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), C2.89(D4⋊6D6), C12.23D4⋊28C2, (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
Generators and relations for C42.179D6
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 >
Subgroups: 640 in 236 conjugacy classes, 99 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C42⋊C2, C4×D4, C4⋊D4, C4.4D4, C4⋊Q8, C4×Dic3, C4⋊Dic3, D6⋊C4, C4×C12, C3×C4⋊C4, S3×C2×C4, C2×D12, C6×Q8, C22.49C24, C4×D12, C4⋊C4⋊7S3, C12⋊D4, C12.23D4, C3×C4⋊Q8, C42.179D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, Q8⋊3S3, S3×C23, C22.49C24, D4⋊6D6, C2×Q8⋊3S3, C42.179D6
►On 96 points
Generators in S96
(1 21 86 47)(2 48 87 22)(3 23 88 37)(4 38 89 24)(5 13 90 39)(6 40 91 14)(7 15 92 41)(8 42 93 16)(9 17 94 43)(10 44 95 18)(11 19 96 45)(12 46 85 20)(25 65 76 50)(26 51 77 66)(27 67 78 52)(28 53 79 68)(29 69 80 54)(30 55 81 70)(31 71 82 56)(32 57 83 72)(33 61 84 58)(34 59 73 62)(35 63 74 60)(36 49 75 64)
(1 52 7 58)(2 59 8 53)(3 54 9 60)(4 49 10 55)(5 56 11 50)(6 51 12 57)(13 31 19 25)(14 26 20 32)(15 33 21 27)(16 28 22 34)(17 35 23 29)(18 30 24 36)(37 80 43 74)(38 75 44 81)(39 82 45 76)(40 77 46 83)(41 84 47 78)(42 79 48 73)(61 86 67 92)(62 93 68 87)(63 88 69 94)(64 95 70 89)(65 90 71 96)(66 85 72 91)
(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 35 92 80)(2 79 93 34)(3 33 94 78)(4 77 95 32)(5 31 96 76)(6 75 85 30)(7 29 86 74)(8 73 87 28)(9 27 88 84)(10 83 89 26)(11 25 90 82)(12 81 91 36)(13 56 45 65)(14 64 46 55)(15 54 47 63)(16 62 48 53)(17 52 37 61)(18 72 38 51)(19 50 39 71)(20 70 40 49)(21 60 41 69)(22 68 42 59)(23 58 43 67)(24 66 44 57)
G:=sub<Sym(96)| (1,21,86,47)(2,48,87,22)(3,23,88,37)(4,38,89,24)(5,13,90,39)(6,40,91,14)(7,15,92,41)(8,42,93,16)(9,17,94,43)(10,44,95,18)(11,19,96,45)(12,46,85,20)(25,65,76,50)(26,51,77,66)(27,67,78,52)(28,53,79,68)(29,69,80,54)(30,55,81,70)(31,71,82,56)(32,57,83,72)(33,61,84,58)(34,59,73,62)(35,63,74,60)(36,49,75,64), (1,52,7,58)(2,59,8,53)(3,54,9,60)(4,49,10,55)(5,56,11,50)(6,51,12,57)(13,31,19,25)(14,26,20,32)(15,33,21,27)(16,28,22,34)(17,35,23,29)(18,30,24,36)(37,80,43,74)(38,75,44,81)(39,82,45,76)(40,77,46,83)(41,84,47,78)(42,79,48,73)(61,86,67,92)(62,93,68,87)(63,88,69,94)(64,95,70,89)(65,90,71,96)(66,85,72,91), (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), (1,35,92,80)(2,79,93,34)(3,33,94,78)(4,77,95,32)(5,31,96,76)(6,75,85,30)(7,29,86,74)(8,73,87,28)(9,27,88,84)(10,83,89,26)(11,25,90,82)(12,81,91,36)(13,56,45,65)(14,64,46,55)(15,54,47,63)(16,62,48,53)(17,52,37,61)(18,72,38,51)(19,50,39,71)(20,70,40,49)(21,60,41,69)(22,68,42,59)(23,58,43,67)(24,66,44,57)>;
G:=Group( (1,21,86,47)(2,48,87,22)(3,23,88,37)(4,38,89,24)(5,13,90,39)(6,40,91,14)(7,15,92,41)(8,42,93,16)(9,17,94,43)(10,44,95,18)(11,19,96,45)(12,46,85,20)(25,65,76,50)(26,51,77,66)(27,67,78,52)(28,53,79,68)(29,69,80,54)(30,55,81,70)(31,71,82,56)(32,57,83,72)(33,61,84,58)(34,59,73,62)(35,63,74,60)(36,49,75,64), (1,52,7,58)(2,59,8,53)(3,54,9,60)(4,49,10,55)(5,56,11,50)(6,51,12,57)(13,31,19,25)(14,26,20,32)(15,33,21,27)(16,28,22,34)(17,35,23,29)(18,30,24,36)(37,80,43,74)(38,75,44,81)(39,82,45,76)(40,77,46,83)(41,84,47,78)(42,79,48,73)(61,86,67,92)(62,93,68,87)(63,88,69,94)(64,95,70,89)(65,90,71,96)(66,85,72,91), (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), (1,35,92,80)(2,79,93,34)(3,33,94,78)(4,77,95,32)(5,31,96,76)(6,75,85,30)(7,29,86,74)(8,73,87,28)(9,27,88,84)(10,83,89,26)(11,25,90,82)(12,81,91,36)(13,56,45,65)(14,64,46,55)(15,54,47,63)(16,62,48,53)(17,52,37,61)(18,72,38,51)(19,50,39,71)(20,70,40,49)(21,60,41,69)(22,68,42,59)(23,58,43,67)(24,66,44,57) );
G=PermutationGroup([[(1,21,86,47),(2,48,87,22),(3,23,88,37),(4,38,89,24),(5,13,90,39),(6,40,91,14),(7,15,92,41),(8,42,93,16),(9,17,94,43),(10,44,95,18),(11,19,96,45),(12,46,85,20),(25,65,76,50),(26,51,77,66),(27,67,78,52),(28,53,79,68),(29,69,80,54),(30,55,81,70),(31,71,82,56),(32,57,83,72),(33,61,84,58),(34,59,73,62),(35,63,74,60),(36,49,75,64)], [(1,52,7,58),(2,59,8,53),(3,54,9,60),(4,49,10,55),(5,56,11,50),(6,51,12,57),(13,31,19,25),(14,26,20,32),(15,33,21,27),(16,28,22,34),(17,35,23,29),(18,30,24,36),(37,80,43,74),(38,75,44,81),(39,82,45,76),(40,77,46,83),(41,84,47,78),(42,79,48,73),(61,86,67,92),(62,93,68,87),(63,88,69,94),(64,95,70,89),(65,90,71,96),(66,85,72,91)], [(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)], [(1,35,92,80),(2,79,93,34),(3,33,94,78),(4,77,95,32),(5,31,96,76),(6,75,85,30),(7,29,86,74),(8,73,87,28),(9,27,88,84),(10,83,89,26),(11,25,90,82),(12,81,91,36),(13,56,45,65),(14,64,46,55),(15,54,47,63),(16,62,48,53),(17,52,37,61),(18,72,38,51),(19,50,39,71),(20,70,40,49),(21,60,41,69),(22,68,42,59),(23,58,43,67),(24,66,44,57)]])
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 |
Matrix representation of C42.179D6 ►in 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] >;
C42.179D6 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