direct product, metabelian, soluble, monomial, A-group
Aliases: D5×C3.A4, C3.(D5×A4), (C2×C10)⋊C18, C15.(C2×A4), (C2×C30).C6, (C3×D5).A4, (C22×D5)⋊C9, C22⋊(C9×D5), C5⋊(C2×C3.A4), (D5×C2×C6).C3, (C2×C6).(C3×D5), (C5×C3.A4)⋊3C2, SmallGroup(360,42)
Series: Derived ►Chief ►Lower central ►Upper central
| C2×C10 — D5×C3.A4 |
Generators and relations for D5×C3.A4
G = < a,b,c,d,e,f | a5=b2=c3=d2=e2=1, f3=c, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >
Smallest permutation representation of D5×C3.A4
►On 90 points
Generators in S90
(1 66 18 23 37)(2 67 10 24 38)(3 68 11 25 39)(4 69 12 26 40)(5 70 13 27 41)(6 71 14 19 42)(7 72 15 20 43)(8 64 16 21 44)(9 65 17 22 45)(28 62 80 50 84)(29 63 81 51 85)(30 55 73 52 86)(31 56 74 53 87)(32 57 75 54 88)(33 58 76 46 89)(34 59 77 47 90)(35 60 78 48 82)(36 61 79 49 83)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 51)(11 52)(12 53)(13 54)(14 46)(15 47)(16 48)(17 49)(18 50)(19 76)(20 77)(21 78)(22 79)(23 80)(24 81)(25 73)(26 74)(27 75)(37 62)(38 63)(39 55)(40 56)(41 57)(42 58)(43 59)(44 60)(45 61)(64 82)(65 83)(66 84)(67 85)(68 86)(69 87)(70 88)(71 89)(72 90)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)(73 76 79)(74 77 80)(75 78 81)(82 85 88)(83 86 89)(84 87 90)
(1 62)(2 63)(4 56)(5 57)(7 59)(8 60)(10 51)(12 53)(13 54)(15 47)(16 48)(18 50)(20 90)(21 82)(23 84)(24 85)(26 87)(27 88)(28 37)(29 38)(31 40)(32 41)(34 43)(35 44)(64 78)(66 80)(67 81)(69 74)(70 75)(72 77)
(2 63)(3 55)(5 57)(6 58)(8 60)(9 61)(10 51)(11 52)(13 54)(14 46)(16 48)(17 49)(19 89)(21 82)(22 83)(24 85)(25 86)(27 88)(29 38)(30 39)(32 41)(33 42)(35 44)(36 45)(64 78)(65 79)(67 81)(68 73)(70 75)(71 76)
(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 81)(82 83 84 85 86 87 88 89 90)
G:=sub<Sym(90)| (1,66,18,23,37)(2,67,10,24,38)(3,68,11,25,39)(4,69,12,26,40)(5,70,13,27,41)(6,71,14,19,42)(7,72,15,20,43)(8,64,16,21,44)(9,65,17,22,45)(28,62,80,50,84)(29,63,81,51,85)(30,55,73,52,86)(31,56,74,53,87)(32,57,75,54,88)(33,58,76,46,89)(34,59,77,47,90)(35,60,78,48,82)(36,61,79,49,83), (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,51)(11,52)(12,53)(13,54)(14,46)(15,47)(16,48)(17,49)(18,50)(19,76)(20,77)(21,78)(22,79)(23,80)(24,81)(25,73)(26,74)(27,75)(37,62)(38,63)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,61)(64,82)(65,83)(66,84)(67,85)(68,86)(69,87)(70,88)(71,89)(72,90), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72)(73,76,79)(74,77,80)(75,78,81)(82,85,88)(83,86,89)(84,87,90), (1,62)(2,63)(4,56)(5,57)(7,59)(8,60)(10,51)(12,53)(13,54)(15,47)(16,48)(18,50)(20,90)(21,82)(23,84)(24,85)(26,87)(27,88)(28,37)(29,38)(31,40)(32,41)(34,43)(35,44)(64,78)(66,80)(67,81)(69,74)(70,75)(72,77), (2,63)(3,55)(5,57)(6,58)(8,60)(9,61)(10,51)(11,52)(13,54)(14,46)(16,48)(17,49)(19,89)(21,82)(22,83)(24,85)(25,86)(27,88)(29,38)(30,39)(32,41)(33,42)(35,44)(36,45)(64,78)(65,79)(67,81)(68,73)(70,75)(71,76), (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)(73,74,75,76,77,78,79,80,81)(82,83,84,85,86,87,88,89,90)>;
G:=Group( (1,66,18,23,37)(2,67,10,24,38)(3,68,11,25,39)(4,69,12,26,40)(5,70,13,27,41)(6,71,14,19,42)(7,72,15,20,43)(8,64,16,21,44)(9,65,17,22,45)(28,62,80,50,84)(29,63,81,51,85)(30,55,73,52,86)(31,56,74,53,87)(32,57,75,54,88)(33,58,76,46,89)(34,59,77,47,90)(35,60,78,48,82)(36,61,79,49,83), (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,51)(11,52)(12,53)(13,54)(14,46)(15,47)(16,48)(17,49)(18,50)(19,76)(20,77)(21,78)(22,79)(23,80)(24,81)(25,73)(26,74)(27,75)(37,62)(38,63)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,61)(64,82)(65,83)(66,84)(67,85)(68,86)(69,87)(70,88)(71,89)(72,90), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72)(73,76,79)(74,77,80)(75,78,81)(82,85,88)(83,86,89)(84,87,90), (1,62)(2,63)(4,56)(5,57)(7,59)(8,60)(10,51)(12,53)(13,54)(15,47)(16,48)(18,50)(20,90)(21,82)(23,84)(24,85)(26,87)(27,88)(28,37)(29,38)(31,40)(32,41)(34,43)(35,44)(64,78)(66,80)(67,81)(69,74)(70,75)(72,77), (2,63)(3,55)(5,57)(6,58)(8,60)(9,61)(10,51)(11,52)(13,54)(14,46)(16,48)(17,49)(19,89)(21,82)(22,83)(24,85)(25,86)(27,88)(29,38)(30,39)(32,41)(33,42)(35,44)(36,45)(64,78)(65,79)(67,81)(68,73)(70,75)(71,76), (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)(73,74,75,76,77,78,79,80,81)(82,83,84,85,86,87,88,89,90) );
