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^[×4], C₄^[×2], C₄^[×2], C₂², C₂²^[×4], C₅, C₈^[×2], C₈^[×2], C₂×C₄, C₂×C₄^[×5], C₂³, D₅^[×2], D₅, C₁₀, C₁₀, C₁₆^[×2], C₁₆^[×2], C₂×C₈, C₂×C₈^[×5], C₂²×C₄, Dic₅^[×2], C₂₀^[×2], D₁₀^[×2], D₁₀^[×2], C₂×C₁₀, C₂×C₁₆^[×2], M₅(2), M₅(2)^[×3], C₂²×C₈, C₅⋊₂C₈^[×2], C₄₀^[×2], C₄×D₅^[×4], C₂×Dic₅, C₂×C₂₀, C₂²×D₅, C₂×M₅(2), C₅⋊₂C₁₆^[×2], C₈₀^[×2], C₈×D₅^[×4], C₂×C₅⋊₂C₈, C₂×C₄₀, C₂×C₄×D₅, D₅×C₁₆^[×2], C₈₀⋊C₂^[×2], C₂₀.₄C₈, C₅×M₅(2), D₅×C₂×C₈, D₅×M₅(2)
Quotients: C₁, C₂^[×7], C₄^[×4], C₂²^[×7], C₈^[×4], C₂×C₄^[×6], C₂³, D₅, C₂×C₈^[×6], C₂²×C₄, D₁₀^[×3], M₅(2)^[×2], C₂²×C₈, C₄×D₅^[×2], C₂²×D₅, C₂×M₅(2), C₈×D₅^[×2], C₂×C₄×D₅, D₅×C₂×C₈, D₅×M₅(2)

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

Generators in S₈₀

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

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

(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)(18 26)(20 28)(22 30)(24 32)(33 41)(35 43)(37 45)(39 47)(50 58)(52 60)(54 62)(56 64)(65 73)(67 75)(69 77)(71 79)

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

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

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