Copied to
clipboard

## G = C2×S3×Dic6order 288 = 25·32

### Direct product of C2, S3 and Dic6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×S3×Dic6
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — S3×Dic3 — C2×S3×Dic3 — C2×S3×Dic6
 Lower central C32 — C3×C6 — C2×S3×Dic6
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×S3×Dic6
G = < a,b,c,d,e | a2=b3=c2=d12=1, e2=d6, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 994 in 331 conjugacy classes, 132 normal (26 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, Q8, C23, C32, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C22×C4, C2×Q8, C3×S3, C3×C6, C3×C6, Dic6, Dic6, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C22×C6, C22×Q8, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C62, C2×Dic6, C2×Dic6, S3×C2×C4, S3×C2×C4, S3×Q8, C22×Dic3, C22×C12, C6×Q8, S3×Dic3, C322Q8, C3×Dic6, S3×C12, C6×Dic3, C6×Dic3, C324Q8, C2×C3⋊Dic3, C6×C12, S3×C2×C6, C22×Dic6, C2×S3×Q8, S3×Dic6, C2×S3×Dic3, C2×C322Q8, C6×Dic6, S3×C2×C12, C2×C324Q8, C2×S3×Dic6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C24, Dic6, C22×S3, C22×Q8, S32, C2×Dic6, S3×Q8, S3×C23, C2×S32, C22×Dic6, C2×S3×Q8, S3×Dic6, C22×S32, C2×S3×Dic6

Smallest permutation representation of C2×S3×Dic6
On 96 points
Generators in S96
(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 25)(11 26)(12 27)(13 86)(14 87)(15 88)(16 89)(17 90)(18 91)(19 92)(20 93)(21 94)(22 95)(23 96)(24 85)(37 53)(38 54)(39 55)(40 56)(41 57)(42 58)(43 59)(44 60)(45 49)(46 50)(47 51)(48 52)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(71 73)(72 74)
(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 29 33)(26 30 34)(27 31 35)(28 32 36)(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 81 77)(74 82 78)(75 83 79)(76 84 80)(85 89 93)(86 90 94)(87 91 95)(88 92 96)
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 61)(8 62)(9 63)(10 64)(11 65)(12 66)(13 38)(14 39)(15 40)(16 41)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 37)(25 78)(26 79)(27 80)(28 81)(29 82)(30 83)(31 84)(32 73)(33 74)(34 75)(35 76)(36 77)(49 93)(50 94)(51 95)(52 96)(53 85)(54 86)(55 87)(56 88)(57 89)(58 90)(59 91)(60 92)
(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 51 7 57)(2 50 8 56)(3 49 9 55)(4 60 10 54)(5 59 11 53)(6 58 12 52)(13 84 19 78)(14 83 20 77)(15 82 21 76)(16 81 22 75)(17 80 23 74)(18 79 24 73)(25 38 31 44)(26 37 32 43)(27 48 33 42)(28 47 34 41)(29 46 35 40)(30 45 36 39)(61 89 67 95)(62 88 68 94)(63 87 69 93)(64 86 70 92)(65 85 71 91)(66 96 72 90)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 4C ··· 4H 4I 4J 4K 4L 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N 12O 12P 12Q 12R order 1 2 2 2 2 2 2 2 3 3 3 4 4 4 ··· 4 4 4 4 4 6 ··· 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 3 3 3 3 2 2 4 2 2 6 ··· 6 18 18 18 18 2 ··· 2 4 4 4 6 6 6 6 2 2 2 2 4 ··· 4 6 6 6 6 12 12 12 12

54 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + - + + + + + - + - + + - image C1 C2 C2 C2 C2 C2 C2 S3 S3 Q8 D6 D6 D6 D6 D6 Dic6 S32 S3×Q8 C2×S32 C2×S32 S3×Dic6 kernel C2×S3×Dic6 S3×Dic6 C2×S3×Dic3 C2×C32⋊2Q8 C6×Dic6 S3×C2×C12 C2×C32⋊4Q8 C2×Dic6 S3×C2×C4 S3×C6 Dic6 C4×S3 C2×Dic3 C2×C12 C22×S3 D6 C2×C4 C6 C4 C22 C2 # reps 1 8 2 2 1 1 1 1 1 4 4 4 3 2 1 8 1 2 2 1 4

Matrix representation of C2×S3×Dic6 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 11 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0 1 0
,
 6 5 0 0 0 0 3 7 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 12 12

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,1,0,0,0,0,11,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[6,3,0,0,0,0,5,7,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12] >;

C2×S3×Dic6 in GAP, Magma, Sage, TeX

C_2\times S_3\times {\rm Dic}_6
% in TeX

G:=Group("C2xS3xDic6");
// GroupNames label

G:=SmallGroup(288,942);
// by ID

G=gap.SmallGroup(288,942);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,346,80,1356,9414]);
// Polycyclic

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

׿
×
𝔽