direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C23.9D6, C24.36D6, C22⋊C4⋊38D6, D6.36(C2×D4), D6⋊C4⋊45C22, (C2×C6).32C24, C6.35(C22×D4), C4⋊Dic3⋊51C22, (C22×S3).94D4, (C22×C4).329D6, C22.126(S3×D4), (C2×C12).572C23, Dic3⋊C4⋊59C22, C6⋊1(C22.D4), C22.71(S3×C23), (C23×C6).58C22, C23.89(C22×S3), C22.72(C4○D12), C6.D4⋊45C22, (S3×C23).95C22, (C22×C6).124C23, C22.67(D4⋊2S3), (C22×S3).151C23, (C22×C12).352C22, (C2×Dic3).178C23, (C22×Dic3).204C22, C2.9(C2×S3×D4), (C2×D6⋊C4)⋊17C2, (S3×C2×C4)⋊65C22, (S3×C22×C4)⋊17C2, C6.12(C2×C4○D4), (C6×C22⋊C4)⋊16C2, (C2×C22⋊C4)⋊11S3, (C2×C4⋊Dic3)⋊19C2, C2.14(C2×C4○D12), (C2×C6).381(C2×D4), C2.9(C2×D4⋊2S3), C3⋊1(C2×C22.D4), (C2×Dic3⋊C4)⋊36C2, (C2×C6).101(C4○D4), (C2×C6.D4)⋊16C2, (C3×C22⋊C4)⋊51C22, (C2×C4).258(C22×S3), (C22×C3⋊D4).10C2, (C2×C3⋊D4).89C22, SmallGroup(192,1047)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 872 in 342 conjugacy classes, 119 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C3, C4 [×10], C22, C22 [×6], C22 [×26], S3 [×4], C6 [×3], C6 [×4], C6 [×2], C2×C4 [×4], C2×C4 [×24], D4 [×8], C23, C23 [×2], C23 [×16], Dic3 [×6], C12 [×4], D6 [×4], D6 [×12], C2×C6, C2×C6 [×6], C2×C6 [×10], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×8], C22×C4 [×2], C22×C4 [×11], C2×D4 [×8], C24, C24, C4×S3 [×8], C2×Dic3 [×6], C2×Dic3 [×6], C3⋊D4 [×8], C2×C12 [×4], C2×C12 [×4], C22×S3 [×6], C22×S3 [×4], C22×C6, C22×C6 [×2], C22×C6 [×6], C2×C22⋊C4, C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C22.D4 [×8], C23×C4, C22×D4, Dic3⋊C4 [×4], C4⋊Dic3 [×4], D6⋊C4 [×4], C6.D4 [×4], C3×C22⋊C4 [×4], S3×C2×C4 [×4], S3×C2×C4 [×4], C22×Dic3 [×3], C2×C3⋊D4 [×4], C2×C3⋊D4 [×4], C22×C12 [×2], S3×C23, C23×C6, C2×C22.D4, C23.9D6 [×8], C2×Dic3⋊C4, C2×C4⋊Dic3, C2×D6⋊C4, C2×C6.D4, C6×C22⋊C4, S3×C22×C4, C22×C3⋊D4, C2×C23.9D6
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×4], C24, C22×S3 [×7], C22.D4 [×4], C22×D4, C2×C4○D4 [×2], C4○D12 [×2], S3×D4 [×2], D4⋊2S3 [×2], S3×C23, C2×C22.D4, C23.9D6 [×4], C2×C4○D12, C2×S3×D4, C2×D4⋊2S3, C2×C23.9D6
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=bd=db, fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >
►On 96 points
Generators in S96
(1 95)(2 96)(3 85)(4 86)(5 87)(6 88)(7 89)(8 90)(9 91)(10 92)(11 93)(12 94)(13 26)(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 33)(21 34)(22 35)(23 36)(24 25)(37 84)(38 73)(39 74)(40 75)(41 76)(42 77)(43 78)(44 79)(45 80)(46 81)(47 82)(48 83)(49 70)(50 71)(51 72)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)
(1 67)(2 33)(3 69)(4 35)(5 71)(6 25)(7 61)(8 27)(9 63)(10 29)(11 65)(12 31)(13 76)(14 90)(15 78)(16 92)(17 80)(18 94)(19 82)(20 96)(21 84)(22 86)(23 74)(24 88)(26 41)(28 43)(30 45)(32 47)(34 37)(36 39)(38 70)(40 72)(42 62)(44 64)(46 66)(48 68)(49 73)(50 87)(51 75)(52 89)(53 77)(54 91)(55 79)(56 93)(57 81)(58 95)(59 83)(60 85)
(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 47)(2 48)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 44)(11 45)(12 46)(13 52)(14 53)(15 54)(16 55)(17 56)(18 57)(19 58)(20 59)(21 60)(22 49)(23 50)(24 51)(25 72)(26 61)(27 62)(28 63)(29 64)(30 65)(31 66)(32 67)(33 68)(34 69)(35 70)(36 71)(73 86)(74 87)(75 88)(76 89)(77 90)(78 91)(79 92)(80 93)(81 94)(82 95)(83 96)(84 85)
