D5xM5(2) - GroupNames

G = D₅×M₅(2) order 320 = 2⁶·5

Direct product of D₅ and M₅(2)

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₅×M₅(2), C₁₆⋊₆D₁₀, C₈₀⋊₇C₂², C₄₀.₆₆C₂³, (D₅×C₁₆)⋊₇C₂, (C₈×D₅).₉C₄, C₈₀⋊C₂⋊₅C₂, (C₄×D₅).₁C₈, C₈.₃₅(C₄×D₅), C₄.₁₅(C₈×D₅), C₅⋊₅(C₂×M₅(2)), C₂₀.₃₆(C₂×C₈), C₄₀.₇₉(C₂×C₄), C₂².₇(C₈×D₅), D₁₀.₁₁(C₂×C₈), (C₂×C₈).₂₇₂D₁₀, (C₅×M₅(2))⋊₅C₂, (C₂×Dic₅).₇C₈, C₂₀.₄C₈⋊₁₃C₂, (C₂²×D₅).₅C₈, C₈.₆₀(C₂²×D₅), C₅⋊₂C₁₆⋊₁₁C₂², C₁₀.₃₉(C₂²×C₈), Dic₅.₁₃(C₂×C₈), (C₈×D₅).₄₇C₂², (C₂×C₄₀).₂₃₀C₂², C₂₀.₁₉₀(C₂²×C₄), C₂.₁₆(D₅×C₂×C₈), (C₂×C₄×D₅).₁₂C₄, (D₅×C₂×C₈).₂₈C₂, C₄.₁₀₅(C₂×C₄×D₅), (C₂×C₅⋊₂C₈).₁₁C₄, (C₂×C₁₀).₂₀(C₂×C₈), C₅⋊₂C₈.₄₄(C₂×C₄), (C₄×D₅).₈₂(C₂×C₄), (C₂×C₄).₁₄₈(C₄×D₅), (C₂×C₂₀).₂₄₅(C₂×C₄), SmallGroup(320,533)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₀ — D₅×M₅(2)

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — C₄₀ — C₈×D₅ — D₅×C₂×C₈ — D₅×M₅(2)

Lower central

C₅ — C₁₀ — D₅×M₅(2)

Upper central

C₁ — C₈ — M₅(2)

Generators and relations for D₅×M₅(2)
G = < a,b,c,d | a⁵=b²=c¹⁶=d²=1, bab=a^-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c⁹ >

Subgroups: 238 in 90 conjugacy classes, 53 normal (41 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₈, C₂×C₄, C₂×C₄, C₂³, D₅, D₅, C₁₀, C₁₀, C₁₆, C₁₆, C₂×C₈, C₂×C₈, C₂²×C₄, Dic₅, C₂₀, D₁₀, D₁₀, C₂×C₁₀, C₂×C₁₆, M₅(2), M₅(2), C₂²×C₈, C₅⋊₂C₈, C₄₀, C₄×D₅, C₂×Dic₅, C₂×C₂₀, C₂²×D₅, C₂×M₅(2), C₅⋊₂C₁₆, C₈₀, C₈×D₅, C₂×C₅⋊₂C₈, C₂×C₄₀, C₂×C₄×D₅, D₅×C₁₆, C₈₀⋊C₂, C₂₀.₄C₈, C₅×M₅(2), D₅×C₂×C₈, D₅×M₅(2)
Quotients: C₁, C₂, C₄, C₂², C₈, C₂×C₄, C₂³, D₅, C₂×C₈, C₂²×C₄, D₁₀, M₅(2), C₂²×C₈, C₄×D₅, C₂²×D₅, C₂×M₅(2), C₈×D₅, C₂×C₄×D₅, D₅×C₂×C₈, D₅×M₅(2)

Smallest permutation representation of D₅×M₅(2)
►On 80 points

Generators in S₈₀

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

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

(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)

(2 10)(4 12)(6 14)(8 16)(17 25)(19 27)(21 29)(23 31)(34 42)(36 44)(38 46)(40 48)(49 57)(51 59)(53 61)(55 63)(65 73)(67 75)(69 77)(71 79)

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

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

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

80 conjugacy classes

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	5A	5B	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J	8K	8L	10A	10B	10C	10D	16A	···	16H	16I	···	16P	20A	20B	20C	20D	20E	20F	40A	···	40H	40I	40J	40K	40L	80A	···	80P
order	1	2	2	2	2	2	4	4	4	4	4	4	5	5	8	8	8	8	8	8	8	8	8	8	8	8	10	10	10	10	16	···	16	16	···	16	20	20	20	20	20	20	40	···	40	40	40	40	40	80	···	80
size	1	1	2	5	5	10	1	1	2	5	5	10	2	2	1	1	1	1	2	2	5	5	5	5	10	10	2	2	4	4	2	···	2	10	···	10	2	2	2	2	4	4	2	···	2	4	4	4	4	4	···	4

80 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4
type	+	+	+	+	+	+							+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₈	C₈	C₈	D₅	D₁₀	D₁₀	M₅(2)	C₄×D₅	C₄×D₅	C₈×D₅	C₈×D₅	D₅×M₅(2)
kernel	D₅×M₅(2)	D₅×C₁₆	C₈₀⋊C₂	C₂₀.₄C₈	C₅×M₅(2)	D₅×C₂×C₈	C₈×D₅	C₂×C₅⋊₂C₈	C₂×C₄×D₅	C₄×D₅	C₂×Dic₅	C₂²×D₅	M₅(2)	C₁₆	C₂×C₈	D₅	C₈	C₂×C₄	C₄	C₂²	C₁
# reps	1	2	2	1	1	1	4	2	2	8	4	4	2	4	2	8	4	4	8	8	8

Matrix representation of D₅×M₅(2) ►in GL₄(𝔽₂₄₁) generated by

1	0	0	0
0	1	0	0
0	0	0	1
0	0	240	189

240	0	0	0
0	240	0	0
0	0	0	1
0	0	1	0

1	54	0	0
111	240	0	0
0	0	1	0
0	0	0	1

1	0	0	0
116	240	0	0
0	0	1	0
0	0	0	1

G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,0,240,0,0,1,189],[240,0,0,0,0,240,0,0,0,0,0,1,0,0,1,0],[1,111,0,0,54,240,0,0,0,0,1,0,0,0,0,1],[1,116,0,0,0,240,0,0,0,0,1,0,0,0,0,1] >;

D₅×M₅(2) in GAP, Magma, Sage, TeX

D_5\times M_5(2)

% in TeX

G:=Group("D5xM5(2)");

// GroupNames label

G:=SmallGroup(320,533);

// by ID

G=gap.SmallGroup(320,533);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,58,80,102,12550]);

// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^16=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^9>;

// generators/relations

G = D5×M5(2) order 320 = 26·5

Direct product of D5 and M5(2)

G = D₅×M₅(2) order 320 = 2⁶·5

Direct product of D₅ and M₅(2)