S3xC5:D5 - GroupNames

G = S₃×C₅⋊D₅ order 300 = 2²·3·5²

Direct product of S₃ and C₅⋊D₅

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

Aliases: S₃×C₅⋊D₅, C₅²⋊₆D₆, C₁₅⋊₁D₁₀, (C₅×S₃)⋊D₅, C₅⋊₁(S₃×D₅), C₅⋊D₁₅⋊₁C₂, (C₅×C₁₅)⋊₂C₂², (S₃×C₅²)⋊₂C₂, C₃⋊₁(C₂×C₅⋊D₅), (C₃×C₅⋊D₅)⋊₁C₂, SmallGroup(300,38)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₅×C₁₅ — S₃×C₅⋊D₅

Chief series

C₁ — C₅ — C₅² — C₅×C₁₅ — C₃×C₅⋊D₅ — S₃×C₅⋊D₅

Lower central

C₅×C₁₅ — S₃×C₅⋊D₅

Upper central

C₁

Generators and relations for S₃×C₅⋊D₅
G = < a,b,c,d,e | a³=b²=c⁵=d⁵=e²=1, bab=a^-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c^-1, ede=d^-1 >

Subgroups: 608 in 80 conjugacy classes, 28 normal (10 characteristic)
C₁, C₂, C₃, C₂², C₅, S₃, S₃, C₆, D₅, C₁₀, D₆, C₁₅, D₁₀, C₅², C₅×S₃, C₃×D₅, D₁₅, C₅⋊D₅, C₅⋊D₅, C₅×C₁₀, S₃×D₅, C₅×C₁₅, C₂×C₅⋊D₅, C₃×C₅⋊D₅, S₃×C₅², C₅⋊D₁₅, S₃×C₅⋊D₅
Quotients: C₁, C₂, C₂², S₃, D₅, D₆, D₁₀, C₅⋊D₅, S₃×D₅, C₂×C₅⋊D₅, S₃×C₅⋊D₅

Smallest permutation representation of S₃×C₅⋊D₅
►On 75 points

Generators in S₇₅

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

(26 51)(27 52)(28 53)(29 54)(30 55)(31 56)(32 57)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 65)(41 66)(42 67)(43 68)(44 69)(45 70)(46 71)(47 72)(48 73)(49 74)(50 75)

(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)

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

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

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

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

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

42 conjugacy classes

class	1	2A	2B	2C	3	5A	···	5L	6	10A	···	10L	15A	···	15L
order	1	2	2	2	3	5	···	5	6	10	···	10	15	···	15
size	1	3	25	75	2	2	···	2	50	6	···	6	4	···	4

42 irreducible representations

dim	1	1	1	1	2	2	2	2	4
type	+	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	S₃	D₅	D₆	D₁₀	S₃×D₅
kernel	S₃×C₅⋊D₅	C₃×C₅⋊D₅	S₃×C₅²	C₅⋊D₁₅	C₅⋊D₅	C₅×S₃	C₅²	C₁₅	C₅
# reps	1	1	1	1	1	12	1	12	12

Matrix representation of S₃×C₅⋊D₅ ►in GL₆(𝔽₃₁)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	29	30	0	0
0	0	3	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	28	30	0	0
0	0	0	0	1	0
0	0	0	0	0	1

0	1	0	0	0	0
30	18	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	13	1
0	0	0	0	17	30

0	1	0	0	0	0
30	18	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	18	19
0	0	0	0	13	0

30	18	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	13	1
0	0	0	0	18	18

G:=sub<GL(6,GF(31))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,29,3,0,0,0,0,30,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,28,0,0,0,0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,30,0,0,0,0,1,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,17,0,0,0,0,1,30],[0,30,0,0,0,0,1,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,13,0,0,0,0,19,0],[30,0,0,0,0,0,18,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,18,0,0,0,0,1,18] >;

S₃×C₅⋊D₅ in GAP, Magma, Sage, TeX

S_3\times C_5\rtimes D_5

% in TeX

G:=Group("S3xC5:D5");

// GroupNames label

G:=SmallGroup(300,38);

// by ID

G=gap.SmallGroup(300,38);

# by ID

G:=PCGroup([5,-2,-2,-3,-5,-5,67,963,6004]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^5=d^5=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;

// generators/relations

G = S3×C5⋊D5 order 300 = 22·3·52

Direct product of S3 and C5⋊D5

G = S₃×C₅⋊D₅ order 300 = 2²·3·5²

Direct product of S₃ and C₅⋊D₅