direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C6×C42⋊C2, (C2×C42)⋊6C6, C42⋊17(C2×C6), (C4×C12)⋊48C22, (C22×C4)⋊12C12, (C22×C12)⋊21C4, C24.35(C2×C6), C2.3(C23×C12), C6.55(C23×C4), (C23×C4).14C6, C4.30(C22×C12), (C23×C12).24C2, (C2×C6).334C24, C23.39(C2×C12), C22.7(C23×C6), C12.188(C22×C4), (C2×C12).706C23, C23.71(C22×C6), (C23×C6).89C22, (C22×C6).467C23, C22.25(C22×C12), (C22×C12).609C22, (C2×C4×C12)⋊5C2, (C6×C4⋊C4)⋊51C2, (C2×C4⋊C4)⋊24C6, C4⋊C4⋊18(C2×C6), C2.1(C6×C4○D4), (C2×C4)⋊11(C2×C12), (C2×C12)⋊40(C2×C4), (C3×C4⋊C4)⋊75C22, C6.220(C2×C4○D4), C22⋊C4.27(C2×C6), (C2×C22⋊C4).15C6, (C6×C22⋊C4).35C2, C22.26(C3×C4○D4), (C2×C6).226(C4○D4), (C2×C4).133(C22×C6), (C22×C6).120(C2×C4), (C2×C6).164(C22×C4), (C22×C4).104(C2×C6), (C3×C22⋊C4).158C22, SmallGroup(192,1403)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 402 in 330 conjugacy classes, 258 normal (18 characteristic)
C1, C2, C2 [×6], C2 [×4], C3, C4 [×8], C4 [×8], C22, C22 [×10], C22 [×12], C6, C6 [×6], C6 [×4], C2×C4 [×36], C2×C4 [×8], C23, C23 [×6], C23 [×4], C12 [×8], C12 [×8], C2×C6, C2×C6 [×10], C2×C6 [×12], C42 [×8], C22⋊C4 [×8], C4⋊C4 [×8], C22×C4 [×2], C22×C4 [×16], C24, C2×C12 [×36], C2×C12 [×8], C22×C6, C22×C6 [×6], C22×C6 [×4], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×8], C23×C4, C4×C12 [×8], C3×C22⋊C4 [×8], C3×C4⋊C4 [×8], C22×C12 [×2], C22×C12 [×16], C23×C6, C2×C42⋊C2, C2×C4×C12 [×2], C6×C22⋊C4 [×2], C6×C4⋊C4 [×2], C3×C42⋊C2 [×8], C23×C12, C6×C42⋊C2
Quotients:
C1, C2 [×15], C3, C4 [×8], C22 [×35], C6 [×15], C2×C4 [×28], C23 [×15], C12 [×8], C2×C6 [×35], C22×C4 [×14], C4○D4 [×4], C24, C2×C12 [×28], C22×C6 [×15], C42⋊C2 [×4], C23×C4, C2×C4○D4 [×2], C22×C12 [×14], C3×C4○D4 [×4], C23×C6, C2×C42⋊C2, C3×C42⋊C2 [×4], C23×C12, C6×C4○D4 [×2], C6×C42⋊C2
Generators and relations
G = < a,b,c,d | a6=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, cd=dc >
►On 96 points
(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 90 26 82)(2 85 27 83)(3 86 28 84)(4 87 29 79)(5 88 30 80)(6 89 25 81)(7 46 13 52)(8 47 14 53)(9 48 15 54)(10 43 16 49)(11 44 17 50)(12 45 18 51)(19 56 96 64)(20 57 91 65)(21 58 92 66)(22 59 93 61)(23 60 94 62)(24 55 95 63)(31 71 39 77)(32 72 40 78)(33 67 41 73)(34 68 42 74)(35 69 37 75)(36 70 38 76)
(1 47 39 58)(2 48 40 59)(3 43 41 60)(4 44 42 55)(5 45 37 56)(6 46 38 57)(7 70 20 81)(8 71 21 82)(9 72 22 83)(10 67 23 84)(11 68 24 79)(12 69 19 80)(13 76 91 89)(14 77 92 90)(15 78 93 85)(16 73 94 86)(17 74 95 87)(18 75 96 88)(25 52 36 65)(26 53 31 66)(27 54 32 61)(28 49 33 62)(29 50 34 63)(30 51 35 64)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 25)(7 91)(8 92)(9 93)(10 94)(11 95)(12 96)(13 20)(14 21)(15 22)(16 23)(17 24)(18 19)(31 39)(32 40)(33 41)(34 42)(35 37)(36 38)(43 49)(44 50)(45 51)(46 52)(47 53)(48 54)(55 63)(56 64)(57 65)(58 66)(59 61)(60 62)(67 86)(68 87)(69 88)(70 89)(71 90)(72 85)(73 84)(74 79)(75 80)(76 81)(77 82)(78 83)
G:=sub<Sym(96)| (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), (1,90,26,82)(2,85,27,83)(3,86,28,84)(4,87,29,79)(5,88,30,80)(6,89,25,81)(7,46,13,52)(8,47,14,53)(9,48,15,54)(10,43,16,49)(11,44,17,50)(12,45,18,51)(19,56,96,64)(20,57,91,65)(21,58,92,66)(22,59,93,61)(23,60,94,62)(24,55,95,63)(31,71,39,77)(32,72,40,78)(33,67,41,73)(34,68,42,74)(35,69,37,75)(36,70,38,76), (1,47,39,58)(2,48,40,59)(3,43,41,60)(4,44,42,55)(5,45,37,56)(6,46,38,57)(7,70,20,81)(8,71,21,82)(9,72,22,83)(10,67,23,84)(11,68,24,79)(12,69,19,80)(13,76,91,89)(14,77,92,90)(15,78,93,85)(16,73,94,86)(17,74,95,87)(18,75,96,88)(25,52,36,65)(26,53,31,66)(27,54,32,61)(28,49,33,62)(29,50,34,63)(30,51,35,64), (1,26)(2,27)(3,28)(4,29)(5,30)(6,25)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,20)(14,21)(15,22)(16,23)(17,24)(18,19)(31,39)(32,40)(33,41)(34,42)(35,37)(36,38)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54)(55,63)(56,64)(57,65)(58,66)(59,61)(60,62)(67,86)(68,87)(69,88)(70,89)(71,90)(72,85)(73,84)(74,79)(75,80)(76,81)(77,82)(78,83)>;
