Copied to
clipboard

## G = C22×S3×Dic3order 288 = 25·32

### Direct product of C22, S3 and Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C22×S3×Dic3
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — S3×Dic3 — C2×S3×Dic3 — C22×S3×Dic3
 Lower central C32 — C22×S3×Dic3
 Upper central C1 — C23

Generators and relations for C22×S3×Dic3
G = < a,b,c,d,e,f | a2=b2=c3=d2=e6=1, f2=e3, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 1298 in 499 conjugacy classes, 228 normal (18 characteristic)
C1, C2, C2 [×6], C2 [×8], C3 [×2], C3, C4 [×8], C22 [×7], C22 [×28], S3 [×8], C6 [×2], C6 [×12], C6 [×15], C2×C4 [×28], C23, C23 [×14], C32, Dic3 [×4], Dic3 [×12], C12 [×4], D6 [×28], C2×C6 [×14], C2×C6 [×35], C22×C4 [×14], C24, C3×S3 [×8], C3×C6, C3×C6 [×6], C4×S3 [×16], C2×Dic3 [×6], C2×Dic3 [×34], C2×C12 [×6], C22×S3 [×14], C22×C6 [×2], C22×C6 [×15], C23×C4, C3×Dic3 [×4], C3⋊Dic3 [×4], S3×C6 [×28], C62 [×7], S3×C2×C4 [×12], C22×Dic3, C22×Dic3 [×15], C22×C12, S3×C23, C23×C6, S3×Dic3 [×16], C6×Dic3 [×6], C2×C3⋊Dic3 [×6], S3×C2×C6 [×14], C2×C62, S3×C22×C4, C23×Dic3, C2×S3×Dic3 [×12], Dic3×C2×C6, C22×C3⋊Dic3, S3×C22×C6, C22×S3×Dic3
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], S3 [×2], C2×C4 [×28], C23 [×15], Dic3 [×8], D6 [×14], C22×C4 [×14], C24, C4×S3 [×4], C2×Dic3 [×28], C22×S3 [×14], C23×C4, S32, S3×C2×C4 [×6], C22×Dic3 [×14], S3×C23 [×2], S3×Dic3 [×4], C2×S32 [×3], S3×C22×C4, C23×Dic3, C2×S3×Dic3 [×6], C22×S32, C22×S3×Dic3

Smallest permutation representation of C22×S3×Dic3
On 96 points
Generators in S96
(1 16)(2 17)(3 18)(4 13)(5 14)(6 15)(7 22)(8 23)(9 24)(10 19)(11 20)(12 21)(25 40)(26 41)(27 42)(28 37)(29 38)(30 39)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)(49 64)(50 65)(51 66)(52 61)(53 62)(54 63)(55 70)(56 71)(57 72)(58 67)(59 68)(60 69)(73 88)(74 89)(75 90)(76 85)(77 86)(78 87)(79 94)(80 95)(81 96)(82 91)(83 92)(84 93)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)
(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(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 57 59)(56 58 60)(61 63 65)(62 64 66)(67 69 71)(68 70 72)(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 64)(2 65)(3 66)(4 61)(5 62)(6 63)(7 70)(8 71)(9 72)(10 67)(11 68)(12 69)(13 52)(14 53)(15 54)(16 49)(17 50)(18 51)(19 58)(20 59)(21 60)(22 55)(23 56)(24 57)(25 88)(26 89)(27 90)(28 85)(29 86)(30 87)(31 94)(32 95)(33 96)(34 91)(35 92)(36 93)(37 76)(38 77)(39 78)(40 73)(41 74)(42 75)(43 82)(44 83)(45 84)(46 79)(47 80)(48 81)
(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 44 4 47)(2 43 5 46)(3 48 6 45)(7 38 10 41)(8 37 11 40)(9 42 12 39)(13 32 16 35)(14 31 17 34)(15 36 18 33)(19 26 22 29)(20 25 23 28)(21 30 24 27)(49 92 52 95)(50 91 53 94)(51 96 54 93)(55 86 58 89)(56 85 59 88)(57 90 60 87)(61 80 64 83)(62 79 65 82)(63 84 66 81)(67 74 70 77)(68 73 71 76)(69 78 72 75)

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

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

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

72 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3A 3B 3C 4A ··· 4H 4I ··· 4P 6A ··· 6N 6O ··· 6U 6V ··· 6AC 12A ··· 12H order 1 2 ··· 2 2 ··· 2 3 3 3 4 ··· 4 4 ··· 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 ··· 1 3 ··· 3 2 2 4 3 ··· 3 9 ··· 9 2 ··· 2 4 ··· 4 6 ··· 6 6 ··· 6

72 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + - + + + - + image C1 C2 C2 C2 C2 C4 S3 S3 D6 Dic3 D6 D6 C4×S3 S32 S3×Dic3 C2×S32 kernel C22×S3×Dic3 C2×S3×Dic3 Dic3×C2×C6 C22×C3⋊Dic3 S3×C22×C6 S3×C2×C6 C22×Dic3 S3×C23 C2×Dic3 C22×S3 C22×S3 C22×C6 C2×C6 C23 C22 C22 # reps 1 12 1 1 1 16 1 1 6 8 6 2 8 1 4 3

Matrix representation of C22×S3×Dic3 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 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 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 1 0 0 0 0 12 0 0 0 0 0 0 0 12 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 12 0
,
 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 8 0 0 0 0 0 5 5

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,5,0,0,0,0,0,5] >;

C22×S3×Dic3 in GAP, Magma, Sage, TeX

C_2^2\times S_3\times {\rm Dic}_3
% in TeX

G:=Group("C2^2xS3xDic3");
// GroupNames label

G:=SmallGroup(288,969);
// by ID

G=gap.SmallGroup(288,969);
# by ID

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

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

׿
×
𝔽