metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12⋊Q8⋊4C2, C4⋊C4.6D6, C24⋊C4⋊17C2, (C2×D4).20D6, (C2×C8).167D6, D4⋊C4.8S3, C12.4(C4○D4), C6.SD16⋊1C2, C4.22(C4○D12), C2.12(D8⋊S3), C6.28(C8⋊C22), C2.Dic12⋊21C2, (C2×Dic3).16D4, D4⋊Dic3.4C2, C2.9(D4.D6), (C6×D4).26C22, C22.167(S3×D4), C4.48(D4⋊2S3), (C2×C24).223C22, (C2×C12).205C23, C23.12D6.2C2, C6.24(C4.4D4), C6.26(C8.C22), C4⋊Dic3.64C22, (C4×Dic3).9C22, (C2×Dic6).52C22, C3⋊2(C42.28C22), C2.14(C23.11D6), (C2×C6).218(C2×D4), (C2×C3⋊C8).11C22, (C3×C4⋊C4).10C22, (C3×D4⋊C4).12C2, (C2×C4).312(C22×S3), SmallGroup(192,324)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12⋊Q8⋊C2
G = < a,b,c,d | a12=b4=d2=1, c2=b2, bab-1=dad=a7, cac-1=a5, cbc-1=b-1, dbd=a3b-1, dcd=a6b2c >
Subgroups: 296 in 100 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×D4, C2×Q8, C3⋊C8, C24, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×C6, C8⋊C4, D4⋊C4, D4⋊C4, Q8⋊C4, C4.4D4, C4⋊Q8, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C6.D4, C3×C4⋊C4, C2×C24, C2×Dic6, C2×Dic6, C6×D4, C42.28C22, C6.SD16, C24⋊C4, C2.Dic12, D4⋊Dic3, C3×D4⋊C4, C12⋊Q8, C23.12D6, C12⋊Q8⋊C2
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C22×S3, C4.4D4, C8⋊C22, C8.C22, C4○D12, S3×D4, D4⋊2S3, C42.28C22, C23.11D6, D8⋊S3, D4.D6, C12⋊Q8⋊C2
Character table of C12⋊Q8⋊C2
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 2 | 8 | 12 | 12 | 24 | 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 | -1 | 2 | 2 | 2 | 0 | 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 | -1 | 2 | 2 | -2 | 0 | 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 | 2 | -2 | -2 | 0 | 2 | -2 | 0 | 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 | 0 | 2 | -2 | -2 | 0 | -2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ14 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ15 | 2 | -2 | -2 | 2 | 0 | 2 | -2 | 2 | 0 | 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 | 2 | -2 | 2 | 0 | 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 |
| ρ17 | 2 | -2 | -2 | 2 | 0 | 2 | 2 | -2 | 0 | 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 |
| ρ18 | 2 | -2 | -2 | 2 | 0 | 2 | 2 | -2 | 0 | 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 | -1 | -2 | 2 | 0 | 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 | -1 | -2 | 2 | 0 | 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 | -1 | -2 | 2 | 0 | 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 | -1 | -2 | 2 | 0 | 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 | 4 | 0 | 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 | -2 | -4 | -4 | 0 | 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 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | -2 | 4 | -4 | 0 | 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 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ27 | 4 | 4 | -4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√6 | √6 | -√6 | √6 | symplectic lifted from D4.D6, Schur index 2 |
