metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.5D6, (C2×D4).19D6, C24⋊C4⋊16C2, (C2×C8).166D6, C6.D8⋊3C2, D4⋊C4⋊15S3, C12.3(C4○D4), C12⋊3D4.4C2, D4⋊Dic3⋊2C2, C2.D24⋊21C2, C2.8(Q8⋊3D6), C4.Dic6⋊2C2, C4.21(C4○D12), C2.11(D8⋊S3), C6.53(C8⋊C22), (C2×Dic3).15D4, (C6×D4).25C22, C22.166(S3×D4), C4.47(D4⋊2S3), (C2×C12).204C23, (C2×C24).222C22, C6.23(C4.4D4), (C2×D12).46C22, C4⋊Dic3.63C22, (C4×Dic3).8C22, C3⋊2(C42.29C22), C2.13(C23.11D6), (C2×C6).217(C2×D4), (C3×C4⋊C4).9C22, (C2×C3⋊C8).10C22, (C3×D4⋊C4)⋊22C2, (C2×C4).311(C22×S3), SmallGroup(192,323)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4.D6
G = < a,b,c,d | a4=b4=c6=d2=1, bab-1=cac-1=dad=a-1, cbc-1=a-1b-1, dbd=ab, dcd=b2c-1 >
Subgroups: 392 in 110 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×D4, C2×D4, C3⋊C8, C24, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C8⋊C4, D4⋊C4, D4⋊C4, C42.C2, C4⋊1D4, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C3×C4⋊C4, C2×C24, C2×D12, C2×C3⋊D4, C6×D4, C42.29C22, C6.D8, C24⋊C4, C2.D24, D4⋊Dic3, C3×D4⋊C4, C4.Dic6, C12⋊3D4, C4⋊C4.D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C22×S3, C4.4D4, C8⋊C22, C4○D12, S3×D4, D4⋊2S3, C42.29C22, C23.11D6, D8⋊S3, Q8⋊3D6, C4⋊C4.D6
Character table of C4⋊C4.D6
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 1 | 1 | 8 | 24 | 2 | 2 | 2 | 8 | 12 | 12 | 24 | 2 | 2 | 2 | 8 | 8 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | -2 | 0 | -1 | 2 | 2 | -2 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 2 | 2 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | -2 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | -2 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | -2 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 2 | -2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ15 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | -2i | 2i | 0 | 0 | -2 | 2 | 0 | 0 | 2i | 2i | -2i | -2i | complex lifted from C4○D4 |
| ρ16 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -2i | 2i | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ17 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 2i | -2i | 0 | 0 | -2 | 2 | 0 | 0 | -2i | -2i | 2i | 2i | complex lifted from C4○D4 |
| ρ18 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2i | -2i | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ19 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √-3 | -√-3 | 2i | -2i | 0 | 0 | 1 | -1 | √3 | -√3 | i | i | -i | -i | complex lifted from C4○D12 |
| ρ20 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√-3 | √-3 | -2i | 2i | 0 | 0 | 1 | -1 | √3 | -√3 | -i | -i | i | i | complex lifted from C4○D12 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √-3 | -√-3 | -2i | 2i | 0 | 0 | 1 | -1 | -√3 | √3 | -i | -i | i | i | complex lifted from C4○D12 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√-3 | √-3 | 2i | -2i | 0 | 0 | 1 | -1 | -√3 | √3 | i | i | -i | -i | complex lifted from C4○D12 |
| ρ23 | 4 | -4 | -4 | 4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ24 | 4 | -4 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ25 | 4 | 4 | 4 | 4 | 0 | 0 | -2 | -4 | -4 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ26 | 4 | -4 | -4 | 4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√6 | √6 | -√6 | √6 | orthogonal lifted from Q8⋊3D6 |
