direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C2×C42, C6⋊1(C2×C42), C12⋊6(C22×C4), C6.2(C23×C4), C3⋊1(C22×C42), (C4×C12)⋊54C22, (C2×C6).15C24, Dic3⋊6(C22×C4), D6.21(C22×C4), (C22×C4).482D6, (C2×C12).874C23, (C4×Dic3)⋊83C22, C22.12(S3×C23), (C22×C6).377C23, C23.322(C22×S3), (C22×S3).252C23, (S3×C23).121C22, (C22×C12).563C22, (C2×Dic3).301C23, (C22×Dic3).242C22, (C2×C4×C12)⋊16C2, (C2×C12)⋊31(C2×C4), C2.1(S3×C22×C4), (C2×C4×Dic3)⋊39C2, C22.67(S3×C2×C4), (S3×C22×C4).24C2, (C2×Dic3)⋊25(C2×C4), (S3×C2×C4).308C22, (C22×S3).73(C2×C4), (C2×C6).145(C22×C4), (C2×C4).816(C22×S3), SmallGroup(192,1030)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C2×C42 |
Generators and relations for S3×C2×C42
G = < a,b,c,d,e | a2=b4=c4=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 888 in 498 conjugacy classes, 303 normal (10 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C2×C4, C2×C4, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C42, C42, C22×C4, C22×C4, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, C2×C42, C2×C42, C23×C4, C4×Dic3, C4×C12, S3×C2×C4, C22×Dic3, C22×C12, S3×C23, C22×C42, S3×C42, C2×C4×Dic3, C2×C4×C12, S3×C22×C4, S3×C2×C42
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C42, C22×C4, C24, C4×S3, C22×S3, C2×C42, C23×C4, S3×C2×C4, S3×C23, C22×C42, S3×C42, S3×C22×C4, S3×C2×C42
►On 96 points
Generators in S96
(1 75)(2 76)(3 73)(4 74)(5 55)(6 56)(7 53)(8 54)(9 31)(10 32)(11 29)(12 30)(13 57)(14 58)(15 59)(16 60)(17 39)(18 40)(19 37)(20 38)(21 96)(22 93)(23 94)(24 95)(25 85)(26 86)(27 87)(28 88)(33 89)(34 90)(35 91)(36 92)(41 67)(42 68)(43 65)(44 66)(45 52)(46 49)(47 50)(48 51)(61 79)(62 80)(63 77)(64 78)(69 84)(70 81)(71 82)(72 83)
(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 33 17 26)(2 34 18 27)(3 35 19 28)(4 36 20 25)(5 29 14 79)(6 30 15 80)(7 31 16 77)(8 32 13 78)(9 60 63 53)(10 57 64 54)(11 58 61 55)(12 59 62 56)(21 70 46 65)(22 71 47 66)(23 72 48 67)(24 69 45 68)(37 88 73 91)(38 85 74 92)(39 86 75 89)(40 87 76 90)(41 94 83 51)(42 95 84 52)(43 96 81 49)(44 93 82 50)
(1 79 68)(2 80 65)(3 77 66)(4 78 67)(5 24 33)(6 21 34)(7 22 35)(8 23 36)(9 82 37)(10 83 38)(11 84 39)(12 81 40)(13 48 25)(14 45 26)(15 46 27)(16 47 28)(17 29 69)(18 30 70)(19 31 71)(20 32 72)(41 74 64)(42 75 61)(43 76 62)(44 73 63)(49 87 59)(50 88 60)(51 85 57)(52 86 58)(53 93 91)(54 94 92)(55 95 89)(56 96 90)
(1 17)(2 18)(3 19)(4 20)(5 45)(6 46)(7 47)(8 48)(9 44)(10 41)(11 42)(12 43)(13 23)(14 24)(15 21)(16 22)(25 36)(26 33)(27 34)(28 35)(29 68)(30 65)(31 66)(32 67)(37 73)(38 74)(39 75)(40 76)(49 56)(50 53)(51 54)(52 55)(57 94)(58 95)(59 96)(60 93)(61 84)(62 81)(63 82)(64 83)(69 79)(70 80)(71 77)(72 78)(85 92)(86 89)(87 90)(88 91)