(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 75 7 81)(2 80 8 74)(3 73 9 79)(4 78 10 84)(5 83 11 77)(6 76 12 82)(13 72 19 66)(14 65 20 71)(15 70 21 64)(16 63 22 69)(17 68 23 62)(18 61 24 67)(25 58 31 52)(26 51 32 57)(27 56 33 50)(28 49 34 55)(29 54 35 60)(30 59 36 53)(37 86 43 92)(38 91 44 85)(39 96 45 90)(40 89 46 95)(41 94 47 88)(42 87 48 93)
G:=sub<Sym(96)| (1,95)(2,96)(3,85)(4,86)(5,87)(6,88)(7,89)(8,90)(9,91)(10,92)(11,93)(12,94)(13,26)(14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,33)(21,34)(22,35)(23,36)(24,25)(37,84)(38,73)(39,74)(40,75)(41,76)(42,77)(43,78)(44,79)(45,80)(46,81)(47,82)(48,83)(49,70)(50,71)(51,72)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69), (1,67)(2,33)(3,69)(4,35)(5,71)(6,25)(7,61)(8,27)(9,63)(10,29)(11,65)(12,31)(13,76)(14,90)(15,78)(16,92)(17,80)(18,94)(19,82)(20,96)(21,84)(22,86)(23,74)(24,88)(26,41)(28,43)(30,45)(32,47)(34,37)(36,39)(38,70)(40,72)(42,62)(44,64)(46,66)(48,68)(49,73)(50,87)(51,75)(52,89)(53,77)(54,91)(55,79)(56,93)(57,81)(58,95)(59,83)(60,85), (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), (1,47)(2,48)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,44)(11,45)(12,46)(13,52)(14,53)(15,54)(16,55)(17,56)(18,57)(19,58)(20,59)(21,60)(22,49)(23,50)(24,51)(25,72)(26,61)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70)(36,71)(73,86)(74,87)(75,88)(76,89)(77,90)(78,91)(79,92)(80,93)(81,94)(82,95)(83,96)(84,85), (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), (1,75,7,81)(2,80,8,74)(3,73,9,79)(4,78,10,84)(5,83,11,77)(6,76,12,82)(13,72,19,66)(14,65,20,71)(15,70,21,64)(16,63,22,69)(17,68,23,62)(18,61,24,67)(25,58,31,52)(26,51,32,57)(27,56,33,50)(28,49,34,55)(29,54,35,60)(30,59,36,53)(37,86,43,92)(38,91,44,85)(39,96,45,90)(40,89,46,95)(41,94,47,88)(42,87,48,93)>;
G:=Group( (1,95)(2,96)(3,85)(4,86)(5,87)(6,88)(7,89)(8,90)(9,91)(10,92)(11,93)(12,94)(13,26)(14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,33)(21,34)(22,35)(23,36)(24,25)(37,84)(38,73)(39,74)(40,75)(41,76)(42,77)(43,78)(44,79)(45,80)(46,81)(47,82)(48,83)(49,70)(50,71)(51,72)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69), (1,67)(2,33)(3,69)(4,35)(5,71)(6,25)(7,61)(8,27)(9,63)(10,29)(11,65)(12,31)(13,76)(14,90)(15,78)(16,92)(17,80)(18,94)(19,82)(20,96)(21,84)(22,86)(23,74)(24,88)(26,41)(28,43)(30,45)(32,47)(34,37)(36,39)(38,70)(40,72)(42,62)(44,64)(46,66)(48,68)(49,73)(50,87)(51,75)(52,89)(53,77)(54,91)(55,79)(56,93)(57,81)(58,95)(59,83)(60,85), (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), (1,47)(2,48)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,44)(11,45)(12,46)(13,52)(14,53)(15,54)(16,55)(17,56)(18,57)(19,58)(20,59)(21,60)(22,49)(23,50)(24,51)(25,72)(26,61)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70)(36,71)(73,86)(74,87)(75,88)(76,89)(77,90)(78,91)(79,92)(80,93)(81,94)(82,95)(83,96)(84,85), (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), (1,75,7,81)(2,80,8,74)(3,73,9,79)(4,78,10,84)(5,83,11,77)(6,76,12,82)(13,72,19,66)(14,65,20,71)(15,70,21,64)(16,63,22,69)(17,68,23,62)(18,61,24,67)(25,58,31,52)(26,51,32,57)(27,56,33,50)(28,49,34,55)(29,54,35,60)(30,59,36,53)(37,86,43,92)(38,91,44,85)(39,96,45,90)(40,89,46,95)(41,94,47,88)(42,87,48,93) );
