metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.138D6, C6.712+ (1+4), C4.4D4⋊7S3, Dic3⋊D4⋊38C2, (C2×D4).108D6, (C2×Q8).106D6, C22⋊C4.72D6, C42⋊2S3⋊35C2, (C2×C6).214C24, Dic3⋊4D4⋊29C2, C23.14D6⋊33C2, Dic3⋊Q8⋊21C2, C2.73(D4⋊6D6), C12.23D4⋊19C2, (C2×C12).630C23, (C4×C12).239C22, D6⋊C4.134C22, (C6×D4).208C22, (C22×C6).44C23, C23.46(C22×S3), (C6×Q8).123C22, Dic3.27(C4○D4), C23.11D6⋊38C2, (C2×D12).161C22, C23.16D6⋊17C2, (C22×S3).94C23, C22.235(S3×C23), Dic3⋊C4.141C22, C3⋊4(C22.49C24), (C2×Dic6).175C22, (C4×Dic3).130C22, (C2×Dic3).251C23, C6.D4.51C22, (C22×Dic3).139C22, C2.73(S3×C4○D4), C6.185(C2×C4○D4), (C3×C4.4D4)⋊8C2, (S3×C2×C4).214C22, (C2×C4).73(C22×S3), (C2×C3⋊D4).57C22, (C3×C22⋊C4).61C22, SmallGroup(192,1229)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 608 in 236 conjugacy classes, 95 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×13], C22, C22 [×12], S3 [×2], C6, C6 [×2], C6 [×2], C2×C4 [×3], C2×C4 [×2], C2×C4 [×14], D4 [×8], Q8 [×2], C23 [×2], C23 [×2], Dic3 [×4], Dic3 [×4], C12 [×5], D6 [×6], C2×C6, C2×C6 [×6], C42, C42 [×4], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×6], C22×C4 [×4], C2×D4, C2×D4 [×5], C2×Q8, C2×Q8, Dic6, C4×S3 [×4], D12, C2×Dic3 [×6], C2×Dic3 [×4], C3⋊D4 [×6], C2×C12 [×3], C2×C12 [×2], C3×D4, C3×Q8, C22×S3 [×2], C22×C6 [×2], C42⋊C2 [×4], C4×D4 [×2], C4⋊D4 [×4], C4.4D4, C4.4D4 [×3], C4⋊Q8, C4×Dic3 [×2], C4×Dic3 [×2], Dic3⋊C4 [×6], D6⋊C4 [×6], C6.D4 [×2], C4×C12, C3×C22⋊C4 [×4], C2×Dic6, S3×C2×C4 [×2], C2×D12, C22×Dic3 [×2], C2×C3⋊D4 [×4], C6×D4, C6×Q8, C22.49C24, C42⋊2S3 [×2], C23.16D6 [×2], Dic3⋊4D4 [×2], Dic3⋊D4 [×2], C23.11D6 [×2], C23.14D6 [×2], Dic3⋊Q8, C12.23D4, C3×C4.4D4, C42.138D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×4], C24, C22×S3 [×7], C2×C4○D4 [×2], 2+ (1+4), S3×C23, C22.49C24, D4⋊6D6, S3×C4○D4 [×2], C42.138D6
Generators and relations
G = < a,b,c,d | a4=b4=c6=1, d2=b2, ab=ba, cac-1=ab2, ad=da, cbc-1=dbd-1=a2b, dcd-1=c-1 >
►On 96 points
(1 27 15 71)(2 48 16 86)(3 29 17 67)(4 44 18 88)(5 25 13 69)(6 46 14 90)(7 42 56 80)(8 23 57 61)(9 38 58 82)(10 19 59 63)(11 40 60 84)(12 21 55 65)(20 96 64 50)(22 92 66 52)(24 94 62 54)(26 34 70 78)(28 36 72 74)(30 32 68 76)(31 87 75 43)(33 89 77 45)(35 85 73 47)(37 93 81 53)(39 95 83 49)(41 91 79 51)
(1 79 73 65)(2 42 74 22)(3 81 75 61)(4 38 76 24)(5 83 77 63)(6 40 78 20)(7 72 52 86)(8 29 53 43)(9 68 54 88)(10 25 49 45)(11 70 50 90)(12 27 51 47)(13 39 33 19)(14 84 34 64)(15 41 35 21)(16 80 36 66)(17 37 31 23)(18 82 32 62)(26 96 46 60)(28 92 48 56)(30 94 44 58)(55 71 91 85)(57 67 93 87)(59 69 95 89)
(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 64 73 84)(2 63 74 83)(3 62 75 82)(4 61 76 81)(5 66 77 80)(6 65 78 79)(7 25 52 45)(8 30 53 44)(9 29 54 43)(10 28 49 48)(11 27 50 47)(12 26 51 46)(13 22 33 42)(14 21 34 41)(15 20 35 40)(16 19 36 39)(17 24 31 38)(18 23 32 37)(55 70 91 90)(56 69 92 89)(57 68 93 88)(58 67 94 87)(59 72 95 86)(60 71 96 85)
G:=sub<Sym(96)| (1,27,15,71)(2,48,16,86)(3,29,17,67)(4,44,18,88)(5,25,13,69)(6,46,14,90)(7,42,56,80)(8,23,57,61)(9,38,58,82)(10,19,59,63)(11,40,60,84)(12,21,55,65)(20,96,64,50)(22,92,66,52)(24,94,62,54)(26,34,70,78)(28,36,72,74)(30,32,68,76)(31,87,75,43)(33,89,77,45)(35,85,73,47)(37,93,81,53)(39,95,83,49)(41,91,79,51), (1,79,73,65)(2,42,74,22)(3,81,75,61)(4,38,76,24)(5,83,77,63)(6,40,78,20)(7,72,52,86)(8,29,53,43)(9,68,54,88)(10,25,49,45)(11,70,50,90)(12,27,51,47)(13,39,33,19)(14,84,34,64)(15,41,35,21)(16,80,36,66)(17,37,31,23)(18,82,32,62)(26,96,46,60)(28,92,48,56)(30,94,44,58)(55,71,91,85)(57,67,93,87)(59,69,95,89), (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), (1,64,73,84)(2,63,74,83)(3,62,75,82)(4,61,76,81)(5,66,77,80)(6,65,78,79)(7,25,52,45)(8,30,53,44)(9,29,54,43)(10,28,49,48)(11,27,50,47)(12,26,51,46)(13,22,33,42)(14,21,34,41)(15,20,35,40)(16,19,36,39)(17,24,31,38)(18,23,32,37)(55,70,91,90)(56,69,92,89)(57,68,93,88)(58,67,94,87)(59,72,95,86)(60,71,96,85)>;
