D5xC2xC20 - GroupNames

G = D₅xC₂xC₂₀ order 400 = 2⁴·5²

Direct product of C₂xC₂₀ and D₅

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

Aliases: D₅xC₂xC₂₀, C₁₀².₂₇C₂², C₂₀:₃(C₂xC₁₀), C₁₀:₂(C₂xC₂₀), (C₁₀xC₂₀):₈C₂, (C₂xC₂₀):₅C₁₀, (C₅xC₂₀):₈C₂², C₅:₂(C₂²xC₂₀), (C₂xDic₅):₅C₁₀, Dic₅:₃(C₂xC₁₀), D₁₀.₈(C₂xC₁₀), (C₂xC₁₀).₄₆D₁₀, C₅²:₁₀(C₂²xC₄), C₂².₉(D₅xC₁₀), (C₁₀xDic₅):₁₁C₂, (C₅xC₁₀).₂₀C₂³, C₁₀.₂(C₂²xC₁₀), (C₂²xD₅).₄C₁₀, C₁₀.₄₁(C₂²xD₅), (C₅xDic₅):₁₀C₂², (D₅xC₁₀).₂₅C₂², C₂.₁(D₅xC₂xC₁₀), (C₅xC₁₀):₉(C₂xC₄), (D₅xC₂xC₁₀).₈C₂, (C₂xC₁₀).₁₁(C₂xC₁₀), SmallGroup(400,182)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₅ — D₅xC₂xC₂₀

Chief series

C₁ — C₅ — C₁₀ — C₅xC₁₀ — D₅xC₁₀ — D₅xC₂xC₁₀ — D₅xC₂xC₂₀

Lower central

C₅ — D₅xC₂xC₂₀

Upper central

C₁ — C₂xC₂₀

Generators and relations for D₅xC₂xC₂₀
G = < a,b,c,d | a²=b²⁰=c⁵=d²=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c^-1 >

Subgroups: 292 in 124 conjugacy classes, 70 normal (22 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₅, C₂xC₄, C₂xC₄, C₂³, D₅, C₁₀, C₁₀, C₁₀, C₂²xC₄, Dic₅, C₂₀, C₂₀, D₁₀, C₂xC₁₀, C₂xC₁₀, C₅², C₄xD₅, C₂xDic₅, C₂xC₂₀, C₂xC₂₀, C₂²xD₅, C₂²xC₁₀, C₅xD₅, C₅xC₁₀, C₅xC₁₀, C₂xC₄xD₅, C₂²xC₂₀, C₅xDic₅, C₅xC₂₀, D₅xC₁₀, C₁₀², D₅xC₂₀, C₁₀xDic₅, C₁₀xC₂₀, D₅xC₂xC₁₀, D₅xC₂xC₂₀
Quotients: C₁, C₂, C₄, C₂², C₅, C₂xC₄, C₂³, D₅, C₁₀, C₂²xC₄, C₂₀, D₁₀, C₂xC₁₀, C₄xD₅, C₂xC₂₀, C₂²xD₅, C₂²xC₁₀, C₅xD₅, C₂xC₄xD₅, C₂²xC₂₀, D₅xC₁₀, D₅xC₂₀, D₅xC₂xC₁₀, D₅xC₂xC₂₀

Smallest permutation representation of D₅xC₂xC₂₀
►On 80 points

Generators in S₈₀

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

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

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

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

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

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

160 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	4F	4G	4H	5A	5B	5C	5D	5E	···	5N	10A	···	10L	10M	···	10AP	10AQ	···	10BF	20A	···	20P	20Q	···	20BD	20BE	···	20BT
order	1	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4	5	5	5	5	5	···	5	10	···	10	10	···	10	10	···	10	20	···	20	20	···	20	20	···	20
size	1	1	1	1	5	5	5	5	1	1	1	1	5	5	5	5	1	1	1	1	2	···	2	1	···	1	2	···	2	5	···	5	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
type	+	+	+	+	+								+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₄	C₅	C₁₀	C₁₀	C₁₀	C₁₀	C₂₀	D₅	D₁₀	D₁₀	C₄xD₅	C₅xD₅	D₅xC₁₀	D₅xC₁₀	D₅xC₂₀
kernel	D₅xC₂xC₂₀	D₅xC₂₀	C₁₀xDic₅	C₁₀xC₂₀	D₅xC₂xC₁₀	D₅xC₁₀	C₂xC₄xD₅	C₄xD₅	C₂xDic₅	C₂xC₂₀	C₂²xD₅	D₁₀	C₂xC₂₀	C₂₀	C₂xC₁₀	C₁₀	C₂xC₄	C₄	C₂²	C₂
# reps	1	4	1	1	1	8	4	16	4	4	4	32	2	4	2	8	8	16	8	32

Matrix representation of D₅xC₂xC₂₀ ►in GL₃(F₄₁) generated by

40	0	0
0	40	0
0	0	40

21	0	0
0	4	0
0	0	4

1	0	0
0	18	0
0	0	16

40	0	0
0	0	25
0	23	0

G:=sub<GL(3,GF(41))| [40,0,0,0,40,0,0,0,40],[21,0,0,0,4,0,0,0,4],[1,0,0,0,18,0,0,0,16],[40,0,0,0,0,23,0,25,0] >;

D₅xC₂xC₂₀ in GAP, Magma, Sage, TeX

D_5\times C_2\times C_{20}

% in TeX

G:=Group("D5xC2xC20");

// GroupNames label

G:=SmallGroup(400,182);

// by ID

G=gap.SmallGroup(400,182);

# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-5,194,11525]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;

// generators/relations

G = D5xC2xC20 order 400 = 24·52

Direct product of C2xC20 and D5

G = D₅xC₂xC₂₀ order 400 = 2⁴·5²

Direct product of C₂xC₂₀ and D₅