G:=Group( (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), (1,90,26,82)(2,85,27,83)(3,86,28,84)(4,87,29,79)(5,88,30,80)(6,89,25,81)(7,46,13,52)(8,47,14,53)(9,48,15,54)(10,43,16,49)(11,44,17,50)(12,45,18,51)(19,56,96,64)(20,57,91,65)(21,58,92,66)(22,59,93,61)(23,60,94,62)(24,55,95,63)(31,71,39,77)(32,72,40,78)(33,67,41,73)(34,68,42,74)(35,69,37,75)(36,70,38,76), (1,47,39,58)(2,48,40,59)(3,43,41,60)(4,44,42,55)(5,45,37,56)(6,46,38,57)(7,70,20,81)(8,71,21,82)(9,72,22,83)(10,67,23,84)(11,68,24,79)(12,69,19,80)(13,76,91,89)(14,77,92,90)(15,78,93,85)(16,73,94,86)(17,74,95,87)(18,75,96,88)(25,52,36,65)(26,53,31,66)(27,54,32,61)(28,49,33,62)(29,50,34,63)(30,51,35,64), (1,26)(2,27)(3,28)(4,29)(5,30)(6,25)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,20)(14,21)(15,22)(16,23)(17,24)(18,19)(31,39)(32,40)(33,41)(34,42)(35,37)(36,38)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54)(55,63)(56,64)(57,65)(58,66)(59,61)(60,62)(67,86)(68,87)(69,88)(70,89)(71,90)(72,85)(73,84)(74,79)(75,80)(76,81)(77,82)(78,83) );
G=PermutationGroup([(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)], [(1,90,26,82),(2,85,27,83),(3,86,28,84),(4,87,29,79),(5,88,30,80),(6,89,25,81),(7,46,13,52),(8,47,14,53),(9,48,15,54),(10,43,16,49),(11,44,17,50),(12,45,18,51),(19,56,96,64),(20,57,91,65),(21,58,92,66),(22,59,93,61),(23,60,94,62),(24,55,95,63),(31,71,39,77),(32,72,40,78),(33,67,41,73),(34,68,42,74),(35,69,37,75),(36,70,38,76)], [(1,47,39,58),(2,48,40,59),(3,43,41,60),(4,44,42,55),(5,45,37,56),(6,46,38,57),(7,70,20,81),(8,71,21,82),(9,72,22,83),(10,67,23,84),(11,68,24,79),(12,69,19,80),(13,76,91,89),(14,77,92,90),(15,78,93,85),(16,73,94,86),(17,74,95,87),(18,75,96,88),(25,52,36,65),(26,53,31,66),(27,54,32,61),(28,49,33,62),(29,50,34,63),(30,51,35,64)], [(1,26),(2,27),(3,28),(4,29),(5,30),(6,25),(7,91),(8,92),(9,93),(10,94),(11,95),(12,96),(13,20),(14,21),(15,22),(16,23),(17,24),(18,19),(31,39),(32,40),(33,41),(34,42),(35,37),(36,38),(43,49),(44,50),(45,51),(46,52),(47,53),(48,54),(55,63),(56,64),(57,65),(58,66),(59,61),(60,62),(67,86),(68,87),(69,88),(70,89),(71,90),(72,85),(73,84),(74,79),(75,80),(76,81),(77,82),(78,83)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 12 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 2 | 8 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 12 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 5 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(13))| [12,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,5,0,0,0,0,5,2,0,0,0,8],[12,0,0,0,0,12,0,0,0,0,8,0,0,0,0,8],[12,0,0,0,0,1,0,0,0,0,12,0,0,0,5,1] >;
120 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3A | 3B | 4A | ··· | 4H | 4I | ··· | 4AB | 6A | ··· | 6N | 6O | ··· | 6V | 12A | ··· | 12P | 12Q | ··· | 12BD |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C6 | C12 | C4○D4 | C3×C4○D4 |
| kernel | C6×C42⋊C2 | C2×C4×C12 | C6×C22⋊C4 | C6×C4⋊C4 | C3×C42⋊C2 | C23×C12 | C2×C42⋊C2 | C22×C12 | C2×C42 | C2×C22⋊C4 | C2×C4⋊C4 | C42⋊C2 | C23×C4 | C22×C4 | C2×C6 | C22 |
| # reps | 1 | 2 | 2 | 2 | 8 | 1 | 2 | 16 | 4 | 4 | 4 | 16 | 2 | 32 | 8 | 16 |
In GAP, Magma, Sage, TeX
C_6\times C_4^2\rtimes C_2
% in TeX
G:=Group("C6xC4^2:C2"); // GroupNames label
G:=SmallGroup(192,1403);
// by ID
G=gap.SmallGroup(192,1403);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,672,701,268]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b*c^2,c*d=d*c>;
// generators/relations