metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.179D6, C6.852+ (1+4), C4⋊Q8⋊17S3, (C4×D12)⋊52C2, C4⋊C4.222D6, C12⋊D4⋊41C2, (C2×Q8).113D6, (C2×C6).278C24, D6⋊C4.53C22, C12.141(C4○D4), C12.23D4⋊28C2, C2.89(D4⋊6D6), (C2×C12).640C23, (C4×C12).219C22, C4.42(Q8⋊3S3), (C6×Q8).145C22, (C2×D12).275C22, C4⋊Dic3.387C22, C22.299(S3×C23), (C22×S3).123C23, C3⋊5(C22.49C24), (C4×Dic3).167C22, (C2×Dic3).275C23, (C3×C4⋊Q8)⋊20C2, C4⋊C4⋊7S3⋊44C2, C6.125(C2×C4○D4), (S3×C2×C4).151C22, C2.33(C2×Q8⋊3S3), (C3×C4⋊C4).221C22, (C2×C4).603(C22×S3), SmallGroup(192,1293)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 640 in 236 conjugacy classes, 99 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×4], C4 [×9], C22, C22 [×12], S3 [×4], C6, C6 [×2], C2×C4, C2×C4 [×6], C2×C4 [×12], D4 [×8], Q8 [×2], C23 [×4], Dic3 [×4], C12 [×4], C12 [×5], D6 [×12], C2×C6, C42, C42 [×4], C22⋊C4 [×12], C4⋊C4 [×4], C4⋊C4 [×2], C22×C4 [×4], C2×D4 [×6], C2×Q8 [×2], C4×S3 [×8], D12 [×8], C2×Dic3 [×4], C2×C12, C2×C12 [×6], C3×Q8 [×2], C22×S3 [×4], C42⋊C2 [×4], C4×D4 [×2], C4⋊D4 [×4], C4.4D4 [×4], C4⋊Q8, C4×Dic3 [×4], C4⋊Dic3 [×2], D6⋊C4 [×12], C4×C12, C3×C4⋊C4 [×4], S3×C2×C4 [×4], C2×D12 [×6], C6×Q8 [×2], C22.49C24, C4×D12 [×2], C4⋊C4⋊7S3 [×4], C12⋊D4 [×4], C12.23D4 [×4], C3×C4⋊Q8, C42.179D6
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), Q8⋊3S3 [×4], S3×C23, C22.49C24, D4⋊6D6, C2×Q8⋊3S3 [×2], C42.179D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=b2, d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, dbd-1=a2b, dcd-1=a2c5 >
►On 96 points
(1 29 37 74)(2 75 38 30)(3 31 39 76)(4 77 40 32)(5 33 41 78)(6 79 42 34)(7 35 43 80)(8 81 44 36)(9 25 45 82)(10 83 46 26)(11 27 47 84)(12 73 48 28)(13 59 96 72)(14 61 85 60)(15 49 86 62)(16 63 87 50)(17 51 88 64)(18 65 89 52)(19 53 90 66)(20 67 91 54)(21 55 92 68)(22 69 93 56)(23 57 94 70)(24 71 95 58)
(1 94 7 88)(2 89 8 95)(3 96 9 90)(4 91 10 85)(5 86 11 92)(6 93 12 87)(13 45 19 39)(14 40 20 46)(15 47 21 41)(16 42 22 48)(17 37 23 43)(18 44 24 38)(25 66 31 72)(26 61 32 67)(27 68 33 62)(28 63 34 69)(29 70 35 64)(30 65 36 71)(49 84 55 78)(50 79 56 73)(51 74 57 80)(52 81 58 75)(53 76 59 82)(54 83 60 77)
(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 66 43 59)(2 58 44 65)(3 64 45 57)(4 56 46 63)(5 62 47 55)(6 54 48 61)(7 72 37 53)(8 52 38 71)(9 70 39 51)(10 50 40 69)(11 68 41 49)(12 60 42 67)(13 29 90 80)(14 79 91 28)(15 27 92 78)(16 77 93 26)(17 25 94 76)(18 75 95 36)(19 35 96 74)(20 73 85 34)(21 33 86 84)(22 83 87 32)(23 31 88 82)(24 81 89 30)
G:=sub<Sym(96)| (1,29,37,74)(2,75,38,30)(3,31,39,76)(4,77,40,32)(5,33,41,78)(6,79,42,34)(7,35,43,80)(8,81,44,36)(9,25,45,82)(10,83,46,26)(11,27,47,84)(12,73,48,28)(13,59,96,72)(14,61,85,60)(15,49,86,62)(16,63,87,50)(17,51,88,64)(18,65,89,52)(19,53,90,66)(20,67,91,54)(21,55,92,68)(22,69,93,56)(23,57,94,70)(24,71,95,58), (1,94,7,88)(2,89,8,95)(3,96,9,90)(4,91,10,85)(5,86,11,92)(6,93,12,87)(13,45,19,39)(14,40,20,46)(15,47,21,41)(16,42,22,48)(17,37,23,43)(18,44,24,38)(25,66,31,72)(26,61,32,67)(27,68,33,62)(28,63,34,69)(29,70,35,64)(30,65,36,71)(49,84,55,78)(50,79,56,73)(51,74,57,80)(52,81,58,75)(53,76,59,82)(54,83,60,77), (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), (1,66,43,59)(2,58,44,65)(3,64,45,57)(4,56,46,63)(5,62,47,55)(6,54,48,61)(7,72,37,53)(8,52,38,71)(9,70,39,51)(10,50,40,69)(11,68,41,49)(12,60,42,67)(13,29,90,80)(14,79,91,28)(15,27,92,78)(16,77,93,26)(17,25,94,76)(18,75,95,36)(19,35,96,74)(20,73,85,34)(21,33,86,84)(22,83,87,32)(23,31,88,82)(24,81,89,30)>;
