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## G = C9×C4○D12order 432 = 24·33

### Direct product of C9 and C4○D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C9×C4○D12
 Chief series C1 — C3 — C32 — C3×C6 — C3×C18 — S3×C18 — S3×C36 — C9×C4○D12
 Lower central C3 — C6 — C9×C4○D12
 Upper central C1 — C36 — C2×C36

Generators and relations for C9×C4○D12
G = < a,b,c,d | a9=b4=d2=1, c6=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c5 >

Subgroups: 260 in 136 conjugacy classes, 69 normal (45 characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×2], S3 [×2], C6 [×2], C6 [×7], C2×C4, C2×C4 [×2], D4 [×3], Q8, C9, C9, C32, Dic3 [×2], C12 [×4], C12 [×4], D6 [×2], C2×C6 [×2], C2×C6 [×3], C4○D4, C18, C18 [×6], C3×S3 [×2], C3×C6, C3×C6, Dic6, C4×S3 [×2], D12, C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×3], C3×D4 [×3], C3×Q8, C3×C9, C36 [×2], C36 [×4], C2×C18, C2×C18 [×3], C3×Dic3 [×2], C3×C12 [×2], S3×C6 [×2], C62, C4○D12, C3×C4○D4, S3×C9 [×2], C3×C18, C3×C18, C2×C36, C2×C36 [×3], D4×C9 [×3], Q8×C9, C3×Dic6, S3×C12 [×2], C3×D12, C3×C3⋊D4 [×2], C6×C12, C9×Dic3 [×2], C3×C36 [×2], S3×C18 [×2], C6×C18, C9×C4○D4, C3×C4○D12, C9×Dic6, S3×C36 [×2], C9×D12, C9×C3⋊D4 [×2], C6×C36, C9×C4○D12
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], C23, C9, D6 [×3], C2×C6 [×7], C4○D4, C18 [×7], C3×S3, C22×S3, C22×C6, C2×C18 [×7], S3×C6 [×3], C4○D12, C3×C4○D4, S3×C9, C22×C18, S3×C2×C6, S3×C18 [×3], C9×C4○D4, C3×C4○D12, S3×C2×C18, C9×C4○D12

Smallest permutation representation of C9×C4○D12
On 72 points
Generators in S72
(1 44 63 9 40 71 5 48 67)(2 45 64 10 41 72 6 37 68)(3 46 65 11 42 61 7 38 69)(4 47 66 12 43 62 8 39 70)(13 53 35 17 57 27 21 49 31)(14 54 36 18 58 28 22 50 32)(15 55 25 19 59 29 23 51 33)(16 56 26 20 60 30 24 52 34)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 16 19 22)(14 17 20 23)(15 18 21 24)(25 28 31 34)(26 29 32 35)(27 30 33 36)(37 46 43 40)(38 47 44 41)(39 48 45 42)(49 52 55 58)(50 53 56 59)(51 54 57 60)(61 70 67 64)(62 71 68 65)(63 72 69 66)
(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)(2 54)(3 53)(4 52)(5 51)(6 50)(7 49)(8 60)(9 59)(10 58)(11 57)(12 56)(13 69)(14 68)(15 67)(16 66)(17 65)(18 64)(19 63)(20 62)(21 61)(22 72)(23 71)(24 70)(25 44)(26 43)(27 42)(28 41)(29 40)(30 39)(31 38)(32 37)(33 48)(34 47)(35 46)(36 45)

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

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

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

162 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 6A 6B 6C ··· 6M 6N 6O 6P 6Q 9A ··· 9F 9G ··· 9L 12A 12B 12C 12D 12E ··· 12R 12S 12T 12U 12V 18A ··· 18F 18G ··· 18AD 18AE ··· 18AP 36A ··· 36L 36M ··· 36AP 36AQ ··· 36BB order 1 2 2 2 2 3 3 3 3 3 4 4 4 4 4 6 6 6 ··· 6 6 6 6 6 9 ··· 9 9 ··· 9 12 12 12 12 12 ··· 12 12 12 12 12 18 ··· 18 18 ··· 18 18 ··· 18 36 ··· 36 36 ··· 36 36 ··· 36 size 1 1 2 6 6 1 1 2 2 2 1 1 2 6 6 1 1 2 ··· 2 6 6 6 6 1 ··· 1 2 ··· 2 1 1 1 1 2 ··· 2 6 6 6 6 1 ··· 1 2 ··· 2 6 ··· 6 1 ··· 1 2 ··· 2 6 ··· 6

162 irreducible representations

 dim 1 1 1 1 1 1 1 1 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 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 C9 C18 C18 C18 C18 C18 S3 D6 D6 C4○D4 C3×S3 S3×C6 S3×C6 C4○D12 C3×C4○D4 S3×C9 S3×C18 S3×C18 C9×C4○D4 C3×C4○D12 C9×C4○D12 kernel C9×C4○D12 C9×Dic6 S3×C36 C9×D12 C9×C3⋊D4 C6×C36 C3×C4○D12 C3×Dic6 S3×C12 C3×D12 C3×C3⋊D4 C6×C12 C4○D12 Dic6 C4×S3 D12 C3⋊D4 C2×C12 C2×C36 C36 C2×C18 C3×C9 C2×C12 C12 C2×C6 C9 C32 C2×C4 C4 C22 C3 C3 C1 # reps 1 1 2 1 2 1 2 2 4 2 4 2 6 6 12 6 12 6 1 2 1 2 2 4 2 4 4 6 12 6 12 8 24

Matrix representation of C9×C4○D12 in GL2(𝔽37) generated by

 9 0 0 9
,
 31 0 0 31
,
 14 0 13 8
,
 29 35 13 8
G:=sub<GL(2,GF(37))| [9,0,0,9],[31,0,0,31],[14,13,0,8],[29,13,35,8] >;

C9×C4○D12 in GAP, Magma, Sage, TeX

C_9\times C_4\circ D_{12}
% in TeX

G:=Group("C9xC4oD12");
// GroupNames label

G:=SmallGroup(432,347);
// by ID

G=gap.SmallGroup(432,347);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,590,192,14118]);
// Polycyclic

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

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