direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C6×C4⋊D4, C4⋊3(C6×D4), C12⋊17(C2×D4), (C2×C12)⋊40D4, (C23×C4)⋊9C6, C22⋊2(C6×D4), C23⋊5(C3×D4), (C22×D4)⋊7C6, (C22×C6)⋊14D4, (C23×C12)⋊14C2, (C6×D4)⋊61C22, C24.13(C2×C6), (C2×C6).342C24, C6.181(C22×D4), (C2×C12).655C23, (C22×C12)⋊65C22, C23.73(C22×C6), (C23×C6).12C22, C22.16(C23×C6), (C22×C6).257C23, C4⋊C4⋊9(C2×C6), C2.5(D4×C2×C6), (D4×C2×C6)⋊19C2, (C2×C4⋊C4)⋊14C6, (C6×C4⋊C4)⋊41C2, (C2×D4)⋊9(C2×C6), (C2×C4)⋊10(C3×D4), (C2×C6)⋊10(C2×D4), C2.5(C6×C4○D4), (C2×C22⋊C4)⋊9C6, (C6×C22⋊C4)⋊29C2, C22⋊C4⋊11(C2×C6), (C3×C4⋊C4)⋊65C22, (C22×C4)⋊20(C2×C6), C6.224(C2×C4○D4), (C2×C4).11(C22×C6), C22.29(C3×C4○D4), (C2×C6).229(C4○D4), (C3×C22⋊C4)⋊65C22, SmallGroup(192,1411)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 706 in 426 conjugacy classes, 194 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C3, C4 [×4], C4 [×6], C22, C22 [×10], C22 [×32], C6 [×3], C6 [×4], C6 [×8], C2×C4 [×12], C2×C4 [×14], D4 [×24], C23, C23 [×10], C23 [×16], C12 [×4], C12 [×6], C2×C6, C2×C6 [×10], C2×C6 [×32], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×6], C22×C4 [×4], C2×D4 [×12], C2×D4 [×12], C24, C24 [×2], C2×C12 [×12], C2×C12 [×14], C3×D4 [×24], C22×C6, C22×C6 [×10], C22×C6 [×16], C2×C22⋊C4 [×2], C2×C4⋊C4, C4⋊D4 [×8], C23×C4, C22×D4, C22×D4 [×2], C3×C22⋊C4 [×8], C3×C4⋊C4 [×4], C22×C12 [×2], C22×C12 [×6], C22×C12 [×4], C6×D4 [×12], C6×D4 [×12], C23×C6, C23×C6 [×2], C2×C4⋊D4, C6×C22⋊C4 [×2], C6×C4⋊C4, C3×C4⋊D4 [×8], C23×C12, D4×C2×C6, D4×C2×C6 [×2], C6×C4⋊D4
Quotients:
C1, C2 [×15], C3, C22 [×35], C6 [×15], D4 [×8], C23 [×15], C2×C6 [×35], C2×D4 [×12], C4○D4 [×2], C24, C3×D4 [×8], C22×C6 [×15], C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, C6×D4 [×12], C3×C4○D4 [×2], C23×C6, C2×C4⋊D4, C3×C4⋊D4 [×4], D4×C2×C6 [×2], C6×C4○D4, C6×C4⋊D4
Generators and relations
G = < a,b,c,d | a6=b4=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
►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 71 32 89)(2 72 33 90)(3 67 34 85)(4 68 35 86)(5 69 36 87)(6 70 31 88)(7 53 92 59)(8 54 93 60)(9 49 94 55)(10 50 95 56)(11 51 96 57)(12 52 91 58)(13 45 19 63)(14 46 20 64)(15 47 21 65)(16 48 22 66)(17 43 23 61)(18 44 24 62)(25 78 39 84)(26 73 40 79)(27 74 41 80)(28 75 42 81)(29 76 37 82)(30 77 38 83)
(1 50 30 44)(2 51 25 45)(3 52 26 46)(4 53 27 47)(5 54 28 48)(6 49 29 43)(7 74 15 68)(8 75 16 69)(9 76 17 70)(10 77 18 71)(11 78 13 72)(12 73 14 67)(19 90 96 84)(20 85 91 79)(21 86 92 80)(22 87 93 81)(23 88 94 82)(24 89 95 83)(31 55 37 61)(32 56 38 62)(33 57 39 63)(34 58 40 64)(35 59 41 65)(36 60 42 66)
(1 41)(2 42)(3 37)(4 38)(5 39)(6 40)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 36)(26 31)(27 32)(28 33)(29 34)(30 35)(43 64)(44 65)(45 66)(46 61)(47 62)(48 63)(49 58)(50 59)(51 60)(52 55)(53 56)(54 57)(67 76)(68 77)(69 78)(70 73)(71 74)(72 75)(79 88)(80 89)(81 90)(82 85)(83 86)(84 87)(91 94)(92 95)(93 96)
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), (1,71,32,89)(2,72,33,90)(3,67,34,85)(4,68,35,86)(5,69,36,87)(6,70,31,88)(7,53,92,59)(8,54,93,60)(9,49,94,55)(10,50,95,56)(11,51,96,57)(12,52,91,58)(13,45,19,63)(14,46,20,64)(15,47,21,65)(16,48,22,66)(17,43,23,61)(18,44,24,62)(25,78,39,84)(26,73,40,79)(27,74,41,80)(28,75,42,81)(29,76,37,82)(30,77,38,83), (1,50,30,44)(2,51,25,45)(3,52,26,46)(4,53,27,47)(5,54,28,48)(6,49,29,43)(7,74,15,68)(8,75,16,69)(9,76,17,70)(10,77,18,71)(11,78,13,72)(12,73,14,67)(19,90,96,84)(20,85,91,79)(21,86,92,80)(22,87,93,81)(23,88,94,82)(24,89,95,83)(31,55,37,61)(32,56,38,62)(33,57,39,63)(34,58,40,64)(35,59,41,65)(36,60,42,66), (1,41)(2,42)(3,37)(4,38)(5,39)(6,40)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,36)(26,31)(27,32)(28,33)(29,34)(30,35)(43,64)(44,65)(45,66)(46,61)(47,62)(48,63)(49,58)(50,59)(51,60)(52,55)(53,56)(54,57)(67,76)(68,77)(69,78)(70,73)(71,74)(72,75)(79,88)(80,89)(81,90)(82,85)(83,86)(84,87)(91,94)(92,95)(93,96)>;