| ρ28 | 4 | 4 | -4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √6 | -√6 | √6 | -√6 | symplectic lifted from D4.D6, Schur index 2 |
| ρ29 | 4 | -4 | 4 | -4 | 0 | -2 | 0 | 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 | -2 | 0 | 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 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 40 89 60)(2 47 90 55)(3 42 91 50)(4 37 92 57)(5 44 93 52)(6 39 94 59)(7 46 95 54)(8 41 96 49)(9 48 85 56)(10 43 86 51)(11 38 87 58)(12 45 88 53)(13 62 74 29)(14 69 75 36)(15 64 76 31)(16 71 77 26)(17 66 78 33)(18 61 79 28)(19 68 80 35)(20 63 81 30)(21 70 82 25)(22 65 83 32)(23 72 84 27)(24 67 73 34)
(1 22 89 83)(2 15 90 76)(3 20 91 81)(4 13 92 74)(5 18 93 79)(6 23 94 84)(7 16 95 77)(8 21 96 82)(9 14 85 75)(10 19 86 80)(11 24 87 73)(12 17 88 78)(25 49 70 41)(26 54 71 46)(27 59 72 39)(28 52 61 44)(29 57 62 37)(30 50 63 42)(31 55 64 47)(32 60 65 40)(33 53 66 45)(34 58 67 38)(35 51 68 43)(36 56 69 48)
(2 8)(4 10)(6 12)(13 74)(14 81)(15 76)(16 83)(17 78)(18 73)(19 80)(20 75)(21 82)(22 77)(23 84)(24 79)(25 34)(26 29)(27 36)(28 31)(30 33)(32 35)(37 60)(38 55)(39 50)(40 57)(41 52)(42 59)(43 54)(44 49)(45 56)(46 51)(47 58)(48 53)(61 64)(62 71)(63 66)(65 68)(67 70)(69 72)(86 92)(88 94)(90 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), (1,40,89,60)(2,47,90,55)(3,42,91,50)(4,37,92,57)(5,44,93,52)(6,39,94,59)(7,46,95,54)(8,41,96,49)(9,48,85,56)(10,43,86,51)(11,38,87,58)(12,45,88,53)(13,62,74,29)(14,69,75,36)(15,64,76,31)(16,71,77,26)(17,66,78,33)(18,61,79,28)(19,68,80,35)(20,63,81,30)(21,70,82,25)(22,65,83,32)(23,72,84,27)(24,67,73,34), (1,22,89,83)(2,15,90,76)(3,20,91,81)(4,13,92,74)(5,18,93,79)(6,23,94,84)(7,16,95,77)(8,21,96,82)(9,14,85,75)(10,19,86,80)(11,24,87,73)(12,17,88,78)(25,49,70,41)(26,54,71,46)(27,59,72,39)(28,52,61,44)(29,57,62,37)(30,50,63,42)(31,55,64,47)(32,60,65,40)(33,53,66,45)(34,58,67,38)(35,51,68,43)(36,56,69,48), (2,8)(4,10)(6,12)(13,74)(14,81)(15,76)(16,83)(17,78)(18,73)(19,80)(20,75)(21,82)(22,77)(23,84)(24,79)(25,34)(26,29)(27,36)(28,31)(30,33)(32,35)(37,60)(38,55)(39,50)(40,57)(41,52)(42,59)(43,54)(44,49)(45,56)(46,51)(47,58)(48,53)(61,64)(62,71)(63,66)(65,68)(67,70)(69,72)(86,92)(88,94)(90,96)>;
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), (1,40,89,60)(2,47,90,55)(3,42,91,50)(4,37,92,57)(5,44,93,52)(6,39,94,59)(7,46,95,54)(8,41,96,49)(9,48,85,56)(10,43,86,51)(11,38,87,58)(12,45,88,53)(13,62,74,29)(14,69,75,36)(15,64,76,31)(16,71,77,26)(17,66,78,33)(18,61,79,28)(19,68,80,35)(20,63,81,30)(21,70,82,25)(22,65,83,32)(23,72,84,27)(24,67,73,34), (1,22,89,83)(2,15,90,76)(3,20,91,81)(4,13,92,74)(5,18,93,79)(6,23,94,84)(7,16,95,77)(8,21,96,82)(9,14,85,75)(10