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## G = C2×C3⋊D15order 180 = 22·32·5

### Direct product of C2 and C3⋊D15

Aliases: C2×C3⋊D15, C6⋊D15, C301S3, C32D30, C156D6, C326D10, C10⋊(C3⋊S3), (C3×C6)⋊2D5, (C3×C30)⋊1C2, (C3×C15)⋊6C22, C52(C2×C3⋊S3), SmallGroup(180,36)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C15 — C2×C3⋊D15
 Chief series C1 — C5 — C15 — C3×C15 — C3⋊D15 — C2×C3⋊D15
 Lower central C3×C15 — C2×C3⋊D15
 Upper central C1 — C2

Generators and relations for C2×C3⋊D15
G = < a,b,c,d | a2=b3=c15=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 420 in 60 conjugacy classes, 27 normal (9 characteristic)
C1, C2, C2, C3, C22, C5, S3, C6, C32, D5, C10, D6, C15, C3⋊S3, C3×C6, D10, D15, C30, C2×C3⋊S3, C3×C15, D30, C3⋊D15, C3×C30, C2×C3⋊D15
Quotients: C1, C2, C22, S3, D5, D6, C3⋊S3, D10, D15, C2×C3⋊S3, D30, C3⋊D15, C2×C3⋊D15

Smallest permutation representation of C2×C3⋊D15
On 90 points
Generators in S90
(1 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 61)(17 62)(18 63)(19 64)(20 65)(21 66)(22 67)(23 68)(24 69)(25 70)(26 71)(27 72)(28 73)(29 74)(30 75)(31 85)(32 86)(33 87)(34 88)(35 89)(36 90)(37 76)(38 77)(39 78)(40 79)(41 80)(42 81)(43 82)(44 83)(45 84)
(1 39 19)(2 40 20)(3 41 21)(4 42 22)(5 43 23)(6 44 24)(7 45 25)(8 31 26)(9 32 27)(10 33 28)(11 34 29)(12 35 30)(13 36 16)(14 37 17)(15 38 18)(46 78 64)(47 79 65)(48 80 66)(49 81 67)(50 82 68)(51 83 69)(52 84 70)(53 85 71)(54 86 72)(55 87 73)(56 88 74)(57 89 75)(58 90 61)(59 76 62)(60 77 63)
(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)
(1 60)(2 59)(3 58)(4 57)(5 56)(6 55)(7 54)(8 53)(9 52)(10 51)(11 50)(12 49)(13 48)(14 47)(15 46)(16 80)(17 79)(18 78)(19 77)(20 76)(21 90)(22 89)(23 88)(24 87)(25 86)(26 85)(27 84)(28 83)(29 82)(30 81)(31 71)(32 70)(33 69)(34 68)(35 67)(36 66)(37 65)(38 64)(39 63)(40 62)(41 61)(42 75)(43 74)(44 73)(45 72)

G:=sub<Sym(90)| (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,61)(17,62)(18,63)(19,64)(20,65)(21,66)(22,67)(23,68)(24,69)(25,70)(26,71)(27,72)(28,73)(29,74)(30,75)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(37,76)(38,77)(39,78)(40,79)(41,80)(42,81)(43,82)(44,83)(45,84), (1,39,19)(2,40,20)(3,41,21)(4,42,22)(5,43,23)(6,44,24)(7,45,25)(8,31,26)(9,32,27)(10,33,28)(11,34,29)(12,35,30)(13,36,16)(14,37,17)(15,38,18)(46,78,64)(47,79,65)(48,80,66)(49,81,67)(50,82,68)(51,83,69)(52,84,70)(53,85,71)(54,86,72)(55,87,73)(56,88,74)(57,89,75)(58,90,61)(59,76,62)(60,77,63), (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), (1,60)(2,59)(3,58)(4,57)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,80)(17,79)(18,78)(19,77)(20,76)(21,90)(22,89)(23,88)(24,87)(25,86)(26,85)(27,84)(28,83)(29,82)(30,81)(31,71)(32,70)(33,69)(34,68)(35,67)(36,66)(37,65)(38,64)(39,63)(40,62)(41,61)(42,75)(43,74)(44,73)(45,72)>;