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), (1,71,32,89)(2,72,33,90)(3,67,34,85)(4,68,35,86)(5,69,36,87)(6,70,31,88)(7,53,92,59)(8,54,93,60)(9,49,94,55)(10,50,95,56)(11,51,96,57)(12,52,91,58)(13,45,19,63)(14,46,20,64)(15,47,21,65)(16,48,22,66)(17,43,23,61)(18,44,24,62)(25,78,39,84)(26,73,40,79)(27,74,41,80)(28,75,42,81)(29,76,37,82)(30,77,38,83), (1,50,30,44)(2,51,25,45)(3,52,26,46)(4,53,27,47)(5,54,28,48)(6,49,29,43)(7,74,15,68)(8,75,16,69)(9,76,17,70)(10,77,18,71)(11,78,13,72)(12,73,14,67)(19,90,96,84)(20,85,91,79)(21,86,92,80)(22,87,93,81)(23,88,94,82)(24,89,95,83)(31,55,37,61)(32,56,38,62)(33,57,39,63)(34,58,40,64)(35,59,41,65)(36,60,42,66), (1,41)(2,42)(3,37)(4,38)(5,39)(6,40)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,36)(26,31)(27,32)(28,33)(29,34)(30,35)(43,64)(44,65)(45,66)(46,61)(47,62)(48,63)(49,58)(50,59)(51,60)(52,55)(53,56)(54,57)(67,76)(68,77)(69,78)(70,73)(71,74)(72,75)(79,88)(80,89)(81,90)(82,85)(83,86)(84,87)(91,94)(92,95)(93,96) );
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)], [(1,71,32,89),(2,72,33,90),(3,67,34,85),(4,68,35,86),(5,69,36,87),(6,70,31,88),(7,53,92,59),(8,54,93,60),(9,49,94,55),(10,50,95,56),(11,51,96,57),(12,52,91,58),(13,45,19,63),(14,46,20,64),(15,47,21,65),(16,48,22,66),(17,43,23,61),(18,44,24,62),(25,78,39,84),(26,73,40,79),(27,74,41,80),(28,75,42,81),(29,76,37,82),(30,77,38,83)], [(1,50,30,44),(2,51,25,45),(3,52,26,46),(4,53,27,47),(5,54,28,48),(6,49,29,43),(7,74,15,68),(8,75,16,69),(9,76,17,70),(10,77,18,71),(11,78,13,72),(12,73,14,67),(19,90,96,84),(20,85,91,79),(21,86,92,80),(22,87,93,81),(23,88,94,82),(24,89,95,83),(31,55,37,61),(32,56,38,62),(33,57,39,63),(34,58,40,64),(35,59,41,65),(36,60,42,66)], [(1,41),(2,42),(3,37),(4,38),(5,39),(6,40),(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24),(25,36),(26,31),(27,32),(28,33),(29,34),(30,35),(43,64),(44,65),(45,66),(46,61),(47,62),(48,63),(49,58),(50,59),(51,60),(52,55),(53,56),(54,57),(67,76),(68,77),(69,78),(70,73),(71,74),(72,75),(79,88),(80,89),(81,90),(82,85),(83,86),(84,87),(91,94),(92,95),(93,96)])
Matrix representation ►G ⊆ GL5(𝔽13)
| 4 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 |
| 0 | 10 | 8 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 1 | 0 | 0 |
| 0 | 11 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(13))| [4,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,9,0,0,0,0,0,9],[12,0,0,0,0,0,5,10,0,0,0,0,8,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,12,11,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,12,0],[1,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,12,0,0,0,0,0,1] >;
84 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 3A | 3B | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 6A | ··· | 6N | 6O | ··· | 6V | 6W | ··· | 6AD | 12A | ··· | 12P | 12Q | ··· | 12X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | C4○D4 | C3×D4 | C3×D4 | C3×C4○D4 |
| kernel | C6×C4⋊D4 | C6×C22⋊C4 | C6×C4⋊C4 | C3×C4⋊D4 | C23×C12 | D4×C2×C6 | C2×C4⋊D4 | C2×C22⋊C4 | C2×C4⋊C4 | C4⋊D4 | C23×C4 | C22×D4 | C2×C12 | C22×C6 | C2×C6 | C2×C4 | C23 | C22 |
| # reps | 1 | 2 | 1 | 8 | 1 | 3 | 2 | 4 | 2 | 16 | 2 | 6 | 4 | 4 | 4 | 8 | 8 | 8 |
In GAP, Magma, Sage, TeX
C_6\times C_4\rtimes D_4
% in TeX
G:=Group("C6xC4:D4"); // GroupNames label
G:=SmallGroup(192,1411);
// by ID
G=gap.SmallGroup(192,1411);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,344,2102]);
// 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,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations