9 58 62 48)(10 59 63 49)(11 39 29 80)(12 40 30 71)(13 31 21 72)(14 32 22 73)(15 33 23 74)(16 34 24 75)(17 35 25 76)(18 36 26 77)(19 37 27 78)(20 38 28 79)

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

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

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

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

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

140 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	···	4N	4O	4P	5A	5B	5C	5D	8A	8B	8C	8D	10A	···	10L	10M	···	10T	10U	···	10AB	20A	···	20P	20Q	···	20BD	20BE	···	20BL	40A	···	40P
order	1	2	2	2	2	2	2	2	4	4	4	4	4	···	4	4	4	5	5	5	5	8	8	8	8	10	···	10	10	···	10	10	···	10	20	···	20	20	···	20	20	···	20	40	···	40
size	1	1	1	1	2	2	4	4	1	1	1	1	2	···	2	4	4	1	1	1	1	4	4	4	4	1	···	1	2	···	2	4	···	4	1	···	1	2	···	2	4	···	4	4	···	4

140 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2
type	+	+	+	+	+												+	+
image	C₁	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₅	C₁₀	C₁₀	C₁₀	C₁₀	C₂₀	C₂₀	C₂₀	D₄	D₄	C₄wrC₂	C₅xD₄	C₅xD₄	C₅xC₄wrC₂
kernel	C₁₀xC₄wrC₂	C₅xC₄wrC₂	C₂xC₄xC₂₀	C₁₀xM₄(2)	C₁₀xC₄oD₄	D₄xC₁₀	Q₈xC₁₀	C₅xC₄oD₄	C₂xC₄wrC₂	C₄wrC₂	C₂xC₄²	C₂xM₄(2)	C₂xC₄oD₄	C₂xD₄	C₂xQ₈	C₄oD₄	C₂xC₂₀	C₂²xC₁₀	C₁₀	C₂xC₄	C₂³	C₂
# reps	1	4	1	1	1	2	2	4	4	16	4	4	4	8	8	16	3	1	8	12	4	32

Matrix representation of C₁₀xC₄wrC₂ ►in GL₄(F₄₁) generated by

4	0	0	0
0	4	0	0
0	0	40	0
0	0	0	40

40	0	0	0
0	40	0	0
0	0	9	0
0	0	0	32

4	39	0	0
28	37	0	0
0	0	0	32
0	0	9	0

40	0	0	0
37	1	0	0
0	0	1	0
0	0	0	9

G:=sub<GL(4,GF(41))| [4,0,0,0,0,4,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,9,0,0,0,0,32],[4,28,0,0,39,37,0,0,0,0,0,9,0,0,32,0],[40,37,0,0,0,1,0,0,0,0,1,0,0,0,0,9] >;

C₁₀xC₄wrC₂ in GAP, Magma, Sage, TeX

C_{10}\times C_4\wr C_2

% in TeX

G:=Group("C10xC4wrC2");

// GroupNames label

G:=SmallGroup(320,921);

// by ID

G=gap.SmallGroup(320,921);

# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,7004,3511,172,124]);

// Polycyclic

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

// generators/relations

G = C10xC4wrC2 order 320 = 26·5

Direct product of C10 and C4wrC2

G = C₁₀xC₄wrC₂ order 320 = 2⁶·5

Direct product of C₁₀ and C₄wrC₂