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G = C3×D365C2order 432 = 24·33

Direct product of C3 and D365C2

direct product, metabelian, supersoluble, monomial

Aliases: C3×D365C2, D3611C6, C12.79D18, Dic1811C6, C62.131D6, (C6×C36)⋊8C2, (C4×D9)⋊9C6, (C2×C12)⋊7D9, C9⋊D47C6, (C2×C36)⋊12C6, (C12×D9)⋊9C2, C4.16(C6×D9), (C3×D36)⋊11C2, C12.75(S3×C6), C36.39(C2×C6), (C6×C12).39S3, D18.6(C2×C6), (C2×C6).23D18, C22.2(C6×D9), (C3×C12).221D6, Dic9.7(C2×C6), C6.52(C22×D9), (C3×Dic18)⋊11C2, (C6×C18).45C22, (C3×C18).41C23, (C3×C36).62C22, C18.18(C22×C6), (C6×D9).12C22, C32.6(C4○D12), (C3×Dic9).14C22, C2.5(C2×C6×D9), C95(C3×C4○D4), (C2×C4)⋊3(C3×D9), C6.22(S3×C2×C6), (C3×C9⋊D4)⋊7C2, (C3×C9)⋊12(C4○D4), (C2×C6).52(S3×C6), C3.1(C3×C4○D12), (C2×C12).25(C3×S3), (C2×C18).31(C2×C6), (C3×C6).155(C22×S3), SmallGroup(432,344)

Series: Derived Chief Lower central Upper central

C1C18 — C3×D365C2
C1C3C9C18C3×C18C6×D9C12×D9 — C3×D365C2
C9C18 — C3×D365C2
C1C12C2×C12

Generators and relations for C3×D365C2
 G = < a,b,c,d | a3=b36=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd=b18c >

Subgroups: 470 in 136 conjugacy classes, 58 normal (42 characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×2], S3 [×2], C6 [×2], C6 [×7], C2×C4, C2×C4 [×2], D4 [×3], Q8, C9, C9, C32, Dic3 [×2], C12 [×4], C12 [×4], D6 [×2], C2×C6 [×2], C2×C6 [×3], C4○D4, D9 [×2], C18, C18 [×4], C3×S3 [×2], C3×C6, C3×C6, Dic6, C4×S3 [×2], D12, C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×3], C3×D4 [×3], C3×Q8, C3×C9, Dic9 [×2], C36 [×2], C36 [×2], D18 [×2], C2×C18, C2×C18, C3×Dic3 [×2], C3×C12 [×2], S3×C6 [×2], C62, C4○D12, C3×C4○D4, C3×D9 [×2], C3×C18, C3×C18, Dic18, C4×D9 [×2], D36, C9⋊D4 [×2], C2×C36, C2×C36, C3×Dic6, S3×C12 [×2], C3×D12, C3×C3⋊D4 [×2], C6×C12, C3×Dic9 [×2], C3×C36 [×2], C6×D9 [×2], C6×C18, D365C2, C3×C4○D12, C3×Dic18, C12×D9 [×2], C3×D36, C3×C9⋊D4 [×2], C6×C36, C3×D365C2
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], C23, D6 [×3], C2×C6 [×7], C4○D4, D9, C3×S3, C22×S3, C22×C6, D18 [×3], S3×C6 [×3], C4○D12, C3×C4○D4, C3×D9, C22×D9, S3×C2×C6, C6×D9 [×3], D365C2, C3×C4○D12, C2×C6×D9, C3×D365C2

Smallest permutation representation of C3×D365C2
On 72 points
Generators in S72
(1 25 13)(2 26 14)(3 27 15)(4 28 16)(5 29 17)(6 30 18)(7 31 19)(8 32 20)(9 33 21)(10 34 22)(11 35 23)(12 36 24)(37 49 61)(38 50 62)(39 51 63)(40 52 64)(41 53 65)(42 54 66)(43 55 67)(44 56 68)(45 57 69)(46 58 70)(47 59 71)(48 60 72)
(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)
(1 58)(2 57)(3 56)(4 55)(5 54)(6 53)(7 52)(8 51)(9 50)(10 49)(11 48)(12 47)(13 46)(14 45)(15 44)(16 43)(17 42)(18 41)(19 40)(20 39)(21 38)(22 37)(23 72)(24 71)(25 70)(26 69)(27 68)(28 67)(29 66)(30 65)(31 64)(32 63)(33 62)(34 61)(35 60)(36 59)
(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)

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

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

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

126 conjugacy classes

class 1 2A2B2C2D3A3B3C3D3E4A4B4C4D4E6A6B6C···6M6N6O6P6Q9A···9I12A12B12C12D12E···12R12S12T12U12V18A···18AA36A···36AJ
order122223333344444666···666669···91212121212···121212121218···1836···36
size1121818112221121818112···2181818182···211112···2181818182···22···2

126 irreducible representations

dim111111111111222222222222222222
type++++++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6S3D6D6C4○D4D9C3×S3D18S3×C6D18S3×C6C3×C4○D4C4○D12C3×D9C6×D9C6×D9D365C2C3×C4○D12C3×D365C2
kernelC3×D365C2C3×Dic18C12×D9C3×D36C3×C9⋊D4C6×C36D365C2Dic18C4×D9D36C9⋊D4C2×C36C6×C12C3×C12C62C3×C9C2×C12C2×C12C12C12C2×C6C2×C6C9C32C2×C4C4C22C3C3C1
# reps112121224242121232643244612612824

Matrix representation of C3×D365C2 in GL2(𝔽37) generated by

100
010
,
220
032
,
032
220
,
10
036
G:=sub<GL(2,GF(37))| [10,0,0,10],[22,0,0,32],[0,22,32,0],[1,0,0,36] >;

C3×D365C2 in GAP, Magma, Sage, TeX

C_3\times D_{36}\rtimes_5C_2
% in TeX

G:=Group("C3xD36:5C2");
// GroupNames label

G:=SmallGroup(432,344);
// by ID

G=gap.SmallGroup(432,344);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,590,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^36=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d=b^18*c>;
// generators/relations

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