direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C6×C42⋊C2, (C2×C42)⋊6C6, C42⋊17(C2×C6), (C22×C4)⋊12C12, (C4×C12)⋊48C22, (C22×C12)⋊21C4, C6.55(C23×C4), C2.3(C23×C12), C24.35(C2×C6), (C23×C4).14C6, (C23×C12).24C2, C4.30(C22×C12), (C2×C6).334C24, C23.39(C2×C12), C22.7(C23×C6), (C2×C12).706C23, C12.188(C22×C4), 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×C12)⋊40(C2×C4), (C2×C4)⋊11(C2×C12), (C3×C4⋊C4)⋊75C22, C6.220(C2×C4○D4), (C2×C22⋊C4).15C6, C22⋊C4.27(C2×C6), (C6×C22⋊C4).35C2, C22.26(C3×C4○D4), (C2×C6).226(C4○D4), (C2×C6).164(C22×C4), (C22×C6).120(C2×C4), (C2×C4).133(C22×C6), (C22×C4).104(C2×C6), (C3×C22⋊C4).158C22, SmallGroup(192,1403)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×C42⋊C2
G = < a,b,c,d | a6=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, cd=dc >
Subgroups: 402 in 330 conjugacy classes, 258 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C23, C23, C23, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C2×C12, C2×C12, C22×C6, C22×C6, C22×C6, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C23×C4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C22×C12, C23×C6, C2×C42⋊C2, C2×C4×C12, C6×C22⋊C4, C6×C4⋊C4, C3×C42⋊C2, C23×C12, C6×C42⋊C2
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, C22×C4, C4○D4, C24, C2×C12, C22×C6, C42⋊C2, C23×C4, C2×C4○D4, C22×C12, C3×C4○D4, C23×C6, C2×C42⋊C2, C3×C42⋊C2, C23×C12, C6×C4○D4, C6×C42⋊C2
►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 83 37 89)(2 84 38 90)(3 79 39 85)(4 80 40 86)(5 81 41 87)(6 82 42 88)(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 63 96 57)(20 64 91 58)(21 65 92 59)(22 66 93 60)(23 61 94 55)(24 62 95 56)(25 70 31 76)(26 71 32 77)(27 72 33 78)(28 67 34 73)(29 68 35 74)(30 69 36 75)
(1 47 32 65)(2 48 33 66)(3 43 34 61)(4 44 35 62)(5 45 36 63)(6 46 31 64)(7 70 20 88)(8 71 21 89)(9 72 22 90)(10 67 23 85)(11 68 24 86)(12 69 19 87)(13 76 91 82)(14 77 92 83)(15 78 93 84)(16 73 94 79)(17 74 95 80)(18 75 96 81)(25 58 42 52)(26 59 37 53)(27 60 38 54)(28 55 39 49)(29 56 40 50)(30 57 41 51)
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 91)(8 92)(9 93)(10 94)(11 95)(12 96)(13 20)(14 21)(15 22)(16 23)(17 24)(18 19)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(43 49)(44 50)(45 51)(46 52)(47 53)(48 54)(55 61)(56 62)(57 63)(58 64)(59 65)(60 66)(67 79)(68 80)(69 81)(70 82)(71 83)(72 84)(73 85)(74 86)(75 87)(76 88)(77 89)(78 90)
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), (1,83,37,89)(2,84,38,90)(3,79,39,85)(4,80,40,86)(5,81,41,87)(6,82,42,88)(7,46,13,52)(8,47,14,53)(9,48,15,54)(10,43,16,49)(11,44,17,50)(12,45,18,51)(19,63,96,57)(20,64,91,58)(21,65,92,59)(22,66,93,60)(23,61,94,55)(24,62,95,56)(25,70,31,76)(26,71,32,77)(27,72,33,78)(28,67,34,73)(29,68,35,74)(30,69,36,75), (1,47,32,65)(2,48,33,66)(3,43,34,61)(4,44,35,62)(5,45,36,63)(6,46,31,64)(7,70,20,88)(8,71,21,89)(9,72,22,90)(10,67,23,85)(11,68,24,86)(12,69,19,87)(13,76,91,82)(14,77,92,83)(15,78,93,84)(16,73,94,79)(17,74,95,80)(18,75,96,81)(25,58,42,52)(26,59,37,53)(27,60,38,54)(28,55,39,49)(29,56,40,50)(30,57,41,51), (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,20)(14,21)(15,22)(16,23)(17,24)(18,19)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54)(55,61)(56,62)(57,63)(58,64)(59,65)(60,66)(67,79)(68,80)(69,81)(70,82)(71,83)(72,84)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)>;
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), (1,83,37,89)(2,84,38,90)(3,79,39,85)(4,80,40,86)(5,81,41,87)(6,82,42,88)(7,46,13,52)(8,47,14,53)(9,48,15,54)(10,43,16,49)(11,44,17,50)(12,45,18,51)(19,63,96,57)(20,64,91,58)(21,65,92,59)(22,66,93,60)(23,61,94,55)(24,62,95,56)(25,70,31,76)(26,71,32,77)(27,72,33,78)(28,67,34,73)(29,68,35,74)(30,69,36,75), (1,47,32,65)(2,48,33,66)(3,43,34,61)(4,44,35,62)(5,45,36,63)(6,46,31,64)(7,70,20,88)(8,71,21,89)(9,72,22,90)(10,67,23,85)(11,68,24,86)(12,69,19,87)(13,76,91,82)(14,77,92,83)(15,78,93,84)(16,73,94,79)(17,74,95,80)(18,75,96,81)(25,58,42,52)(26,59,37,53)(27,60,38,54)(28,55,39,49)(29,56,40,50)(30,57,41,51), (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,20)(14,21)(15,22)(16,23)(17,24)(18,19)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54)(55,61)(56,62)(57,63)(58,64)(59,65)(60,66)(67,79)(68,80)(69,81)(70,82)(71,83)(72,84)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90) );
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)], [(1,83,37,89),(2,84,38,90),(3,79,39,85),(4,80,40,86),(5,81,41,87),(6,82,42,88),(7,46,13,52),(8,47,14,53),(9,48,15,54),(10,43,16,49),(11,44,17,50),(12,45,18,51),(19,63,96,57),(20,64,91,58),(21,65,92,59),(22,66,93,60),(23,61,94,55),(24,62,95,56),(25,70,31,76),(26,71,32,77),(27,72,33,78),(28,67,34,73),(29,68,35,74),(30,69,36,75)], [(1,47,32,65),(2,48,33,66),(3,43,34,61),(4,44,35,62),(5,45,36,63),(6,46,31,64),(7,70,20,88),(8,71,21,89),(9,72,22,90),(10,67,23,85),(11,68,24,86),(12,69,19,87),(13,76,91,82),(14,77,92,83),(15,78,93,84),(16,73,94,79),(17,74,95,80),(18,75,96,81),(25,58,42,52),(26,59,37,53),(27,60,38,54),(28,55,39,49),(29,56,40,50),(30,57,41,51)], [(1,37),(2,38),(3,39),(4,40),(5,41),(6,42),(7,91),(8,92),(9,93),(10,94),(11,95),(12,96),(13,20),(14,21),(15,22),(16,23),(17,24),(18,19),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(43,49),(44,50),(45,51),(46,52),(47,53),(48,54),(55,61),(56,62),(57,63),(58,64),(59,65),(60,66),(67,79),(68,80),(69,81),(70,82),(71,83),(72,84),(73,85),(74,86),(75,87),(76,88),(77,89),(78,90)]])
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 |
Matrix representation of C6×C42⋊C2 ►in 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] >;
C6×C42⋊C2 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