G=PermutationGroup([(1,95),(2,96),(3,85),(4,86),(5,87),(6,88),(7,89),(8,90),(9,91),(10,92),(11,93),(12,94),(13,26),(14,27),(15,28),(16,29),(17,30),(18,31),(19,32),(20,33),(21,34),(22,35),(23,36),(24,25),(37,84),(38,73),(39,74),(40,75),(41,76),(42,77),(43,78),(44,79),(45,80),(46,81),(47,82),(48,83),(49,70),(50,71),(51,72),(52,61),(53,62),(54,63),(55,64),(56,65),(57,66),(58,67),(59,68),(60,69)], [(1,67),(2,33),(3,69),(4,35),(5,71),(6,25),(7,61),(8,27),(9,63),(10,29),(11,65),(12,31),(13,76),(14,90),(15,78),(16,92),(17,80),(18,94),(19,82),(20,96),(21,84),(22,86),(23,74),(24,88),(26,41),(28,43),(30,45),(32,47),(34,37),(36,39),(38,70),(40,72),(42,62),(44,64),(46,66),(48,68),(49,73),(50,87),(51,75),(52,89),(53,77),(54,91),(55,79),(56,93),(57,81),(58,95),(59,83),(60,85)], [(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)], [(1,47),(2,48),(3,37),(4,38),(5,39),(6,40),(7,41),(8,42),(9,43),(10,44),(11,45),(12,46),(13,52),(14,53),(15,54),(16,55),(17,56),(18,57),(19,58),(20,59),(21,60),(22,49),(23,50),(24,51),(25,72),(26,61),(27,62),(28,63),(29,64),(30,65),(31,66),(32,67),(33,68),(34,69),(35,70),(36,71),(73,86),(74,87),(75,88),(76,89),(77,90),(78,91),(79,92),(80,93),(81,94),(82,95),(83,96),(84,85)], [(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)], [(1,75,7,81),(2,80,8,74),(3,73,9,79),(4,78,10,84),(5,83,11,77),(6,76,12,82),(13,72,19,66),(14,65,20,71),(15,70,21,64),(16,63,22,69),(17,68,23,62),(18,61,24,67),(25,58,31,52),(26,51,32,57),(27,56,33,50),(28,49,34,55),(29,54,35,60),(30,59,36,53),(37,86,43,92),(38,91,44,85),(39,96,45,90),(40,89,46,95),(41,94,47,88),(42,87,48,93)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 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 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 11 | 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 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 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 | 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 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 8 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,11,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,0,0,8],[8,8,0,0,0,0,0,5,0,0,0,0,0,0,1,1,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,0,5] >;
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 6A | ··· | 6G | 6H | 6I | 6J | 6K | 12A | ··· | 12H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | C4○D4 | C4○D12 | S3×D4 | D4⋊2S3 |
| kernel | C2×C23.9D6 | C23.9D6 | C2×Dic3⋊C4 | C2×C4⋊Dic3 | C2×D6⋊C4 | C2×C6.D4 | C6×C22⋊C4 | S3×C22×C4 | C22×C3⋊D4 | C2×C22⋊C4 | C22×S3 | C22⋊C4 | C22×C4 | C24 | C2×C6 | C22 | C22 | C22 |
| # reps | 1 | 8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 1 | 8 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_2\times C_2^3._9D_6
% in TeX
G:=Group("C2xC2^3.9D6"); // GroupNames label
G:=SmallGroup(192,1047);
// by ID
G=gap.SmallGroup(192,1047);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,100,675,297,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=f^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,e*b*e^-1=b*d=d*b,f*b*f^-1=b*c*d,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^5>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