| ρ27 | 4 | -4 | -4 | 4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √6 | -√6 | √6 | -√6 | orthogonal lifted from Q8⋊3D6 |
| ρ28 | 4 | 4 | -4 | -4 | 0 | 0 | -2 | 4 | -4 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4⋊2S3, Schur index 2 |
| ρ29 | 4 | -4 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-6 | -√-6 | -√-6 | √-6 | complex lifted from D8⋊S3 |
| ρ30 | 4 | -4 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-6 | √-6 | √-6 | -√-6 | complex lifted from D8⋊S3 |
►On 96 points
Generators in S96
(1 41 73 63)(2 64 74 42)(3 37 75 65)(4 66 76 38)(5 39 77 61)(6 62 78 40)(7 85 51 19)(8 20 52 86)(9 87 53 21)(10 22 54 88)(11 89 49 23)(12 24 50 90)(13 57 79 35)(14 36 80 58)(15 59 81 31)(16 32 82 60)(17 55 83 33)(18 34 84 56)(25 47 91 69)(26 70 92 48)(27 43 93 71)(28 72 94 44)(29 45 95 67)(30 68 96 46)
(1 69 57 19)(2 8 58 26)(3 71 59 21)(4 10 60 28)(5 67 55 23)(6 12 56 30)(7 63 25 13)(9 65 27 15)(11 61 29 17)(14 48 64 86)(16 44 66 88)(18 46 62 90)(20 80 70 42)(22 82 72 38)(24 84 68 40)(31 87 75 43)(32 94 76 54)(33 89 77 45)(34 96 78 50)(35 85 73 47)(36 92 74 52)(37 93 81 53)(39 95 83 49)(41 91 79 51)
(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 3)(2 58)(4 56)(6 60)(7 87)(8 48)(9 85)(10 46)(11 89)(12 44)(13 81)(14 42)(15 79)(16 40)(17 83)(18 38)(19 53)(20 92)(21 51)(22 96)(23 49)(24 94)(25 43)(26 86)(27 47)(28 90)(29 45)(30 88)(31 35)(32 78)(34 76)(36 74)(37 63)(39 61)(41 65)(50 72)(52 70)(54 68)(57 59)(62 82)(64 80)(66 84)(67 95)(69 93)(71 91)(73 75)
G:=sub<Sym(96)| (1,41,73,63)(2,64,74,42)(3,37,75,65)(4,66,76,38)(5,39,77,61)(6,62,78,40)(7,85,51,19)(8,20,52,86)(9,87,53,21)(10,22,54,88)(11,89,49,23)(12,24,50,90)(13,57,79,35)(14,36,80,58)(15,59,81,31)(16,32,82,60)(17,55,83,33)(18,34,84,56)(25,47,91,69)(26,70,92,48)(27,43,93,71)(28,72,94,44)(29,45,95,67)(30,68,96,46), (1,69,57,19)(2,8,58,26)(3,71,59,21)(4,10,60,28)(5,67,55,23)(6,12,56,30)(7,63,25,13)(9,65,27,15)(11,61,29,17)(14,48,64,86)(16,44,66,88)(18,46,62,90)(20,80,70,42)(22,82,72,38)(24,84,68,40)(31,87,75,43)(32,94,76,54)(33,89,77,45)(34,96,78,50)(35,85,73,47)(36,92,74,52)(37,93,81,53)(39,95,83,49)(41,91,79,51), (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), (1,3)(2,58)(4,56)(6,60)(7,87)(8,48)(9,85)(10,46)(11,89)(12,44)(13,81)(14,42)(15,79)(16,40)(17,83)(18,38)(19,53)(20,92)(21,51)(22,96)(23,49)(24,94)(25,43)(26,86)(27,47)(28,90)(29,45)(30,88)(31,35)(32,78)(34,76)(36,74)(37,63)(39,61)(41,65)(50,72)(52,70)(54,68)(57,59)(62,82)(64,80)(66,84)(67,95)(69,93)(71,91)(73,75)>;
