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G = C9×C4.Dic3order 432 = 24·33

Direct product of C9 and C4.Dic3

direct product, metacyclic, supersoluble, monomial

Aliases: C9×C4.Dic3, C12.1C36, C36.79D6, C36.9Dic3, C62.18C12, C3⋊C85C18, C4.(C9×Dic3), (C2×C6).6C36, (C3×C36).9C4, C6.6(C2×C36), (C6×C18).2C4, (C3×C9)⋊7M4(2), C4.15(S3×C18), (C6×C36).17C2, (C6×C12).31C6, (C2×C12).5C18, (C2×C36).18S3, C32(C9×M4(2)), C12.119(S3×C6), C12.15(C2×C18), (C3×C12).16C12, C22.(C9×Dic3), C6.30(C6×Dic3), C2.3(Dic3×C18), (C2×C18).2Dic3, (C3×C36).53C22, C12.19(C3×Dic3), C18.18(C2×Dic3), C32.3(C3×M4(2)), (C9×C3⋊C8)⋊12C2, (C3×C3⋊C8).8C6, (C2×C4).2(S3×C9), (C3×C4.Dic3).C3, (C3×C12).90(C2×C6), (C2×C12).40(C3×S3), (C3×C6).52(C2×C12), (C3×C18).28(C2×C4), (C2×C6).9(C3×Dic3), C3.4(C3×C4.Dic3), SmallGroup(432,127)

Series: Derived Chief Lower central Upper central

C1C6 — C9×C4.Dic3
C1C3C6C3×C6C3×C12C3×C36C9×C3⋊C8 — C9×C4.Dic3
C3C6 — C9×C4.Dic3
C1C36C2×C36

Generators and relations for C9×C4.Dic3
 G = < a,b,c,d | a9=b4=1, c6=b2, d2=b2c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c5 >

Subgroups: 116 in 76 conjugacy classes, 45 normal (39 characteristic)
C1, C2, C2, C3 [×2], C3, C4 [×2], C22, C6 [×2], C6 [×5], C8 [×2], C2×C4, C9, C9, C32, C12 [×4], C12 [×2], C2×C6 [×2], C2×C6, M4(2), C18, C18 [×4], C3×C6, C3×C6, C3⋊C8 [×2], C24 [×2], C2×C12 [×2], C2×C12, C3×C9, C36 [×2], C36 [×2], C2×C18, C2×C18, C3×C12 [×2], C62, C4.Dic3, C3×M4(2), C3×C18, C3×C18, C72 [×2], C2×C36, C2×C36, C3×C3⋊C8 [×2], C6×C12, C3×C36 [×2], C6×C18, C9×M4(2), C3×C4.Dic3, C9×C3⋊C8 [×2], C6×C36, C9×C4.Dic3
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, C9, Dic3 [×2], C12 [×2], D6, C2×C6, M4(2), C18 [×3], C3×S3, C2×Dic3, C2×C12, C36 [×2], C2×C18, C3×Dic3 [×2], S3×C6, C4.Dic3, C3×M4(2), S3×C9, C2×C36, C6×Dic3, C9×Dic3 [×2], S3×C18, C9×M4(2), C3×C4.Dic3, Dic3×C18, C9×C4.Dic3

Smallest permutation representation of C9×C4.Dic3
On 72 points
Generators in S72
(1 30 24 9 26 20 5 34 16)(2 31 13 10 27 21 6 35 17)(3 32 14 11 28 22 7 36 18)(4 33 15 12 29 23 8 25 19)(37 68 54 41 72 58 45 64 50)(38 69 55 42 61 59 46 65 51)(39 70 56 43 62 60 47 66 52)(40 71 57 44 63 49 48 67 53)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 22 19 16)(14 23 20 17)(15 24 21 18)(25 34 31 28)(26 35 32 29)(27 36 33 30)(37 40 43 46)(38 41 44 47)(39 42 45 48)(49 52 55 58)(50 53 56 59)(51 54 57 60)(61 64 67 70)(62 65 68 71)(63 66 69 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 48 10 45 7 42 4 39)(2 41 11 38 8 47 5 44)(3 46 12 43 9 40 6 37)(13 58 22 55 19 52 16 49)(14 51 23 60 20 57 17 54)(15 56 24 53 21 50 18 59)(25 66 34 63 31 72 28 69)(26 71 35 68 32 65 29 62)(27 64 36 61 33 70 30 67)

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

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

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

162 conjugacy classes

class 1 2A2B3A3B3C3D3E4A4B4C6A6B6C···6M8A8B8C8D9A···9F9G···9L12A12B12C12D12E···12R18A···18F18G···18AD24A···24H36A···36L36M···36AP72A···72X
order12233333444666···688889···99···91212121212···1218···1818···1824···2436···3636···3672···72
size11211222112112···266661···12···211112···21···12···26···61···12···26···6

162 irreducible representations

dim111111111111111222222222222222222
type++++-+-
imageC1C2C2C3C4C4C6C6C9C12C12C18C18C36C36S3Dic3D6Dic3M4(2)C3×S3C3×Dic3S3×C6C3×Dic3C4.Dic3C3×M4(2)S3×C9C9×Dic3S3×C18C9×Dic3C9×M4(2)C3×C4.Dic3C9×C4.Dic3
kernelC9×C4.Dic3C9×C3⋊C8C6×C36C3×C4.Dic3C3×C36C6×C18C3×C3⋊C8C6×C12C4.Dic3C3×C12C62C3⋊C8C2×C12C12C2×C6C2×C36C36C36C2×C18C3×C9C2×C12C12C12C2×C6C9C32C2×C4C4C4C22C3C3C1
# reps12122242644126121211112222244666612824

Matrix representation of C9×C4.Dic3 in GL2(𝔽37) generated by

90
09
,
310
06
,
80
023
,
019
120
G:=sub<GL(2,GF(37))| [9,0,0,9],[31,0,0,6],[8,0,0,23],[0,12,19,0] >;

C9×C4.Dic3 in GAP, Magma, Sage, TeX

C_9\times C_4.{\rm Dic}_3
% in TeX

G:=Group("C9xC4.Dic3");
// GroupNames label

G:=SmallGroup(432,127);
// by ID

G=gap.SmallGroup(432,127);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-3,84,1037,142,192,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^9=b^4=1,c^6=b^2,d^2=b^2*c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations

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