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## G = C3×C4.D12order 288 = 25·32

### Direct product of C3 and C4.D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×C4.D12
 Chief series C1 — C3 — C6 — C2×C6 — C62 — S3×C2×C6 — S3×C2×C12 — C3×C4.D12
 Lower central C3 — C2×C6 — C3×C4.D12
 Upper central C1 — C2×C6 — C3×C4⋊C4

Generators and relations for C3×C4.D12
G = < a,b,c,d | a3=b4=c12=1, d2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b2c-1 >

Subgroups: 370 in 161 conjugacy classes, 70 normal (38 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×4], S3 [×2], C6 [×6], C6 [×5], C2×C4, C2×C4 [×2], C2×C4 [×5], Q8 [×2], C23, C32, Dic3 [×3], C12 [×4], C12 [×11], D6 [×2], D6 [×2], C2×C6 [×2], C2×C6 [×5], C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C22×C4, C2×Q8, C3×S3 [×2], C3×C6 [×3], Dic6 [×2], C4×S3 [×2], C2×Dic3, C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×4], C2×C12 [×8], C3×Q8 [×2], C22×S3, C22×C6, C22⋊Q8, C3×Dic3 [×3], C3×C12 [×2], C3×C12 [×2], S3×C6 [×2], S3×C6 [×2], C62, C4⋊Dic3 [×2], D6⋊C4 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], C3×C4⋊C4 [×3], C2×Dic6, S3×C2×C4, C22×C12, C6×Q8, C3×Dic6 [×2], S3×C12 [×2], C6×Dic3, C6×Dic3 [×2], C6×C12, C6×C12 [×2], S3×C2×C6, C4.D12, C3×C22⋊Q8, C3×C4⋊Dic3 [×2], C3×D6⋊C4 [×2], C32×C4⋊C4, C6×Dic6, S3×C2×C12, C3×C4.D12
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], Q8 [×2], C23, D6 [×3], C2×C6 [×7], C2×D4, C2×Q8, C4○D4, C3×S3, D12 [×2], C3×D4 [×2], C3×Q8 [×2], C22×S3, C22×C6, C22⋊Q8, S3×C6 [×3], C2×D12, D42S3, S3×Q8, C6×D4, C6×Q8, C3×C4○D4, C3×D12 [×2], S3×C2×C6, C4.D12, C3×C22⋊Q8, C6×D12, C3×D42S3, C3×S3×Q8, C3×C4.D12

Smallest permutation representation of C3×C4.D12
On 96 points
Generators in S96
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 41 45)(38 42 46)(39 43 47)(40 44 48)(49 53 57)(50 54 58)(51 55 59)(52 56 60)(61 69 65)(62 70 66)(63 71 67)(64 72 68)(73 77 81)(74 78 82)(75 79 83)(76 80 84)(85 93 89)(86 94 90)(87 95 91)(88 96 92)
(1 70 96 29)(2 30 85 71)(3 72 86 31)(4 32 87 61)(5 62 88 33)(6 34 89 63)(7 64 90 35)(8 36 91 65)(9 66 92 25)(10 26 93 67)(11 68 94 27)(12 28 95 69)(13 54 80 48)(14 37 81 55)(15 56 82 38)(16 39 83 57)(17 58 84 40)(18 41 73 59)(19 60 74 42)(20 43 75 49)(21 50 76 44)(22 45 77 51)(23 52 78 46)(24 47 79 53)
(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)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 49 96 43)(2 42 85 60)(3 59 86 41)(4 40 87 58)(5 57 88 39)(6 38 89 56)(7 55 90 37)(8 48 91 54)(9 53 92 47)(10 46 93 52)(11 51 94 45)(12 44 95 50)(13 36 80 65)(14 64 81 35)(15 34 82 63)(16 62 83 33)(17 32 84 61)(18 72 73 31)(19 30 74 71)(20 70 75 29)(21 28 76 69)(22 68 77 27)(23 26 78 67)(24 66 79 25)

G:=sub<Sym(96)| (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48)(49,53,57)(50,54,58)(51,55,59)(52,56,60)(61,69,65)(62,70,66)(63,71,67)(64,72,68)(73,77,81)(74,78,82)(75,79,83)(76,80,84)(85,93,89)(86,94,90)(87,95,91)(88,96,92), (1,70,96,29)(2,30,85,71)(3,72,86,31)(4,32,87,61)(5,62,88,33)(6,34,89,63)(7,64,90,35)(8,36,91,65)(9,66,92,25)(10,26,93,67)(11,68,94,27)(12,28,95,69)(13,54,80,48)(14,37,81,55)(15,56,82,38)(16,39,83,57)(17,58,84,40)(18,41,73,59)(19,60,74,42)(20,43,75,49)(21,50,76,44)(22,45,77,51)(23,52,78,46)(24,47,79,53), (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)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,49,96,43)(2,42,85,60)(3,59,86,41)(4,40,87,58)(5,57,88,39)(6,38,89,56)(7,55,90,37)(8,48,91,54)(9,53,92,47)(10,46,93,52)(11,51,94,45)(12,44,95,50)(13,36,80,65)(14,64,81,35)(15,34,82,63)(16,62,83,33)(17,32,84,61)(18,72,73,31)(19,30,74,71)(20,70,75,29)(21,28,76,69)(22,68,77,27)(23,26,78,67)(24,66,79,25)>;