G:=Group( (1,27,15,71)(2,48,16,86)(3,29,17,67)(4,44,18,88)(5,25,13,69)(6,46,14,90)(7,42,56,80)(8,23,57,61)(9,38,58,82)(10,19,59,63)(11,40,60,84)(12,21,55,65)(20,96,64,50)(22,92,66,52)(24,94,62,54)(26,34,70,78)(28,36,72,74)(30,32,68,76)(31,87,75,43)(33,89,77,45)(35,85,73,47)(37,93,81,53)(39,95,83,49)(41,91,79,51), (1,79,73,65)(2,42,74,22)(3,81,75,61)(4,38,76,24)(5,83,77,63)(6,40,78,20)(7,72,52,86)(8,29,53,43)(9,68,54,88)(10,25,49,45)(11,70,50,90)(12,27,51,47)(13,39,33,19)(14,84,34,64)(15,41,35,21)(16,80,36,66)(17,37,31,23)(18,82,32,62)(26,96,46,60)(28,92,48,56)(30,94,44,58)(55,71,91,85)(57,67,93,87)(59,69,95,89), (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), (1,64,73,84)(2,63,74,83)(3,62,75,82)(4,61,76,81)(5,66,77,80)(6,65,78,79)(7,25,52,45)(8,30,53,44)(9,29,54,43)(10,28,49,48)(11,27,50,47)(12,26,51,46)(13,22,33,42)(14,21,34,41)(15,20,35,40)(16,19,36,39)(17,24,31,38)(18,23,32,37)(55,70,91,90)(56,69,92,89)(57,68,93,88)(58,67,94,87)(59,72,95,86)(60,71,96,85) );
G=PermutationGroup([(1,27,15,71),(2,48,16,86),(3,29,17,67),(4,44,18,88),(5,25,13,69),(6,46,14,90),(7,42,56,80),(8,23,57,61),(9,38,58,82),(10,19,59,63),(11,40,60,84),(12,21,55,65),(20,96,64,50),(22,92,66,52),(24,94,62,54),(26,34,70,78),(28,36,72,74),(30,32,68,76),(31,87,75,43),(33,89,77,45),(35,85,73,47),(37,93,81,53),(39,95,83,49),(41,91,79,51)], [(1,79,73,65),(2,42,74,22),(3,81,75,61),(4,38,76,24),(5,83,77,63),(6,40,78,20),(7,72,52,86),(8,29,53,43),(9,68,54,88),(10,25,49,45),(11,70,50,90),(12,27,51,47),(13,39,33,19),(14,84,34,64),(15,41,35,21),(16,80,36,66),(17,37,31,23),(18,82,32,62),(26,96,46,60),(28,92,48,56),(30,94,44,58),(55,71,91,85),(57,67,93,87),(59,69,95,89)], [(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)], [(1,64,73,84),(2,63,74,83),(3,62,75,82),(4,61,76,81),(5,66,77,80),(6,65,78,79),(7,25,52,45),(8,30,53,44),(9,29,54,43),(10,28,49,48),(11,27,50,47),(12,26,51,46),(13,22,33,42),(14,21,34,41),(15,20,35,40),(16,19,36,39),(17,24,31,38),(18,23,32,37),(55,70,91,90),(56,69,92,89),(57,68,93,88),(58,67,94,87),(59,72,95,86),(60,71,96,85)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 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 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(6,GF(13))| [0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0] >;
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4O | 4P | 4Q | 6A | 6B | 6C | 6D | 6E | 12A | ··· | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 4 | ··· | 4 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | C4○D4 | 2+ (1+4) | D4⋊6D6 | S3×C4○D4 |
| kernel | C42.138D6 | C42⋊2S3 | C23.16D6 | Dic3⋊4D4 | Dic3⋊D4 | C23.11D6 | C23.14D6 | Dic3⋊Q8 | C12.23D4 | C3×C4.4D4 | C4.4D4 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | Dic3 | C6 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 8 | 1 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{138}D_6 % in TeX
G:=Group("C4^2.138D6"); // GroupNames label
G:=SmallGroup(192,1229);
// by ID
G=gap.SmallGroup(192,1229);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,758,387,100,346,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=b^2,a*b=b*a,c*a*c^-1=a*b^2,a*d=d*a,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations