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## G = C3×D4.D9order 432 = 24·33

### Direct product of C3 and D4.D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C36 — C3×D4.D9
 Chief series C1 — C3 — C9 — C18 — C36 — C3×C36 — C3×Dic18 — C3×D4.D9
 Lower central C9 — C18 — C36 — C3×D4.D9
 Upper central C1 — C6 — C12 — C3×D4

Generators and relations for C3×D4.D9
G = < a,b,c,d,e | a3=b4=c2=d9=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >

Subgroups: 230 in 76 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C8, D4, Q8, C9, C9, C32, Dic3, C12, C12, C2×C6, SD16, C18, C18, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C3×D4, C3×D4, C3×Q8, C3×C9, Dic9, C36, C36, C2×C18, C3×Dic3, C3×C12, C62, D4.S3, C3×SD16, C3×C18, C3×C18, C9⋊C8, Dic18, D4×C9, D4×C9, C3×C3⋊C8, C3×Dic6, D4×C32, C3×Dic9, C3×C36, C6×C18, D4.D9, C3×D4.S3, C3×C9⋊C8, C3×Dic18, D4×C3×C9, C3×D4.D9
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, SD16, D9, C3×S3, C3⋊D4, C3×D4, D18, S3×C6, D4.S3, C3×SD16, C3×D9, C9⋊D4, C3×C3⋊D4, C6×D9, D4.D9, C3×D4.S3, C3×C9⋊D4, C3×D4.D9

Smallest permutation representation of C3×D4.D9
On 72 points
Generators in S72
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)
(1 28 10 19)(2 29 11 20)(3 30 12 21)(4 31 13 22)(5 32 14 23)(6 33 15 24)(7 34 16 25)(8 35 17 26)(9 36 18 27)(37 55 46 64)(38 56 47 65)(39 57 48 66)(40 58 49 67)(41 59 50 68)(42 60 51 69)(43 61 52 70)(44 62 53 71)(45 63 54 72)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)
(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 52 10 43)(2 51 11 42)(3 50 12 41)(4 49 13 40)(5 48 14 39)(6 47 15 38)(7 46 16 37)(8 54 17 45)(9 53 18 44)(19 70 28 61)(20 69 29 60)(21 68 30 59)(22 67 31 58)(23 66 32 57)(24 65 33 56)(25 64 34 55)(26 72 35 63)(27 71 36 62)

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

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

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

81 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 6F ··· 6M 8A 8B 9A ··· 9I 12A 12B 12C 12D 12E 12F 12G 18A ··· 18I 18J ··· 18AA 24A 24B 24C 24D 36A ··· 36I order 1 2 2 3 3 3 3 3 4 4 6 6 6 6 6 6 ··· 6 8 8 9 ··· 9 12 12 12 12 12 12 12 18 ··· 18 18 ··· 18 24 24 24 24 36 ··· 36 size 1 1 4 1 1 2 2 2 2 36 1 1 2 2 2 4 ··· 4 18 18 2 ··· 2 2 2 4 4 4 36 36 2 ··· 2 4 ··· 4 18 18 18 18 4 ··· 4

81 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + - - image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D6 SD16 D9 C3×S3 C3×D4 C3⋊D4 D18 S3×C6 C3×SD16 C3×D9 C9⋊D4 C3×C3⋊D4 C6×D9 C3×C9⋊D4 D4.S3 D4.D9 C3×D4.S3 C3×D4.D9 kernel C3×D4.D9 C3×C9⋊C8 C3×Dic18 D4×C3×C9 D4.D9 C9⋊C8 Dic18 D4×C9 D4×C32 C3×C18 C3×C12 C3×C9 C3×D4 C3×D4 C18 C3×C6 C12 C12 C9 D4 C6 C6 C4 C2 C32 C3 C3 C1 # reps 1 1 1 1 2 2 2 2 1 1 1 2 3 2 2 2 3 2 4 6 6 4 6 12 1 3 2 6

Matrix representation of C3×D4.D9 in GL4(𝔽73) generated by

 64 0 0 0 0 64 0 0 0 0 1 0 0 0 0 1
,
 72 0 0 0 0 72 0 0 0 0 6 3 0 0 12 67
,
 72 0 0 0 0 1 0 0 0 0 6 3 0 0 37 67
,
 2 0 0 0 0 37 0 0 0 0 1 0 0 0 0 1
,
 0 37 0 0 2 0 0 0 0 0 30 18 0 0 27 43
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,0,72,0,0,0,0,6,12,0,0,3,67],[72,0,0,0,0,1,0,0,0,0,6,37,0,0,3,67],[2,0,0,0,0,37,0,0,0,0,1,0,0,0,0,1],[0,2,0,0,37,0,0,0,0,0,30,27,0,0,18,43] >;

C3×D4.D9 in GAP, Magma, Sage, TeX

C_3\times D_4.D_9
% in TeX

G:=Group("C3xD4.D9");
// GroupNames label

G:=SmallGroup(432,148);
// by ID

G=gap.SmallGroup(432,148);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,197,1011,514,80,10085,292,14118]);
// Polycyclic

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

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