G:=Group( (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48)(49,53,57)(50,54,58)(51,55,59)(52,56,60)(61,69,65)(62,70,66)(63,71,67)(64,72,68)(73,77,81)(74,78,82)(75,79,83)(76,80,84)(85,93,89)(86,94,90)(87,95,91)(88,96,92), (1,70,96,29)(2,30,85,71)(3,72,86,31)(4,32,87,61)(5,62,88,33)(6,34,89,63)(7,64,90,35)(8,36,91,65)(9,66,92,25)(10,26,93,67)(11,68,94,27)(12,28,95,69)(13,54,80,48)(14,37,81,55)(15,56,82,38)(16,39,83,57)(17,58,84,40)(18,41,73,59)(19,60,74,42)(20,43,75,49)(21,50,76,44)(22,45,77,51)(23,52,78,46)(24,47,79,53), (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)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,49,96,43)(2,42,85,60)(3,59,86,41)(4,40,87,58)(5,57,88,39)(6,38,89,56)(7,55,90,37)(8,48,91,54)(9,53,92,47)(10,46,93,52)(11,51,94,45)(12,44,95,50)(13,36,80,65)(14,64,81,35)(15,34,82,63)(16,62,83,33)(17,32,84,61)(18,72,73,31)(19,30,74,71)(20,70,75,29)(21,28,76,69)(22,68,77,27)(23,26,78,67)(24,66,79,25) );

G=PermutationGroup([(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,17,21),(14,18,22),(15,19,23),(16,20,24),(25,33,29),(26,34,30),(27,35,31),(28,36,32),(37,41,45),(38,42,46),(39,43,47),(40,44,48),(49,53,57),(50,54,58),(51,55,59),(52,56,60),(61,69,65),(62,70,66),(63,71,67),(64,72,68),(73,77,81),(74,78,82),(75,79,83),(76,80,84),(85,93,89),(86,94,90),(87,95,91),(88,96,92)], [(1,70,96,29),(2,30,85,71),(3,72,86,31),(4,32,87,61),(5,62,88,33),(6,34,89,63),(7,64,90,35),(8,36,91,65),(9,66,92,25),(10,26,93,67),(11,68,94,27),(12,28,95,69),(13,54,80,48),(14,37,81,55),(15,56,82,38),(16,39,83,57),(17,58,84,40),(18,41,73,59),(19,60,74,42),(20,43,75,49),(21,50,76,44),(22,45,77,51),(23,52,78,46),(24,47,79,53)], [(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),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,49,96,43),(2,42,85,60),(3,59,86,41),(4,40,87,58),(5,57,88,39),(6,38,89,56),(7,55,90,37),(8,48,91,54),(9,53,92,47),(10,46,93,52),(11,51,94,45),(12,44,95,50),(13,36,80,65),(14,64,81,35),(15,34,82,63),(16,62,83,33),(17,32,84,61),(18,72,73,31),(19,30,74,71),(20,70,75,29),(21,28,76,69),(22,68,77,27),(23,26,78,67),(24,66,79,25)])

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 6G ··· 6O 6P 6Q 6R 6S 12A 12B 12C 12D 12E ··· 12Z 12AA 12AB 12AC 12AD 12AE 12AF 12AG 12AH order 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 6 6 6 12 12 12 12 12 ··· 12 12 12 12 12 12 12 12 12 size 1 1 1 1 6 6 1 1 2 2 2 2 2 4 4 6 6 12 12 1 ··· 1 2 ··· 2 6 6 6 6 2 2 2 2 4 ··· 4 6 6 6 6 12 12 12 12

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + - + + - - image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 S3 D4 Q8 D6 C4○D4 C3×S3 D12 C3×D4 C3×Q8 S3×C6 C3×C4○D4 C3×D12 D4⋊2S3 S3×Q8 C3×D4⋊2S3 C3×S3×Q8 kernel C3×C4.D12 C3×C4⋊Dic3 C3×D6⋊C4 C32×C4⋊C4 C6×Dic6 S3×C2×C12 C4.D12 C4⋊Dic3 D6⋊C4 C3×C4⋊C4 C2×Dic6 S3×C2×C4 C3×C4⋊C4 C3×C12 S3×C6 C2×C12 C3×C6 C4⋊C4 C12 C12 D6 C2×C4 C6 C4 C6 C6 C2 C2 # reps 1 2 2 1 1 1 2 4 4 2 2 2 1 2 2 3 2 2 4 4 4 6 4 8 1 1 2 2

Matrix representation of C3×C4.D12 in GL4(𝔽13) generated by

 3 0 0 0 0 3 0 0 0 0 1 0 0 0 0 1
,
 12 0 0 0 0 12 0 0 0 0 4 2 0 0 11 9
,
 6 0 0 0 0 11 0 0 0 0 0 1 0 0 1 0
,
 0 11 0 0 6 0 0 0 0 0 3 6 0 0 7 10
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,4,11,0,0,2,9],[6,0,0,0,0,11,0,0,0,0,0,1,0,0,1,0],[0,6,0,0,11,0,0,0,0,0,3,7,0,0,6,10] >;

C3×C4.D12 in GAP, Magma, Sage, TeX

C_3\times C_4.D_{12}
% in TeX

G:=Group("C3xC4.D12");
// GroupNames label

G:=SmallGroup(288,668);
// by ID

G=gap.SmallGroup(288,668);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,344,590,555,394,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^4=c^12=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations

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