,19,86,80)(11,24,87,73)(12,17,88,78)(25,49,70,41)(26,54,71,46)(27,59,72,39)(28,52,61,44)(29,57,62,37)(30,50,63,42)(31,55,64,47)(32,60,65,40)(33,53,66,45)(34,58,67,38)(35,51,68,43)(36,56,69,48), (2,8)(4,10)(6,12)(13,74)(14,81)(15,76)(16,83)(17,78)(18,73)(19,80)(20,75)(21,82)(22,77)(23,84)(24,79)(25,34)(26,29)(27,36)(28,31)(30,33)(32,35)(37,60)(38,55)(39,50)(40,57)(41,52)(42,59)(43,54)(44,49)(45,56)(46,51)(47,58)(48,53)(61,64)(62,71)(63,66)(65,68)(67,70)(69,72)(86,92)(88,94)(90,96) );
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)], [(1,40,89,60),(2,47,90,55),(3,42,91,50),(4,37,92,57),(5,44,93,52),(6,39,94,59),(7,46,95,54),(8,41,96,49),(9,48,85,56),(10,43,86,51),(11,38,87,58),(12,45,88,53),(13,62,74,29),(14,69,75,36),(15,64,76,31),(16,71,77,26),(17,66,78,33),(18,61,79,28),(19,68,80,35),(20,63,81,30),(21,70,82,25),(22,65,83,32),(23,72,84,27),(24,67,73,34)], [(1,22,89,83),(2,15,90,76),(3,20,91,81),(4,13,92,74),(5,18,93,79),(6,23,94,84),(7,16,95,77),(8,21,96,82),(9,14,85,75),(10,19,86,80),(11,24,87,73),(12,17,88,78),(25,49,70,41),(26,54,71,46),(27,59,72,39),(28,52,61,44),(29,57,62,37),(30,50,63,42),(31,55,64,47),(32,60,65,40),(33,53,66,45),(34,58,67,38),(35,51,68,43),(36,56,69,48)], [(2,8),(4,10),(6,12),(13,74),(14,81),(15,76),(16,83),(17,78),(18,73),(19,80),(20,75),(21,82),(22,77),(23,84),(24,79),(25,34),(26,29),(27,36),(28,31),(30,33),(32,35),(37,60),(38,55),(39,50),(40,57),(41,52),(42,59),(43,54),(44,49),(45,56),(46,51),(47,58),(48,53),(61,64),(62,71),(63,66),(65,68),(67,70),(69,72),(86,92),(88,94),(90,96)]])
Matrix representation of C12⋊Q8⋊C2 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 46 | 71 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 42 | 62 | 42 | 62 |
| 0 | 0 | 11 | 31 | 11 | 31 |
| 0 | 0 | 42 | 62 | 31 | 11 |
| 0 | 0 | 11 | 31 | 62 | 42 |
| 1 | 19 | 0 | 0 | 0 | 0 |
| 46 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 65 |
| 0 | 0 | 0 | 0 | 26 | 39 |
| 0 | 0 | 39 | 8 | 0 | 0 |
| 0 | 0 | 47 | 34 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 46 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,1,0,0,0,0,72,72,0,0],[46,0,0,0,0,0,71,27,0,0,0,0,0,0,42,11,42,11,0,0,62,31,62,31,0,0,42,11,31,62,0,0,62,31,11,42],[1,46,0,0,0,0,19,72,0,0,0,0,0,0,0,0,39,47,0,0,0,0,8,34,0,0,34,26,0,0,0,0,65,39,0,0],[1,46,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;
C12⋊Q8⋊C2 in GAP, Magma, Sage, TeX
C_{12}\rtimes Q_8\rtimes C_2 % in TeX
G:=Group("C12:Q8:C2"); // GroupNames label
G:=SmallGroup(192,324);
// by ID
G=gap.SmallGroup(192,324);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,120,1094,135,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^4=d^2=1,c^2=b^2,b*a*b^-1=d*a*d=a^7,c*a*c^-1=a^5,c*b*c^-1=b^-1,d*b*d=a^3*b^-1,d*c*d=a^6*b^2*c>;
// generators/relations
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