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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, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C4○D4, D9, C18, C18, C3×S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C3×C9, Dic9, C36, C36, D18, C2×C18, C2×C18, C3×Dic3, C3×C12, S3×C6, C62, C4○D12, C3×C4○D4, C3×D9, C3×C18, C3×C18, Dic18, C4×D9, D36, C9⋊D4, C2×C36, C2×C36, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C6×C12, C3×Dic9, C3×C36, C6×D9, C6×C18, D365C2, C3×C4○D12, C3×Dic18, C12×D9, C3×D36, C3×C9⋊D4, C6×C36, C3×D365C2
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C4○D4, D9, C3×S3, C22×S3, C22×C6, D18, S3×C6, C4○D12, C3×C4○D4, C3×D9, C22×D9, S3×C2×C6, C6×D9, 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 40)(2 39)(3 38)(4 37)(5 72)(6 71)(7 70)(8 69)(9 68)(10 67)(11 66)(12 65)(13 64)(14 63)(15 62)(16 61)(17 60)(18 59)(19 58)(20 57)(21 56)(22 55)(23 54)(24 53)(25 52)(26 51)(27 50)(28 49)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)
(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), (1,40)(2,39)(3,38)(4,37)(5,72)(6,71)(7,70)(8,69)(9,68)(10,67)(11,66)(12,65)(13,64)(14,63)(15,62)(16,61)(17,60)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41), (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), (1,40)(2,39)(3,38)(4,37)(5,72)(6,71)(7,70)(8,69)(9,68)(10,67)(11,66)(12,65)(13,64)(14,63)(15,62)(16,61)(17,60)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41), (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)], [(1,40),(2,39),(3,38),(4,37),(5,72),(6,71),(7,70),(8,69),(9,68),(10,67),(11,66),(12,65),(13,64),(14,63),(15,62),(16,61),(17,60),(18,59),(19,58),(20,57),(21,56),(22,55),(23,54),(24,53),(25,52),(26,51),(27,50),(28,49),(29,48),(30,47),(31,46),(32,45),(33,44),(34,43),(35,42),(36,41)], [(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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