Copied to
clipboard

## G = C3×D6.D4order 288 = 25·32

### Direct product of C3 and D6.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×D6.D4
 Chief series C1 — C3 — C6 — C2×C6 — C62 — S3×C2×C6 — S3×C2×C12 — C3×D6.D4
 Lower central C3 — C2×C6 — C3×D6.D4
 Upper central C1 — C2×C6 — C3×C4⋊C4

Generators and relations for C3×D6.D4
G = < a,b,c,d,e | a3=b6=c2=d4=1, e2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=ece-1=b3c, ede-1=d-1 >

Subgroups: 434 in 169 conjugacy classes, 62 normal (58 characteristic)
C1, C2 [×3], C2 [×3], C3 [×2], C3, C4 [×5], C22, C22 [×7], S3 [×3], C6 [×6], C6 [×6], C2×C4 [×3], C2×C4 [×4], D4 [×2], C23 [×2], C32, Dic3 [×2], C12 [×12], D6 [×2], D6 [×5], C2×C6 [×2], C2×C6 [×8], C22⋊C4 [×3], C4⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3 [×3], C3×C6 [×3], C4×S3 [×2], D12 [×2], C2×Dic3 [×2], C2×C12 [×6], C2×C12 [×7], C3×D4 [×2], C22×S3 [×2], C22×C6 [×2], C22.D4, C3×Dic3 [×2], C3×C12 [×3], S3×C6 [×2], S3×C6 [×5], C62, Dic3⋊C4, D6⋊C4 [×3], C3×C22⋊C4 [×3], C3×C4⋊C4 [×2], C3×C4⋊C4 [×2], S3×C2×C4, C2×D12, C22×C12, C6×D4, S3×C12 [×2], C3×D12 [×2], C6×Dic3 [×2], C6×C12 [×3], S3×C2×C6 [×2], D6.D4, C3×C22.D4, C3×Dic3⋊C4, C3×D6⋊C4 [×3], C32×C4⋊C4, S3×C2×C12, C6×D12, C3×D6.D4
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], C2×D4, C4○D4 [×2], C3×S3, C3×D4 [×2], C22×S3, C22×C6, C22.D4, S3×C6 [×3], C4○D12, S3×D4, Q83S3, C6×D4, C3×C4○D4 [×2], S3×C2×C6, D6.D4, C3×C22.D4, C3×C4○D12, C3×S3×D4, C3×Q83S3, C3×D6.D4

Smallest permutation representation of C3×D6.D4
On 96 points
Generators in S96
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 48)(49 51 53)(50 52 54)(55 59 57)(56 60 58)(61 65 63)(62 66 64)(67 71 69)(68 72 70)(73 77 75)(74 78 76)(79 83 81)(80 84 82)(85 89 87)(86 90 88)(91 95 93)(92 96 94)
(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 64)(2 63)(3 62)(4 61)(5 66)(6 65)(7 46)(8 45)(9 44)(10 43)(11 48)(12 47)(13 60)(14 59)(15 58)(16 57)(17 56)(18 55)(19 77)(20 76)(21 75)(22 74)(23 73)(24 78)(25 72)(26 71)(27 70)(28 69)(29 68)(30 67)(31 89)(32 88)(33 87)(34 86)(35 85)(36 90)(37 84)(38 83)(39 82)(40 81)(41 80)(42 79)(49 95)(50 94)(51 93)(52 92)(53 91)(54 96)
(1 28 16 20)(2 29 17 21)(3 30 18 22)(4 25 13 23)(5 26 14 24)(6 27 15 19)(7 82 91 89)(8 83 92 90)(9 84 93 85)(10 79 94 86)(11 80 95 87)(12 81 96 88)(31 43 39 50)(32 44 40 51)(33 45 41 52)(34 46 42 53)(35 47 37 54)(36 48 38 49)(55 77 62 70)(56 78 63 71)(57 73 64 72)(58 74 65 67)(59 75 66 68)(60 76 61 69)
(1 54 4 51)(2 49 5 52)(3 50 6 53)(7 58 10 55)(8 59 11 56)(9 60 12 57)(13 44 16 47)(14 45 17 48)(15 46 18 43)(19 34 22 31)(20 35 23 32)(21 36 24 33)(25 40 28 37)(26 41 29 38)(27 42 30 39)(61 96 64 93)(62 91 65 94)(63 92 66 95)(67 79 70 82)(68 80 71 83)(69 81 72 84)(73 85 76 88)(74 86 77 89)(75 87 78 90)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G 6A ··· 6F 6G ··· 6O 6P 6Q 6R 6S 6T 6U 12A 12B 12C 12D 12E ··· 12Z 12AA 12AB 12AC 12AD 12AE 12AF order 1 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 6 6 6 6 6 12 12 12 12 12 ··· 12 12 12 12 12 12 12 size 1 1 1 1 6 6 12 1 1 2 2 2 2 2 4 4 6 6 12 1 ··· 1 2 ··· 2 6 6 6 6 12 12 2 2 2 2 4 ··· 4 6 6 6 6 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 4 4 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 S3 D4 D6 C4○D4 C3×S3 C3×D4 S3×C6 C4○D12 C3×C4○D4 C3×C4○D12 S3×D4 Q8⋊3S3 C3×S3×D4 C3×Q8⋊3S3 kernel C3×D6.D4 C3×Dic3⋊C4 C3×D6⋊C4 C32×C4⋊C4 S3×C2×C12 C6×D12 D6.D4 Dic3⋊C4 D6⋊C4 C3×C4⋊C4 S3×C2×C4 C2×D12 C3×C4⋊C4 S3×C6 C2×C12 C3×C6 C4⋊C4 D6 C2×C4 C6 C6 C2 C6 C6 C2 C2 # reps 1 1 3 1 1 1 2 2 6 2 2 2 1 2 3 4 2 4 6 4 8 8 1 1 2 2

Matrix representation of C3×D6.D4 in GL4(𝔽13) generated by

 9 0 0 0 0 9 0 0 0 0 1 0 0 0 0 1
,
 10 0 0 0 10 4 0 0 0 0 1 0 0 0 0 1
,
 4 8 0 0 3 9 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 12 12 0 0 0 0 5 3 0 0 0 8
,
 5 0 0 0 8 8 0 0 0 0 5 3 0 0 5 8
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[10,10,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[4,3,0,0,8,9,0,0,0,0,1,0,0,0,0,1],[1,12,0,0,0,12,0,0,0,0,5,0,0,0,3,8],[5,8,0,0,0,8,0,0,0,0,5,5,0,0,3,8] >;

C3×D6.D4 in GAP, Magma, Sage, TeX

C_3\times D_6.D_4
% in TeX

G:=Group("C3xD6.D4");
// GroupNames label

G:=SmallGroup(288,665);
// by ID

G=gap.SmallGroup(288,665);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,176,590,555,268,9414]);
// Polycyclic

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

׿
×
𝔽