C5xC8:D5 - GroupNames

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

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

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

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

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

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

130 conjugacy classes

class	1	2A	2B	4A	4B	4C	5A	5B	5C	5D	5E	···	5N	8A	8B	8C	8D	10A	10B	10C	10D	10E	···	10N	10O	10P	10Q	10R	20A	···	20H	20I	···	20AB	20AC	20AD	20AE	20AF	40A	···	40AV	40AW	···	40BD
order	1	2	2	4	4	4	5	5	5	5	5	···	5	8	8	8	8	10	10	10	10	10	···	10	10	10	10	10	20	···	20	20	···	20	20	20	20	20	40	···	40	40	···	40
size	1	1	10	1	1	10	1	1	1	1	2	···	2	2	2	10	10	1	1	1	1	2	···	2	10	10	10	10	1	···	1	2	···	2	10	10	10	10	2	···	2	10	···	10

130 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2
type	+	+	+	+									+		+
image	C₁	C₂	C₂	C₂	C₄	C₄	C₅	C₁₀	C₁₀	C₁₀	C₂₀	C₂₀	D₅	M₄(2)	D₁₀	C₄×D₅	C₅×D₅	C₈⋊D₅	C₅×M₄(2)	D₅×C₁₀	D₅×C₂₀	C₅×C₈⋊D₅
kernel	C₅×C₈⋊D₅	C₅×C₅⋊₂C₈	C₅×C₄₀	D₅×C₂₀	C₅×Dic₅	D₅×C₁₀	C₈⋊D₅	C₅⋊₂C₈	C₄₀	C₄×D₅	Dic₅	D₁₀	C₄₀	C₅²	C₂₀	C₁₀	C₈	C₅	C₅	C₄	C₂	C₁
# reps	1	1	1	1	2	2	4	4	4	4	8	8	2	2	2	4	8	8	8	8	16	32

Matrix representation of C₅×C₈⋊D₅ ►in GL₂(𝔽₄₁) generated by

10	0
0	10

27	0
0	14

10	0
0	37

0	37
10	0

G:=sub<GL(2,GF(41))| [10,0,0,10],[27,0,0,14],[10,0,0,37],[0,10,37,0] >;

C₅×C₈⋊D₅ in GAP, Magma, Sage, TeX

C_5\times C_8\rtimes D_5

% in TeX

G:=Group("C5xC8:D5");

// GroupNames label

G:=SmallGroup(400,77);

// by ID

G=gap.SmallGroup(400,77);

# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-5,505,127,69,11525]);

// Polycyclic

G:=Group<a,b,c,d|a^5=b^8=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^5,d*c*d=c^-1>;

// generators/relations

Export

Subgroup lattice of C₅×C₈⋊D₅ in TeX

G = C5×C8⋊D5 order 400 = 24·52

Direct product of C5 and C8⋊D5

G = C₅×C₈⋊D₅ order 400 = 2⁴·5²

Direct product of C₅ and C₈⋊D₅