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G = C32×C4.Dic3order 432 = 24·33

Direct product of C32 and C4.Dic3

direct product, metabelian, supersoluble, monomial

Aliases: C32×C4.Dic3, C62.22C12, C12.15C62, C3316M4(2), C62.14Dic3, C6.7(C6×C12), (C6×C12).56S3, C12.1(C3×C12), (C6×C12).33C6, (C3×C62).7C4, C12.124(S3×C6), (C3×C12).18C12, C4.(C32×Dic3), (C3×C12).238D6, (C32×C12).8C4, C6.36(C6×Dic3), (C3×C12).31Dic3, C12.21(C3×Dic3), C32(C32×M4(2)), C22.(C32×Dic3), C3210(C3×M4(2)), (C32×C12).90C22, C3⋊C85(C3×C6), (C3×C3⋊C8)⋊12C6, C4.15(S3×C3×C6), (C3×C6×C12).7C2, C2.3(Dic3×C3×C6), (C32×C3⋊C8)⋊19C2, (C3×C6).57(C2×C12), (C2×C12).45(C3×S3), (C2×C6).14(C3×C12), (C3×C12).95(C2×C6), (C2×C12).10(C3×C6), (C2×C4).2(S3×C32), (C32×C6).65(C2×C4), (C3×C6).77(C2×Dic3), (C2×C6).11(C3×Dic3), SmallGroup(432,470)

Series: Derived Chief Lower central Upper central

C1C6 — C32×C4.Dic3
C1C3C6C12C3×C12C32×C12C32×C3⋊C8 — C32×C4.Dic3
C3C6 — C32×C4.Dic3
C1C3×C12C6×C12

Generators and relations for C32×C4.Dic3
 G = < a,b,c,d,e | a3=b3=c4=1, d6=c2, e2=c2d3, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d5 >

Subgroups: 296 in 184 conjugacy classes, 90 normal (26 characteristic)
C1, C2, C2, C3, C3 [×4], C3 [×4], C4 [×2], C22, C6, C6 [×4], C6 [×17], C8 [×2], C2×C4, C32, C32 [×4], C32 [×4], C12 [×2], C12 [×8], C12 [×8], C2×C6, C2×C6 [×4], C2×C6 [×4], M4(2), C3×C6, C3×C6 [×4], C3×C6 [×17], C3⋊C8 [×2], C24 [×8], C2×C12, C2×C12 [×4], C2×C12 [×4], C33, C3×C12 [×2], C3×C12 [×8], C3×C12 [×8], C62, C62 [×4], C62 [×4], C4.Dic3, C3×M4(2) [×4], C32×C6, C32×C6, C3×C3⋊C8 [×8], C3×C24 [×2], C6×C12, C6×C12 [×4], C6×C12 [×4], C32×C12 [×2], C3×C62, C3×C4.Dic3 [×4], C32×M4(2), C32×C3⋊C8 [×2], C3×C6×C12, C32×C4.Dic3
Quotients: C1, C2 [×3], C3 [×4], C4 [×2], C22, S3, C6 [×12], C2×C4, C32, Dic3 [×2], C12 [×8], D6, C2×C6 [×4], M4(2), C3×S3 [×4], C3×C6 [×3], C2×Dic3, C2×C12 [×4], C3×Dic3 [×8], C3×C12 [×2], S3×C6 [×4], C62, C4.Dic3, C3×M4(2) [×4], S3×C32, C6×Dic3 [×4], C6×C12, C32×Dic3 [×2], S3×C3×C6, C3×C4.Dic3 [×4], C32×M4(2), Dic3×C3×C6, C32×C4.Dic3

Smallest permutation representation of C32×C4.Dic3
On 72 points
Generators in S72
(1 27 17)(2 28 18)(3 29 19)(4 30 20)(5 31 21)(6 32 22)(7 33 23)(8 34 24)(9 35 13)(10 36 14)(11 25 15)(12 26 16)(37 63 60)(38 64 49)(39 65 50)(40 66 51)(41 67 52)(42 68 53)(43 69 54)(44 70 55)(45 71 56)(46 72 57)(47 61 58)(48 62 59)
(1 31 13)(2 32 14)(3 33 15)(4 34 16)(5 35 17)(6 36 18)(7 25 19)(8 26 20)(9 27 21)(10 28 22)(11 29 23)(12 30 24)(37 71 52)(38 72 53)(39 61 54)(40 62 55)(41 63 56)(42 64 57)(43 65 58)(44 66 59)(45 67 60)(46 68 49)(47 69 50)(48 70 51)
(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 46 10 43 7 40 4 37)(2 39 11 48 8 45 5 42)(3 44 12 41 9 38 6 47)(13 49 22 58 19 55 16 52)(14 54 23 51 20 60 17 57)(15 59 24 56 21 53 18 50)(25 62 34 71 31 68 28 65)(26 67 35 64 32 61 29 70)(27 72 36 69 33 66 30 63)

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

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

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

162 conjugacy classes

class 1 2A2B3A···3H3I···3Q4A4B4C6A···6H6I···6AQ8A8B8C8D12A···12P12Q···12BH24A···24AF
order1223···33···34446···66···6888812···1212···1224···24
size1121···12···21121···12···266661···12···26···6

162 irreducible representations

dim1111111111222222222222
type++++-+-
imageC1C2C2C3C4C4C6C6C12C12S3Dic3D6Dic3M4(2)C3×S3C3×Dic3S3×C6C3×Dic3C4.Dic3C3×M4(2)C3×C4.Dic3
kernelC32×C4.Dic3C32×C3⋊C8C3×C6×C12C3×C4.Dic3C32×C12C3×C62C3×C3⋊C8C6×C12C3×C12C62C6×C12C3×C12C3×C12C62C33C2×C12C12C12C2×C6C32C32C3
# reps121822168161611112888841632

Matrix representation of C32×C4.Dic3 in GL3(𝔽73) generated by

6400
010
001
,
800
0640
0064
,
100
0460
0027
,
100
0240
003
,
100
001
0460
G:=sub<GL(3,GF(73))| [64,0,0,0,1,0,0,0,1],[8,0,0,0,64,0,0,0,64],[1,0,0,0,46,0,0,0,27],[1,0,0,0,24,0,0,0,3],[1,0,0,0,0,46,0,1,0] >;

C32×C4.Dic3 in GAP, Magma, Sage, TeX

C_3^2\times C_4.{\rm Dic}_3
% in TeX

G:=Group("C3^2xC4.Dic3");
// GroupNames label

G:=SmallGroup(432,470);
// by ID

G=gap.SmallGroup(432,470);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,252,1037,102,14118]);
// Polycyclic

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

׿
×
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