G:=sub<Sym(96)| (1,75)(2,76)(3,73)(4,74)(5,55)(6,56)(7,53)(8,54)(9,31)(10,32)(11,29)(12,30)(13,57)(14,58)(15,59)(16,60)(17,39)(18,40)(19,37)(20,38)(21,96)(22,93)(23,94)(24,95)(25,85)(26,86)(27,87)(28,88)(33,89)(34,90)(35,91)(36,92)(41,67)(42,68)(43,65)(44,66)(45,52)(46,49)(47,50)(48,51)(61,79)(62,80)(63,77)(64,78)(69,84)(70,81)(71,82)(72,83), (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), (1,33,17,26)(2,34,18,27)(3,35,19,28)(4,36,20,25)(5,29,14,79)(6,30,15,80)(7,31,16,77)(8,32,13,78)(9,60,63,53)(10,57,64,54)(11,58,61,55)(12,59,62,56)(21,70,46,65)(22,71,47,66)(23,72,48,67)(24,69,45,68)(37,88,73,91)(38,85,74,92)(39,86,75,89)(40,87,76,90)(41,94,83,51)(42,95,84,52)(43,96,81,49)(44,93,82,50), (1,79,68)(2,80,65)(3,77,66)(4,78,67)(5,24,33)(6,21,34)(7,22,35)(8,23,36)(9,82,37)(10,83,38)(11,84,39)(12,81,40)(13,48,25)(14,45,26)(15,46,27)(16,47,28)(17,29,69)(18,30,70)(19,31,71)(20,32,72)(41,74,64)(42,75,61)(43,76,62)(44,73,63)(49,87,59)(50,88,60)(51,85,57)(52,86,58)(53,93,91)(54,94,92)(55,95,89)(56,96,90), (1,17)(2,18)(3,19)(4,20)(5,45)(6,46)(7,47)(8,48)(9,44)(10,41)(11,42)(12,43)(13,23)(14,24)(15,21)(16,22)(25,36)(26,33)(27,34)(28,35)(29,68)(30,65)(31,66)(32,67)(37,73)(38,74)(39,75)(40,76)(49,56)(50,53)(51,54)(52,55)(57,94)(58,95)(59,96)(60,93)(61,84)(62,81)(63,82)(64,83)(69,79)(70,80)(71,77)(72,78)(85,92)(86,89)(87,90)(88,91)>;
G:=Group( (1,75)(2,76)(3,73)(4,74)(5,55)(6,56)(7,53)(8,54)(9,31)(10,32)(11,29)(12,30)(13,57)(14,58)(15,59)(16,60)(17,39)(18,40)(19,37)(20,38)(21,96)(22,93)(23,94)(24,95)(25,85)(26,86)(27,87)(28,88)(33,89)(34,90)(35,91)(36,92)(41,67)(42,68)(43,65)(44,66)(45,52)(46,49)(47,50)(48,51)(61,79)(62,80)(63,77)(64,78)(69,84)(70,81)(71,82)(72,83), (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), (1,33,17,26)(2,34,18,27)(3,35,19,28)(4,36,20,25)(5,29,14,79)(6,30,15,80)(7,31,16,77)(8,32,13,78)(9,60,63,53)(10,57,64,54)(11,58,61,55)(12,59,62,56)(21,70,46,65)(22,71,47,66)(23,72,48,67)(24,69,45,68)(37,88,73,91)(38,85,74,92)(39,86,75,89)(40,87,76,90)(41,94,83,51)(42,95,84,52)(43,96,81,49)(44,93,82,50), (1,79,68)(2,80,65)(3,77,66)(4,78,67)(5,24,33)(6,21,34)(7,22,35)(8,23,36)(9,82,37)(10,83,38)(11,84,39)(12,81,40)(13,48,25)(14,45,26)(15,46,27)(16,47,28)(17,29,69)(18,30,70)(19,31,71)(20,32,72)(41,74,64)(42,75,61)(43,76,62)(44,73,63)(49,87,59)(50,88,60)(51,85,57)(52,86,58)(53,93,91)(54,94,92)(55,95,89)(56,96,90), (1,17)(2,18)(3,19)(4,20)(5,45)(6,46)(7,47)(8,48)(9,44)(10,41)(11,42)(12,43)(13,23)(14,24)(15,21)(16,22)(25,36)(26,33)(27,34)(28,35)(29,68)(30,65)(31,66)(32,67)(37,73)(38,74)(39,75)(40,76)(49,56)(50,53)(51,54)(52,55)(57,94)(58,95)(59,96)(60,93)(61,84)(62,81)(63,82)(64,83)(69,79)(70,80)(71,77)(72,78)(85,92)(86,89)(87,90)(88,91) );