G:=Group( (1,29,37,74)(2,75,38,30)(3,31,39,76)(4,77,40,32)(5,33,41,78)(6,79,42,34)(7,35,43,80)(8,81,44,36)(9,25,45,82)(10,83,46,26)(11,27,47,84)(12,73,48,28)(13,59,96,72)(14,61,85,60)(15,49,86,62)(16,63,87,50)(17,51,88,64)(18,65,89,52)(19,53,90,66)(20,67,91,54)(21,55,92,68)(22,69,93,56)(23,57,94,70)(24,71,95,58), (1,94,7,88)(2,89,8,95)(3,96,9,90)(4,91,10,85)(5,86,11,92)(6,93,12,87)(13,45,19,39)(14,40,20,46)(15,47,21,41)(16,42,22,48)(17,37,23,43)(18,44,24,38)(25,66,31,72)(26,61,32,67)(27,68,33,62)(28,63,34,69)(29,70,35,64)(30,65,36,71)(49,84,55,78)(50,79,56,73)(51,74,57,80)(52,81,58,75)(53,76,59,82)(54,83,60,77), (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), (1,66,43,59)(2,58,44,65)(3,64,45,57)(4,56,46,63)(5,62,47,55)(6,54,48,61)(7,72,37,53)(8,52,38,71)(9,70,39,51)(10,50,40,69)(11,68,41,49)(12,60,42,67)(13,29,90,80)(14,79,91,28)(15,27,92,78)(16,77,93,26)(17,25,94,76)(18,75,95,36)(19,35,96,74)(20,73,85,34)(21,33,86,84)(22,83,87,32)(23,31,88,82)(24,81,89,30) );
G=PermutationGroup([(1,29,37,74),(2,75,38,30),(3,31,39,76),(4,77,40,32),(5,33,41,78),(6,79,42,34),(7,35,43,80),(8,81,44,36),(9,25,45,82),(10,83,46,26),(11,27,47,84),(12,73,48,28),(13,59,96,72),(14,61,85,60),(15,49,86,62),(16,63,87,50),(17,51,88,64),(18,65,89,52),(19,53,90,66),(20,67,91,54),(21,55,92,68),(22,69,93,56),(23,57,94,70),(24,71,95,58)], [(1,94,7,88),(2,89,8,95),(3,96,9,90),(4,91,10,85),(5,86,11,92),(6,93,12,87),(13,45,19,39),(14,40,20,46),(15,47,21,41),(16,42,22,48),(17,37,23,43),(18,44,24,38),(25,66,31,72),(26,61,32,67),(27,68,33,62),(28,63,34,69),(29,70,35,64),(30,65,36,71),(49,84,55,78),(50,79,56,73),(51,74,57,80),(52,81,58,75),(53,76,59,82),(54,83,60,77)], [(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)], [(1,66,43,59),(2,58,44,65),(3,64,45,57),(4,56,46,63),(5,62,47,55),(6,54,48,61),(7,72,37,53),(8,52,38,71),(9,70,39,51),(10,50,40,69),(11,68,41,49),(12,60,42,67),(13,29,90,80),(14,79,91,28),(15,27,92,78),(16,77,93,26),(17,25,94,76),(18,75,95,36),(19,35,96,74),(20,73,85,34),(21,33,86,84),(22,83,87,32),(23,31,88,82),(24,81,89,30)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 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 | 1 | 3 |
| 0 | 0 | 0 | 0 | 8 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 3 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 10 |
| 0 | 0 | 0 | 0 | 5 | 1 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 8 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 11 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 8 | 12 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,8,0,0,0,0,3,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,3,0,0,0,0,0,8,0,0,0,0,0,0,12,5,0,0,0,0,10,1],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,8,0,0,0,0,0,8,5,0,0,0,0,0,0,8,0,0,0,0,0,11,5],[12,0,0,0,0,0,12,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,1,8,0,0,0,0,0,12] >;
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | ··· | 4Q | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | C4○D4 | 2+ (1+4) | Q8⋊3S3 | D4⋊6D6 |
| kernel | C42.179D6 | C4×D12 | C4⋊C4⋊7S3 | C12⋊D4 | C12.23D4 | C3×C4⋊Q8 | C4⋊Q8 | C42 | C4⋊C4 | C2×Q8 | C12 | C6 | C4 | C2 |
| # reps | 1 | 2 | 4 | 4 | 4 | 1 | 1 | 1 | 4 | 2 | 8 | 1 | 4 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{179}D_6 % in TeX
G:=Group("C4^2.179D6"); // GroupNames label
G:=SmallGroup(192,1293);
// by ID
G=gap.SmallGroup(192,1293);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,387,100,675,570,185,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=b^2,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b,d*c*d^-1=a^2*c^5>;
// generators/relations