G:=Group( (1,41,73,63)(2,64,74,42)(3,37,75,65)(4,66,76,38)(5,39,77,61)(6,62,78,40)(7,85,51,19)(8,20,52,86)(9,87,53,21)(10,22,54,88)(11,89,49,23)(12,24,50,90)(13,57,79,35)(14,36,80,58)(15,59,81,31)(16,32,82,60)(17,55,83,33)(18,34,84,56)(25,47,91,69)(26,70,92,48)(27,43,93,71)(28,72,94,44)(29,45,95,67)(30,68,96,46), (1,69,57,19)(2,8,58,26)(3,71,59,21)(4,10,60,28)(5,67,55,23)(6,12,56,30)(7,63,25,13)(9,65,27,15)(11,61,29,17)(14,48,64,86)(16,44,66,88)(18,46,62,90)(20,80,70,42)(22,82,72,38)(24,84,68,40)(31,87,75,43)(32,94,76,54)(33,89,77,45)(34,96,78,50)(35,85,73,47)(36,92,74,52)(37,93,81,53)(39,95,83,49)(41,91,79,51), (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), (1,3)(2,58)(4,56)(6,60)(7,87)(8,48)(9,85)(10,46)(11,89)(12,44)(13,81)(14,42)(15,79)(16,40)(17,83)(18,38)(19,53)(20,92)(21,51)(22,96)(23,49)(24,94)(25,43)(26,86)(27,47)(28,90)(29,45)(30,88)(31,35)(32,78)(34,76)(36,74)(37,63)(39,61)(41,65)(50,72)(52,70)(54,68)(57,59)(62,82)(64,80)(66,84)(67,95)(69,93)(71,91)(73,75) );
G=PermutationGroup([[(1,41,73,63),(2,64,74,42),(3,37,75,65),(4,66,76,38),(5,39,77,61),(6,62,78,40),(7,85,51,19),(8,20,52,86),(9,87,53,21),(10,22,54,88),(11,89,49,23),(12,24,50,90),(13,57,79,35),(14,36,80,58),(15,59,81,31),(16,32,82,60),(17,55,83,33),(18,34,84,56),(25,47,91,69),(26,70,92,48),(27,43,93,71),(28,72,94,44),(29,45,95,67),(30,68,96,46)], [(1,69,57,19),(2,8,58,26),(3,71,59,21),(4,10,60,28),(5,67,55,23),(6,12,56,30),(7,63,25,13),(9,65,27,15),(11,61,29,17),(14,48,64,86),(16,44,66,88),(18,46,62,90),(20,80,70,42),(22,82,72,38),(24,84,68,40),(31,87,75,43),(32,94,76,54),(33,89,77,45),(34,96,78,50),(35,85,73,47),(36,92,74,52),(37,93,81,53),(39,95,83,49),(41,91,79,51)], [(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)], [(1,3),(2,58),(4,56),(6,60),(7,87),(8,48),(9,85),(10,46),(11,89),(12,44),(13,81),(14,42),(15,79),(16,40),(17,83),(18,38),(19,53),(20,92),(21,51),(22,96),(23,49),(24,94),(25,43),(26,86),(27,47),(28,90),(29,45),(30,88),(31,35),(32,78),(34,76),(36,74),(37,63),(39,61),(41,65),(50,72),(52,70),(54,68),(57,59),(62,82),(64,80),(66,84),(67,95),(69,93),(71,91),(73,75)]])
Matrix representation of C4⋊C4.D6 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 71 | 0 |
| 0 | 0 | 0 | 1 | 0 | 71 |
| 0 | 0 | 1 | 0 | 72 | 0 |
| 0 | 0 | 0 | 1 | 0 | 72 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 9 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 66 | 59 | 28 | 14 |
| 0 | 0 | 14 | 7 | 59 | 14 |
| 0 | 0 | 7 | 66 | 7 | 14 |
| 0 | 0 | 7 | 14 | 59 | 66 |
| 72 | 67 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 61 | 12 |
| 0 | 0 | 0 | 12 | 61 | 49 |
| 0 | 0 | 6 | 6 | 61 | 0 |
| 0 | 0 | 67 | 0 | 0 | 61 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 24 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 72 | 1 | 1 |
| 0 | 0 | 0 | 1 | 0 | 72 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0,1,0,0,71,0,72,0,0,0,0,71,0,72],[46,9,0,0,0,0,0,27,0,0,0,0,0,0,66,14,7,7,0,0,59,7,66,14,0,0,28,59,7,59,0,0,14,14,14,66],[72,0,0,0,0,0,67,1,0,0,0,0,0,0,12,0,6,67,0,0,0,12,6,0,0,0,61,61,61,0,0,0,12,49,0,61],[1,24,0,0,0,0,0,72,0,0,0,0,0,0,72,0,72,0,0,0,72,1,72,1,0,0,0,0,1,0,0,0,0,0,1,72] >;
C4⋊C4.D6 in GAP, Magma, Sage, TeX
C_4\rtimes C_4.D_6
% in TeX
G:=Group("C4:C4.D6"); // GroupNames label
G:=SmallGroup(192,323);
// by ID
G=gap.SmallGroup(192,323);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,1094,135,100,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=d^2=1,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^-1*b^-1,d*b*d=a*b,d*c*d=b^2*c^-1>;
// generators/relations
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