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## G = D5×C3.A4order 360 = 23·32·5

### Direct product of D5 and C3.A4

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

 Derived series C1 — C2×C10 — D5×C3.A4
 Chief series C1 — C5 — C2×C10 — C2×C30 — C5×C3.A4 — D5×C3.A4
 Lower central C2×C10 — D5×C3.A4
 Upper central C1 — C3

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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