G:=Group( (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,61)(17,62)(18,63)(19,64)(20,65)(21,66)(22,67)(23,68)(24,69)(25,70)(26,71)(27,72)(28,73)(29,74)(30,75)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(37,76)(38,77)(39,78)(40,79)(41,80)(42,81)(43,82)(44,83)(45,84), (1,39,19)(2,40,20)(3,41,21)(4,42,22)(5,43,23)(6,44,24)(7,45,25)(8,31,26)(9,32,27)(10,33,28)(11,34,29)(12,35,30)(13,36,16)(14,37,17)(15,38,18)(46,78,64)(47,79,65)(48,80,66)(49,81,67)(50,82,68)(51,83,69)(52,84,70)(53,85,71)(54,86,72)(55,87,73)(56,88,74)(57,89,75)(58,90,61)(59,76,62)(60,77,63), (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), (1,60)(2,59)(3,58)(4,57)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,80)(17,79)(18,78)(19,77)(20,76)(21,90)(22,89)(23,88)(24,87)(25,86)(26,85)(27,84)(28,83)(29,82)(30,81)(31,71)(32,70)(33,69)(34,68)(35,67)(36,66)(37,65)(38,64)(39,63)(40,62)(41,61)(42,75)(43,74)(44,73)(45,72) );

G=PermutationGroup([[(1,46),(2,47),(3,48),(4,49),(5,50),(6,51),(7,52),(8,53),(9,54),(10,55),(11,56),(12,57),(13,58),(14,59),(15,60),(16,61),(17,62),(18,63),(19,64),(20,65),(21,66),(22,67),(23,68),(24,69),(25,70),(26,71),(27,72),(28,73),(29,74),(30,75),(31,85),(32,86),(33,87),(34,88),(35,89),(36,90),(37,76),(38,77),(39,78),(40,79),(41,80),(42,81),(43,82),(44,83),(45,84)], [(1,39,19),(2,40,20),(3,41,21),(4,42,22),(5,43,23),(6,44,24),(7,45,25),(8,31,26),(9,32,27),(10,33,28),(11,34,29),(12,35,30),(13,36,16),(14,37,17),(15,38,18),(46,78,64),(47,79,65),(48,80,66),(49,81,67),(50,82,68),(51,83,69),(52,84,70),(53,85,71),(54,86,72),(55,87,73),(56,88,74),(57,89,75),(58,90,61),(59,76,62),(60,77,63)], [(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)], [(1,60),(2,59),(3,58),(4,57),(5,56),(6,55),(7,54),(8,53),(9,52),(10,51),(11,50),(12,49),(13,48),(14,47),(15,46),(16,80),(17,79),(18,78),(19,77),(20,76),(21,90),(22,89),(23,88),(24,87),(25,86),(26,85),(27,84),(28,83),(29,82),(30,81),(31,71),(32,70),(33,69),(34,68),(35,67),(36,66),(37,65),(38,64),(39,63),(40,62),(41,61),(42,75),(43,74),(44,73),(45,72)]])

C2×C3⋊D15 is a maximal subgroup of
C30.D6  C327D20  C15⋊D12  C6.D30  C3⋊D60  D62D15  C60⋊S3  C62⋊D5  C2×D5×C3⋊S3  C2×S3×D15
C2×C3⋊D15 is a maximal quotient of
C12.D15  C60⋊S3  C62⋊D5

48 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 5A 5B 6A 6B 6C 6D 10A 10B 15A ··· 15P 30A ··· 30P order 1 2 2 2 3 3 3 3 5 5 6 6 6 6 10 10 15 ··· 15 30 ··· 30 size 1 1 45 45 2 2 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2

48 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 S3 D5 D6 D10 D15 D30 kernel C2×C3⋊D15 C3⋊D15 C3×C30 C30 C3×C6 C15 C32 C6 C3 # reps 1 2 1 4 2 4 2 16 16

Matrix representation of C2×C3⋊D15 in GL6(𝔽31)

 30 0 0 0 0 0 0 30 0 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 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 29 25 0 0 0 0 16 1 0 0 0 0 0 0 19 3 0 0 0 0 28 11
,
 13 13 0 0 0 0 18 30 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 22 27 0 0 0 0 4 12
,
 27 22 0 0 0 0 12 4 0 0 0 0 0 0 1 0 0 0 0 0 15 30 0 0 0 0 0 0 18 18 0 0 0 0 1 13

G:=sub<GL(6,GF(31))| [30,0,0,0,0,0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,29,16,0,0,0,0,25,1,0,0,0,0,0,0,19,28,0,0,0,0,3,11],[13,18,0,0,0,0,13,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,22,4,0,0,0,0,27,12],[27,12,0,0,0,0,22,4,0,0,0,0,0,0,1,15,0,0,0,0,0,30,0,0,0,0,0,0,18,1,0,0,0,0,18,13] >;

C2×C3⋊D15 in GAP, Magma, Sage, TeX

C_2\times C_3\rtimes D_{15}
% in TeX

G:=Group("C2xC3:D15");
// GroupNames label

G:=SmallGroup(180,36);
// by ID

G=gap.SmallGroup(180,36);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-5,122,483,3604]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^3=c^15=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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