C10xC4oD4 - GroupNames

G = C₁₀xC₄oD₄ order 160 = 2⁵·5

Direct product of C₁₀ and C₄oD₄

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

Aliases: C₁₀xC₄oD₄, C₂₀.₅₅C₂³, C₁₀.₁₈C₂⁴, D₄:₃(C₂xC₁₀), (C₂xD₄):₇C₁₀, (C₂xQ₈):₆C₁₀, Q₈:₃(C₂xC₁₀), (D₄xC₁₀):₁₆C₂, (C₂²xC₄):₆C₁₀, (Q₈xC₁₀):₁₃C₂, (C₂²xC₂₀):₁₃C₂, (C₂xC₂₀):₁₆C₂², (C₅xD₄):₁₂C₂², (C₂xC₁₀).₆C₂³, C₂.₃(C₂³xC₁₀), C₄.₈(C₂²xC₁₀), (C₅xQ₈):₁₁C₂², C₂³.₁₁(C₂xC₁₀), C₂².₁(C₂²xC₁₀), (C₂²xC₁₀).₃₀C₂², (C₂xC₂₀)o(C₅xD₄), (C₂xC₂₀)o(C₅xQ₈), (C₂xC₄):₅(C₂xC₁₀), (C₂xC₂₀)o(Q₈xC₁₀), SmallGroup(160,231)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂ — C₁₀xC₄oD₄

Chief series

C₁ — C₂ — C₁₀ — C₂xC₁₀ — C₅xD₄ — C₅xC₄oD₄ — C₁₀xC₄oD₄

Lower central

C₁ — C₂ — C₁₀xC₄oD₄

Upper central

C₁ — C₂xC₂₀ — C₁₀xC₄oD₄

Generators and relations for C₁₀xC₄oD₄
G = < a,b,c,d | a¹⁰=b⁴=d²=1, c²=b², ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b²c >

Subgroups: 188 in 164 conjugacy classes, 140 normal (12 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₅, C₂xC₄, C₂xC₄, D₄, Q₈, C₂³, C₁₀, C₁₀, C₁₀, C₂²xC₄, C₂xD₄, C₂xQ₈, C₄oD₄, C₂₀, C₂xC₁₀, C₂xC₁₀, C₂xC₁₀, C₂xC₄oD₄, C₂xC₂₀, C₂xC₂₀, C₅xD₄, C₅xQ₈, C₂²xC₁₀, C₂²xC₂₀, D₄xC₁₀, Q₈xC₁₀, C₅xC₄oD₄, C₁₀xC₄oD₄
Quotients: C₁, C₂, C₂², C₅, C₂³, C₁₀, C₄oD₄, C₂⁴, C₂xC₁₀, C₂xC₄oD₄, C₂²xC₁₀, C₅xC₄oD₄, C₂³xC₁₀, C₁₀xC₄oD₄

Smallest permutation representation of C₁₀xC₄oD₄
►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 41 28 33)(2 42 29 34)(3 43 30 35)(4 44 21 36)(5 45 22 37)(6 46 23 38)(7 47 24 39)(8 48 25 40)(9 49 26 31)(10 50 27 32)(11 55 80 63)(12 56 71 64)(13 57 72 65)(14 58 73 66)(15 59 74 67)(16 60 75 68)(17 51 76 69)(18 52 77 70)(19 53 78 61)(20 54 79 62)

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

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

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

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

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

C₁₀xC₄oD₄ is a maximal subgroup of
C₄oD₄:Dic₅ C₂₀.(C₂xD₄) (D₄xC₁₀).₂₄C₄ (D₄xC₁₀):₂₁C₄ (D₄xC₁₀).₂₉C₄ (C₅xD₄):₁₄D₄ (C₅xD₄).₃₂D₄ (D₄xC₁₀):₂₂C₄ C₂₀.₇₆C₂⁴ C₂₀.C₂⁴ C₁₀.₁₀₄2_-¹⁺⁴ C₁₀.₁₀₅2_-¹⁺⁴ C₁₀.₁₀₆2_-¹⁺⁴ (C₂xC₂₀):₁₅D₄ C₁₀.₁₄₅2₊¹⁺⁴ C₁₀.₁₄₆2₊¹⁺⁴ C₁₀.₁₀₇2_-¹⁺⁴ (C₂xC₂₀):₁₇D₄ C₁₀.₁₄₇2₊¹⁺⁴ C₁₀.₁₄₈2₊¹⁺⁴ C₁₀.C₂⁵
C₁₀xC₄oD₄ is a maximal quotient of
D₄xC₂xC₂₀ Q₈xC₂xC₂₀

100 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	4A	4B	4C	4D	4E	···	4J	5A	5B	5C	5D	10A	···	10L	10M	···	10AJ	20A	···	20P	20Q	···	20AN
order	1	2	2	2	2	···	2	4	4	4	4	4	···	4	5	5	5	5	10	···	10	10	···	10	20	···	20	20	···	20
size	1	1	1	1	2	···	2	1	1	1	1	2	···	2	1	1	1	1	1	···	1	2	···	2	1	···	1	2	···	2

100 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2
type	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₅	C₁₀	C₁₀	C₁₀	C₁₀	C₄oD₄	C₅xC₄oD₄
kernel	C₁₀xC₄oD₄	C₂²xC₂₀	D₄xC₁₀	Q₈xC₁₀	C₅xC₄oD₄	C₂xC₄oD₄	C₂²xC₄	C₂xD₄	C₂xQ₈	C₄oD₄	C₁₀	C₂
# reps	1	3	3	1	8	4	12	12	4	32	4	16

Matrix representation of C₁₀xC₄oD₄ ►in GL₃(F₄₁) generated by

40	0	0
0	10	0
0	0	10

40	0	0
0	9	0
0	0	9

1	0	0
0	0	1
0	40	0

1	0	0
0	0	40
0	40	0

G:=sub<GL(3,GF(41))| [40,0,0,0,10,0,0,0,10],[40,0,0,0,9,0,0,0,9],[1,0,0,0,0,40,0,1,0],[1,0,0,0,0,40,0,40,0] >;

C₁₀xC₄oD₄ in GAP, Magma, Sage, TeX

C_{10}\times C_4\circ D_4

% in TeX

G:=Group("C10xC4oD4");

// GroupNames label

G:=SmallGroup(160,231);

// by ID

G=gap.SmallGroup(160,231);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-2,985,374]);

// Polycyclic

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

// generators/relations

G = C10xC4oD4 order 160 = 25·5

Direct product of C10 and C4oD4

G = C₁₀xC₄oD₄ order 160 = 2⁵·5

Direct product of C₁₀ and C₄oD₄