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## G = C3×C4.Dic9order 432 = 24·33

### Direct product of C3 and C4.Dic9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C3×C4.Dic9
 Chief series C1 — C3 — C9 — C18 — C36 — C3×C36 — C3×C9⋊C8 — C3×C4.Dic9
 Lower central C9 — C18 — C3×C4.Dic9
 Upper central C1 — C12 — C2×C12

Generators and relations for C3×C4.Dic9
G = < a,b,c,d | a3=b4=1, c18=b2, d2=c9, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c17 >

Subgroups: 158 in 76 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C8, C2×C4, C9, C9, C32, C12, C12, C2×C6, C2×C6, M4(2), C18, C18, C3×C6, C3×C6, C3⋊C8, C24, C2×C12, C2×C12, C3×C9, C36, C36, C2×C18, C2×C18, C3×C12, C62, C4.Dic3, C3×M4(2), C3×C18, C3×C18, C9⋊C8, C2×C36, C2×C36, C3×C3⋊C8, C6×C12, C3×C36, C6×C18, C4.Dic9, C3×C4.Dic3, C3×C9⋊C8, C6×C36, C3×C4.Dic9
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, Dic3, C12, D6, C2×C6, M4(2), D9, C3×S3, C2×Dic3, C2×C12, Dic9, D18, C3×Dic3, S3×C6, C4.Dic3, C3×M4(2), C3×D9, C2×Dic9, C6×Dic3, C3×Dic9, C6×D9, C4.Dic9, C3×C4.Dic3, C6×Dic9, C3×C4.Dic9

Smallest permutation representation of C3×C4.Dic9
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 10 19 28)(2 11 20 29)(3 12 21 30)(4 13 22 31)(5 14 23 32)(6 15 24 33)(7 16 25 34)(8 17 26 35)(9 18 27 36)(37 64 55 46)(38 65 56 47)(39 66 57 48)(40 67 58 49)(41 68 59 50)(42 69 60 51)(43 70 61 52)(44 71 62 53)(45 72 63 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 55 10 64 19 37 28 46)(2 72 11 45 20 54 29 63)(3 53 12 62 21 71 30 44)(4 70 13 43 22 52 31 61)(5 51 14 60 23 69 32 42)(6 68 15 41 24 50 33 59)(7 49 16 58 25 67 34 40)(8 66 17 39 26 48 35 57)(9 47 18 56 27 65 36 38)

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

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

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

126 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 3E 4A 4B 4C 6A 6B 6C ··· 6M 8A 8B 8C 8D 9A ··· 9I 12A 12B 12C 12D 12E ··· 12R 18A ··· 18AA 24A ··· 24H 36A ··· 36AJ order 1 2 2 3 3 3 3 3 4 4 4 6 6 6 ··· 6 8 8 8 8 9 ··· 9 12 12 12 12 12 ··· 12 18 ··· 18 24 ··· 24 36 ··· 36 size 1 1 2 1 1 2 2 2 1 1 2 1 1 2 ··· 2 18 18 18 18 2 ··· 2 1 1 1 1 2 ··· 2 2 ··· 2 18 ··· 18 2 ··· 2

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + - + - + - image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 S3 Dic3 D6 Dic3 M4(2) D9 C3×S3 Dic9 D18 C3×Dic3 S3×C6 Dic9 C3×Dic3 C3×M4(2) C4.Dic3 C3×D9 C3×Dic9 C6×D9 C3×Dic9 C4.Dic9 C3×C4.Dic3 C3×C4.Dic9 kernel C3×C4.Dic9 C3×C9⋊C8 C6×C36 C4.Dic9 C3×C36 C6×C18 C9⋊C8 C2×C36 C36 C2×C18 C6×C12 C3×C12 C3×C12 C62 C3×C9 C2×C12 C2×C12 C12 C12 C12 C12 C2×C6 C2×C6 C9 C32 C2×C4 C4 C4 C22 C3 C3 C1 # reps 1 2 1 2 2 2 4 2 4 4 1 1 1 1 2 3 2 3 3 2 2 3 2 4 4 6 6 6 6 12 8 24

Matrix representation of C3×C4.Dic9 in GL2(𝔽37) generated by

 26 0 0 26
,
 31 0 0 6
,
 13 0 0 17
,
 0 6 1 0
G:=sub<GL(2,GF(37))| [26,0,0,26],[31,0,0,6],[13,0,0,17],[0,1,6,0] >;

C3×C4.Dic9 in GAP, Magma, Sage, TeX

C_3\times C_4.{\rm Dic}_9
% in TeX

G:=Group("C3xC4.Dic9");
// GroupNames label

G:=SmallGroup(432,125);
// by ID

G=gap.SmallGroup(432,125);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,80,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^4=1,c^18=b^2,d^2=c^9,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^17>;
// generators/relations

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