G=PermutationGroup([[(1,66,18,23,37),(2,67,10,24,38),(3,68,11,25,39),(4,69,12,26,40),(5,70,13,27,41),(6,71,14,19,42),(7,72,15,20,43),(8,64,16,21,44),(9,65,17,22,45),(28,62,80,50,84),(29,63,81,51,85),(30,55,73,52,86),(31,56,74,53,87),(32,57,75,54,88),(33,58,76,46,89),(34,59,77,47,90),(35,60,78,48,82),(36,61,79,49,83)], [(1,28),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,35),(9,36),(10,51),(11,52),(12,53),(13,54),(14,46),(15,47),(16,48),(17,49),(18,50),(19,76),(20,77),(21,78),(22,79),(23,80),(24,81),(25,73),(26,74),(27,75),(37,62),(38,63),(39,55),(40,56),(41,57),(42,58),(43,59),(44,60),(45,61),(64,82),(65,83),(66,84),(67,85),(68,86),(69,87),(70,88),(71,89),(72,90)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(28,31,34),(29,32,35),(30,33,36),(37,40,43),(38,41,44),(39,42,45),(46,49,52),(47,50,53),(48,51,54),(55,58,61),(56,59,62),(57,60,63),(64,67,70),(65,68,71),(66,69,72),(73,76,79),(74,77,80),(75,78,81),(82,85,88),(83,86,89),(84,87,90)], [(1,62),(2,63),(4,56),(5,57),(7,59),(8,60),(10,51),(12,53),(13,54),(15,47),(16,48),(18,50),(20,90),(21,82),(23,84),(24,85),(26,87),(27,88),(28,37),(29,38),(31,40),(32,41),(34,43),(35,44),(64,78),(66,80),(67,81),(69,74),(70,75),(72,77)], [(2,63),(3,55),(5,57),(6,58),(8,60),(9,61),(10,51),(11,52),(13,54),(14,46),(16,48),(17,49),(19,89),(21,82),(22,83),(24,85),(25,86),(27,88),(29,38),(30,39),(32,41),(33,42),(35,44),(36,45),(64,78),(65,79),(67,81),(68,73),(70,75),(71,76)], [(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),(73,74,75,76,77,78,79,80,81),(82,83,84,85,86,87,88,89,90)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 9A | ··· | 9F | 10A | 10B | 15A | 15B | 15C | 15D | 18A | ··· | 18F | 30A | 30B | 30C | 30D | 45A | ··· | 45L |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 10 | 10 | 15 | 15 | 15 | 15 | 18 | ··· | 18 | 30 | 30 | 30 | 30 | 45 | ··· | 45 |
| size | 1 | 3 | 5 | 15 | 1 | 1 | 2 | 2 | 3 | 3 | 5 | 5 | 15 | 15 | 4 | ··· | 4 | 6 | 6 | 2 | 2 | 2 | 2 | 20 | ··· | 20 | 6 | 6 | 6 | 6 | 8 | ··· | 8 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 |
| type | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | D5 | C3×D5 | C9×D5 | A4 | C2×A4 | C3.A4 | C2×C3.A4 | D5×A4 | D5×C3.A4 |
| kernel | D5×C3.A4 | C5×C3.A4 | D5×C2×C6 | C2×C30 | C22×D5 | C2×C10 | C3.A4 | C2×C6 | C22 | C3×D5 | C15 | D5 | C5 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 2 | 4 | 12 | 1 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of D5×C3.A4 ►in GL5(𝔽181)
| 13 | 1 | 0 | 0 | 0 |
| 180 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 13 | 0 | 0 | 0 |
| 0 | 180 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 132 | 0 | 0 |
| 0 | 0 | 0 | 132 | 0 |
| 0 | 0 | 0 | 0 | 132 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 180 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 132 | 180 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 180 | 0 | 0 |
| 0 | 0 | 0 | 180 | 0 |
| 0 | 0 | 48 | 49 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 133 | 132 | 179 |
| 0 | 0 | 133 | 0 | 0 |
| 0 | 0 | 156 | 114 | 48 |
G:=sub<GL(5,GF(181))| [13,180,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,13,180,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,132,0,0,0,0,0,132,0,0,0,0,0,132],[1,0,0,0,0,0,1,0,0,0,0,0,180,0,0,0,0,0,1,132,0,0,0,0,180],[1,0,0,0,0,0,1,0,0,0,0,0,180,0,48,0,0,0,180,49,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,133,133,156,0,0,132,0,114,0,0,179,0,48] >;
D5×C3.A4 in GAP, Magma, Sage, TeX
D_5\times C_3.A_4
% in TeX
G:=Group("D5xC3.A4"); // GroupNames label
G:=SmallGroup(360,42);
// by ID
G=gap.SmallGroup(360,42);
# by ID
G:=PCGroup([6,-2,-3,-3,-2,2,-5,43,657,280,10373]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^5=b^2=c^3=d^2=e^2=1,f^3=c,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,f*d*f^-1=d*e=e*d,f*e*f^-1=d>;
// generators/relations
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