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), C23.322(C22×S3), (C22×C6).377C23, (S3×C23).121C22, (C22×S3).252C23, (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×C4).816(C22×S3), (C2×C6).145(C22×C4), SmallGroup(192,1030)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C2×C42 |
Subgroups: 888 in 498 conjugacy classes, 303 normal (10 characteristic)
C1, C2 [×7], C2 [×8], C3, C4 [×12], C4 [×12], C22, C22 [×6], C22 [×28], S3 [×8], C6 [×7], C2×C4 [×18], C2×C4 [×66], C23, C23 [×14], Dic3 [×12], C12 [×12], D6 [×28], C2×C6, C2×C6 [×6], C42 [×4], C42 [×12], C22×C4 [×3], C22×C4 [×39], C24, C4×S3 [×48], C2×Dic3 [×18], C2×C12 [×18], C22×S3 [×14], C22×C6, C2×C42, C2×C42 [×11], C23×C4 [×3], C4×Dic3 [×12], C4×C12 [×4], S3×C2×C4 [×36], C22×Dic3 [×3], C22×C12 [×3], S3×C23, C22×C42, S3×C42 [×8], C2×C4×Dic3 [×3], C2×C4×C12, S3×C22×C4 [×3], S3×C2×C42
Quotients:
C1, C2 [×15], C4 [×24], C22 [×35], S3, C2×C4 [×84], C23 [×15], D6 [×7], C42 [×16], C22×C4 [×42], C24, C4×S3 [×12], C22×S3 [×7], C2×C42 [×12], C23×C4 [×3], S3×C2×C4 [×18], S3×C23, C22×C42, S3×C42 [×4], S3×C22×C4 [×3], S3×C2×C42
Generators and relations
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 >
►On 96 points
Generators in S96
(1 72)(2 69)(3 70)(4 71)(5 89)(6 90)(7 91)(8 92)(9 18)(10 19)(11 20)(12 17)(13 96)(14 93)(15 94)(16 95)(21 60)(22 57)(23 58)(24 59)(25 53)(26 54)(27 55)(28 56)(29 41)(30 42)(31 43)(32 44)(33 50)(34 51)(35 52)(36 49)(37 66)(38 67)(39 68)(40 65)(45 86)(46 87)(47 88)(48 85)(61 79)(62 80)(63 77)(64 78)(73 81)(74 82)(75 83)(76 84)
(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 18 25)(2 34 19 26)(3 35 20 27)(4 36 17 28)(5 66 23 79)(6 67 24 80)(7 68 21 77)(8 65 22 78)(9 53 72 50)(10 54 69 51)(11 55 70 52)(12 56 71 49)(13 73 46 31)(14 74 47 32)(15 75 48 29)(16 76 45 30)(37 58 61 89)(38 59 62 90)(39 60 63 91)(40 57 64 92)(41 94 83 85)(42 95 84 86)(43 96 81 87)(44 93 82 88)
(1 79 30)(2 80 31)(3 77 32)(4 78 29)(5 16 33)(6 13 34)(7 14 35)(8 15 36)(9 37 84)(10 38 81)(11 39 82)(12 40 83)(17 65 75)(18 66 76)(19 67 73)(20 68 74)(21 47 27)(22 48 28)(23 45 25)(24 46 26)(41 71 64)(42 72 61)(43 69 62)(44 70 63)(49 92 94)(50 89 95)(51 90 96)(52 91 93)(53 58 86)(54 59 87)(55 60 88)(56 57 85)
(1 18)(2 19)(3 20)(4 17)(5 45)(6 46)(7 47)(8 48)(9 72)(10 69)(11 70)(12 71)(13 24)(14 21)(15 22)(16 23)(25 33)(26 34)(27 35)(28 36)(29 65)(30 66)(31 67)(32 68)(37 42)(38 43)(39 44)(40 41)(49 56)(50 53)(51 54)(52 55)(57 94)(58 95)(59 96)(60 93)(61 84)(62 81)(63 82)(64 83)(73 80)(74 77)(75 78)(76 79)(85 92)(86 89)(87 90)(88 91)
G:=sub<Sym(96)| (1,72)(2,69)(3,70)(4,71)(5,89)(6,90)(7,91)(8,92)(9,18)(10,19)(11,20)(12,17)(13,96)(14,93)(15,94)(16,95)(21,60)(22,57)(23,58)(24,59)(25,53)(26,54)(27,55)(28,56)(29,41)(30,42)(31,43)(32,44)(33,50)(34,51)(35,52)(36,49)(37,66)(38,67)(39,68)(40,65)(45,86)(46,87)(47,88)(48,85)(61,79)(62,80)(63,77)(64,78)(73,81)(74,82)(75,83)(76,84), (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), (1,33,18,25)(2,34,19,26)(3,35,20,27)(4,36,17,28)(5,66,23,79)(6,67,24,80)(7,68,21,77)(8,65,22,78)(9,53,72,50)(10,54,69,51)(11,55,70,52)(12,56,71,49)(13,73,46,31)(14,74,47,32)(15,75,48,29)(16,76,45,30)(37,58,61,89)(38,59,62,90)(39,60,63,91)(40,57,64,92)(41,94,83,85)(42,95,84,86)(43,96,81,87)(44,93,82,88), (1,79,30)(2,80,31)(3,77,32)(4,78,29)(5,16,33)(6,13,34)(7,14,35)(8,15,36)(9,37,84)(10,38,81)(11,39,82)(12,40,83)(17,65,75)(18,66,76)(19,67,73)(20,68,74)(21,47,27)(22,48,28)(23,45,25)(24,46,26)(41,71,64)(42,72,61)(43,69,62)(44,70,63)(49,92,94)(50,89,95)(51,90,96)(52,91,93)(53,58,86)(54,59,87)(55,60,88)(56,57,85), (1,18)(2,19)(3,20)(4,17)(5,45)(6,46)(7,47)(8,48)(9,72)(10,69)(11,70)(12,71)(13,24)(14,21)(15,22)(16,23)(25,33)(26,34)(27,35)(28,36)(29,65)(30,66)(31,67)(32,68)(37,42)(38,43)(39,44)(40,41)(49,56)(50,53)(51,54)(52,55)(57,94)(58,95)(59,96)(60,93)(61,84)(62,81)(63,82)(64,83)(73,80)(74,77)(75,78)(76,79)(85,92)(86,89)(87,90)(88,91)>;