G=PermutationGroup([[(1,75),(2,76),(3,73),(4,74),(5,55),(6,56),(7,53),(8,54),(9,31),(10,32),(11,29),(12,30),(13,57),(14,58),(15,59),(16,60),(17,39),(18,40),(19,37),(20,38),(21,96),(22,93),(23,94),(24,95),(25,85),(26,86),(27,87),(28,88),(33,89),(34,90),(35,91),(36,92),(41,67),(42,68),(43,65),(44,66),(45,52),(46,49),(47,50),(48,51),(61,79),(62,80),(63,77),(64,78),(69,84),(70,81),(71,82),(72,83)], [(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)], [(1,33,17,26),(2,34,18,27),(3,35,19,28),(4,36,20,25),(5,29,14,79),(6,30,15,80),(7,31,16,77),(8,32,13,78),(9,60,63,53),(10,57,64,54),(11,58,61,55),(12,59,62,56),(21,70,46,65),(22,71,47,66),(23,72,48,67),(24,69,45,68),(37,88,73,91),(38,85,74,92),(39,86,75,89),(40,87,76,90),(41,94,83,51),(42,95,84,52),(43,96,81,49),(44,93,82,50)], [(1,79,68),(2,80,65),(3,77,66),(4,78,67),(5,24,33),(6,21,34),(7,22,35),(8,23,36),(9,82,37),(10,83,38),(11,84,39),(12,81,40),(13,48,25),(14,45,26),(15,46,27),(16,47,28),(17,29,69),(18,30,70),(19,31,71),(20,32,72),(41,74,64),(42,75,61),(43,76,62),(44,73,63),(49,87,59),(50,88,60),(51,85,57),(52,86,58),(53,93,91),(54,94,92),(55,95,89),(56,96,90)], [(1,17),(2,18),(3,19),(4,20),(5,45),(6,46),(7,47),(8,48),(9,44),(10,41),(11,42),(12,43),(13,23),(14,24),(15,21),(16,22),(25,36),(26,33),(27,34),(28,35),(29,68),(30,65),(31,66),(32,67),(37,73),(38,74),(39,75),(40,76),(49,56),(50,53),(51,54),(52,55),(57,94),(58,95),(59,96),(60,93),(61,84),(62,81),(63,82),(64,83),(69,79),(70,80),(71,77),(72,78),(85,92),(86,89),(87,90),(88,91)]])
96 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 3 | 4A | ··· | 4X | 4Y | ··· | 4AV | 6A | ··· | 6G | 12A | ··· | 12X |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D6 | D6 | C4×S3 |
| kernel | S3×C2×C42 | S3×C42 | C2×C4×Dic3 | C2×C4×C12 | S3×C22×C4 | S3×C2×C4 | C2×C42 | C42 | C22×C4 | C2×C4 |
| # reps | 1 | 8 | 3 | 1 | 3 | 48 | 1 | 4 | 3 | 24 |
Matrix representation of S3×C2×C42 ►in GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 5 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 8 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 1 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[5,0,0,0,0,5,0,0,0,0,1,0,0,0,0,1],[8,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,12,12],[1,0,0,0,0,12,0,0,0,0,0,1,0,0,1,0] >;
S3×C2×C42 in GAP, Magma, Sage, TeX
S_3\times C_2\times C_4^2
% in TeX
G:=Group("S3xC2xC4^2"); // GroupNames label
G:=SmallGroup(192,1030);
// by ID
G=gap.SmallGroup(192,1030);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,80,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^4=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations