Copied to
clipboard

## G = C3×C12.Q8order 288 = 25·32

### Direct product of C3 and C12.Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C3×C12.Q8
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C6×C12 — C6×C3⋊C8 — C3×C12.Q8
 Lower central C3 — C6 — C12 — C3×C12.Q8
 Upper central C1 — C2×C6 — C2×C12 — C3×C4⋊C4

Generators and relations for C3×C12.Q8
G = < a,b,c,d | a3=b4=c12=1, d2=bc6, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c-1 >

Subgroups: 178 in 85 conjugacy classes, 46 normal (42 characteristic)
C1, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×2], C22, C6 [×6], C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], C32, Dic3, C12 [×4], C12 [×7], C2×C6 [×2], C2×C6, C4⋊C4, C4⋊C4, C2×C8, C3×C6 [×3], C3⋊C8 [×2], C24 [×2], C2×Dic3, C2×C12 [×2], C2×C12 [×6], C4.Q8, C3×Dic3, C3×C12 [×2], C3×C12, C62, C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4 [×2], C3×C4⋊C4 [×2], C2×C24, C3×C3⋊C8 [×2], C6×Dic3, C6×C12, C6×C12, C12.Q8, C3×C4.Q8, C6×C3⋊C8, C3×C4⋊Dic3, C32×C4⋊C4, C3×C12.Q8
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, D4, Q8, C12 [×2], D6, C2×C6, C4⋊C4, SD16 [×2], C3×S3, Dic6, C4×S3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C4.Q8, S3×C6, Dic3⋊C4, D4.S3, Q82S3, C3×C4⋊C4, C3×SD16 [×2], C3×Dic6, S3×C12, C3×C3⋊D4, C12.Q8, C3×C4.Q8, C3×Dic3⋊C4, C3×D4.S3, C3×Q82S3, C3×C12.Q8

Smallest permutation representation of C3×C12.Q8
On 96 points
Generators in S96
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(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 45 41)(38 46 42)(39 47 43)(40 48 44)(49 57 53)(50 58 54)(51 59 55)(52 60 56)(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 89 93)(86 90 94)(87 91 95)(88 92 96)
(1 84 96 16)(2 17 85 73)(3 74 86 18)(4 19 87 75)(5 76 88 20)(6 21 89 77)(7 78 90 22)(8 23 91 79)(9 80 92 24)(10 13 93 81)(11 82 94 14)(12 15 95 83)(25 63 56 45)(26 46 57 64)(27 65 58 47)(28 48 59 66)(29 67 60 37)(30 38 49 68)(31 69 50 39)(32 40 51 70)(33 71 52 41)(34 42 53 72)(35 61 54 43)(36 44 55 62)
(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 52 78 47 96 33 22 65)(2 40 23 57 85 70 79 26)(3 50 80 45 86 31 24 63)(4 38 13 55 87 68 81 36)(5 60 82 43 88 29 14 61)(6 48 15 53 89 66 83 34)(7 58 84 41 90 27 16 71)(8 46 17 51 91 64 73 32)(9 56 74 39 92 25 18 69)(10 44 19 49 93 62 75 30)(11 54 76 37 94 35 20 67)(12 42 21 59 95 72 77 28)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G ··· 6O 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12Z 12AA 12AB 12AC 12AD 24A ··· 24H order 1 2 2 2 3 3 3 3 3 4 4 4 4 4 4 6 ··· 6 6 ··· 6 8 8 8 8 12 12 12 12 12 ··· 12 12 12 12 12 24 ··· 24 size 1 1 1 1 1 1 2 2 2 2 2 4 4 12 12 1 ··· 1 2 ··· 2 6 6 6 6 2 2 2 2 4 ··· 4 12 12 12 12 6 ··· 6

72 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 4 4 4 4 type + + + + + - + + - - + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 S3 Q8 D4 D6 SD16 C3×S3 Dic6 C4×S3 C3×Q8 C3⋊D4 C3×D4 S3×C6 C3×SD16 C3×Dic6 S3×C12 C3×C3⋊D4 D4.S3 Q8⋊2S3 C3×D4.S3 C3×Q8⋊2S3 kernel C3×C12.Q8 C6×C3⋊C8 C3×C4⋊Dic3 C32×C4⋊C4 C12.Q8 C3×C3⋊C8 C2×C3⋊C8 C4⋊Dic3 C3×C4⋊C4 C3⋊C8 C3×C4⋊C4 C3×C12 C62 C2×C12 C3×C6 C4⋊C4 C12 C12 C12 C2×C6 C2×C6 C2×C4 C6 C4 C4 C22 C6 C6 C2 C2 # reps 1 1 1 1 2 4 2 2 2 8 1 1 1 1 4 2 2 2 2 2 2 2 8 4 4 4 1 1 2 2

Matrix representation of C3×C12.Q8 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 8 0 0 0 0 8
,
 1 71 0 0 1 72 0 0 0 0 72 0 0 0 0 72
,
 19 69 0 0 17 54 0 0 0 0 3 0 0 0 24 24
,
 0 12 0 0 67 12 0 0 0 0 72 63 0 0 0 1
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,8],[1,1,0,0,71,72,0,0,0,0,72,0,0,0,0,72],[19,17,0,0,69,54,0,0,0,0,3,24,0,0,0,24],[0,67,0,0,12,12,0,0,0,0,72,0,0,0,63,1] >;

C3×C12.Q8 in GAP, Magma, Sage, TeX

C_3\times C_{12}.Q_8
% in TeX

G:=Group("C3xC12.Q8");
// GroupNames label

G:=SmallGroup(288,242);
// by ID

G=gap.SmallGroup(288,242);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,168,1037,92,1271,102,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^4=c^12=1,d^2=b*c^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^-1>;
// generators/relations

׿
×
𝔽