G:=Group( (1,72)(2,69)(3,70)(4,71)(5,89)(6,90)(7,91)(8,92)(9,18)(10,19)(11,20)(12,17)(13,96)(14,93)(15,94)(16,95)(21,60)(22,57)(23,58)(24,59)(25,53)(26,54)(27,55)(28,56)(29,41)(30,42)(31,43)(32,44)(33,50)(34,51)(35,52)(36,49)(37,66)(38,67)(39,68)(40,65)(45,86)(46,87)(47,88)(48,85)(61,79)(62,80)(63,77)(64,78)(73,81)(74,82)(75,83)(76,84), (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), (1,33,18,25)(2,34,19,26)(3,35,20,27)(4,36,17,28)(5,66,23,79)(6,67,24,80)(7,68,21,77)(8,65,22,78)(9,53,72,50)(10,54,69,51)(11,55,70,52)(12,56,71,49)(13,73,46,31)(14,74,47,32)(15,75,48,29)(16,76,45,30)(37,58,61,89)(38,59,62,90)(39,60,63,91)(40,57,64,92)(41,94,83,85)(42,95,84,86)(43,96,81,87)(44,93,82,88), (1,79,30)(2,80,31)(3,77,32)(4,78,29)(5,16,33)(6,13,34)(7,14,35)(8,15,36)(9,37,84)(10,38,81)(11,39,82)(12,40,83)(17,65,75)(18,66,76)(19,67,73)(20,68,74)(21,47,27)(22,48,28)(23,45,25)(24,46,26)(41,71,64)(42,72,61)(43,69,62)(44,70,63)(49,92,94)(50,89,95)(51,90,96)(52,91,93)(53,58,86)(54,59,87)(55,60,88)(56,57,85), (1,18)(2,19)(3,20)(4,17)(5,45)(6,46)(7,47)(8,48)(9,72)(10,69)(11,70)(12,71)(13,24)(14,21)(15,22)(16,23)(25,33)(26,34)(27,35)(28,36)(29,65)(30,66)(31,67)(32,68)(37,42)(38,43)(39,44)(40,41)(49,56)(50,53)(51,54)(52,55)(57,94)(58,95)(59,96)(60,93)(61,84)(62,81)(63,82)(64,83)(73,80)(74,77)(75,78)(76,79)(85,92)(86,89)(87,90)(88,91) );
G=PermutationGroup([(1,72),(2,69),(3,70),(4,71),(5,89),(6,90),(7,91),(8,92),(9,18),(10,19),(11,20),(12,17),(13,96),(14,93),(15,94),(16,95),(21,60),(22,57),(23,58),(24,59),(25,53),(26,54),(27,55),(28,56),(29,41),(30,42),(31,43),(32,44),(33,50),(34,51),(35,52),(36,49),(37,66),(38,67),(39,68),(40,65),(45,86),(46,87),(47,88),(48,85),(61,79),(62,80),(63,77),(64,78),(73,81),(74,82),(75,83),(76,84)], [(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)], [(1,33,18,25),(2,34,19,26),(3,35,20,27),(4,36,17,28),(5,66,23,79),(6,67,24,80),(7,68,21,77),(8,65,22,78),(9,53,72,50),(10,54,69,51),(11,55,70,52),(12,56,71,49),(13,73,46,31),(14,74,47,32),(15,75,48,29),(16,76,45,30),(37,58,61,89),(38,59,62,90),(39,60,63,91),(40,57,64,92),(41,94,83,85),(42,95,84,86),(43,96,81,87),(44,93,82,88)], [(1,79,30),(2,80,31),(3,77,32),(4,78,29),(5,16,33),(6,13,34),(7,14,35),(8,15,36),(9,37,84),(10,38,81),(11,39,82),(12,40,83),(17,65,75),(18,66,76),(19,67,73),(20,68,74),(21,47,27),(22,48,28),(23,45,25),(24,46,26),(41,71,64),(42,72,61),(43,69,62),(44,70,63),(49,92,94),(50,89,95),(51,90,96),(52,91,93),(53,58,86),(54,59,87),(55,60,88),(56,57,85)], [(1,18),(2,19),(3,20),(4,17),(5,45),(6,46),(7,47),(8,48),(9,72),(10,69),(11,70),(12,71),(13,24),(14,21),(15,22),(16,23),(25,33),(26,34),(27,35),(28,36),(29,65),(30,66),(31,67),(32,68),(37,42),(38,43),(39,44),(40,41),(49,56),(50,53),(51,54),(52,55),(57,94),(58,95),(59,96),(60,93),(61,84),(62,81),(63,82),(64,83),(73,80),(74,77),(75,78),(76,79),(85,92),(86,89),(87,90),(88,91)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