metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.114D6, C6.202+ 1+4, (C4×D4)⋊21S3, (D4×C12)⋊23C2, C4⋊C4.319D6, (C4×Dic6)⋊34C2, (C2×D4).220D6, C23.9D6⋊8C2, C12⋊3D4.7C2, Dic3⋊5D4⋊16C2, (C22×C4).64D6, C42⋊7S3⋊19C2, C4.16(C4○D12), C23.12D6⋊9C2, (C2×C6).103C24, D6⋊C4.87C22, C22⋊C4.116D6, Dic6⋊C4⋊16C2, C12.111(C4○D4), C2.21(D4⋊6D6), (C2×C12).701C23, (C4×C12).158C22, C23.11D6⋊8C2, (C6×D4).263C22, Dic3.35(C4○D4), (C2×D12).139C22, C23.28D6⋊18C2, Dic3⋊C4.66C22, (C22×S3).37C23, C4⋊Dic3.301C22, C23.110(C22×S3), (C22×C6).173C23, C22.128(S3×C23), (C2×Dic3).44C23, (C4×Dic3).76C22, (C22×C12).365C22, C3⋊1(C22.53C24), (C2×Dic6).145C22, C6.D4.107C22, (C4×C3⋊D4)⋊45C2, C2.26(S3×C4○D4), C6.45(C2×C4○D4), C2.52(C2×C4○D12), (S3×C2×C4).202C22, (C3×C4⋊C4).332C22, (C2×C4).286(C22×S3), (C2×C3⋊D4).116C22, (C3×C22⋊C4).127C22, SmallGroup(192,1118)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.114D6
G = < a,b,c,d | a4=b4=1, c6=d2=b2, ab=ba, cac-1=a-1b2, dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=c5 >
Subgroups: 600 in 236 conjugacy classes, 97 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C4×D4, C4×D4, C4×Q8, C22.D4, C4.4D4, C4⋊1D4, C4×Dic3, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C2×C3⋊D4, C22×C12, C6×D4, C22.53C24, C4×Dic6, C42⋊7S3, C23.9D6, C23.11D6, Dic6⋊C4, Dic3⋊5D4, C4×C3⋊D4, C23.28D6, C23.12D6, C12⋊3D4, D4×C12, C42.114D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, C4○D12, S3×C23, C22.53C24, C2×C4○D12, D4⋊6D6, S3×C4○D4, C42.114D6
►On 96 points
(1 24 48 93)(2 88 37 19)(3 14 38 95)(4 90 39 21)(5 16 40 85)(6 92 41 23)(7 18 42 87)(8 94 43 13)(9 20 44 89)(10 96 45 15)(11 22 46 91)(12 86 47 17)(25 77 50 67)(26 62 51 84)(27 79 52 69)(28 64 53 74)(29 81 54 71)(30 66 55 76)(31 83 56 61)(32 68 57 78)(33 73 58 63)(34 70 59 80)(35 75 60 65)(36 72 49 82)
(1 73 7 79)(2 74 8 80)(3 75 9 81)(4 76 10 82)(5 77 11 83)(6 78 12 84)(13 59 19 53)(14 60 20 54)(15 49 21 55)(16 50 22 56)(17 51 23 57)(18 52 24 58)(25 91 31 85)(26 92 32 86)(27 93 33 87)(28 94 34 88)(29 95 35 89)(30 96 36 90)(37 64 43 70)(38 65 44 71)(39 66 45 72)(40 67 46 61)(41 68 47 62)(42 69 48 63)
(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 6 7 12)(2 11 8 5)(3 4 9 10)(13 16 19 22)(14 21 20 15)(17 24 23 18)(25 53 31 59)(26 58 32 52)(27 51 33 57)(28 56 34 50)(29 49 35 55)(30 54 36 60)(37 46 43 40)(38 39 44 45)(41 42 47 48)(61 80 67 74)(62 73 68 79)(63 78 69 84)(64 83 70 77)(65 76 71 82)(66 81 72 75)(85 88 91 94)(86 93 92 87)(89 96 95 90)
G:=sub<Sym(96)| (1,24,48,93)(2,88,37,19)(3,14,38,95)(4,90,39,21)(5,16,40,85)(6,92,41,23)(7,18,42,87)(8,94,43,13)(9,20,44,89)(10,96,45,15)(11,22,46,91)(12,86,47,17)(25,77,50,67)(26,62,51,84)(27,79,52,69)(28,64,53,74)(29,81,54,71)(30,66,55,76)(31,83,56,61)(32,68,57,78)(33,73,58,63)(34,70,59,80)(35,75,60,65)(36,72,49,82), (1,73,7,79)(2,74,8,80)(3,75,9,81)(4,76,10,82)(5,77,11,83)(6,78,12,84)(13,59,19,53)(14,60,20,54)(15,49,21,55)(16,50,22,56)(17,51,23,57)(18,52,24,58)(25,91,31,85)(26,92,32,86)(27,93,33,87)(28,94,34,88)(29,95,35,89)(30,96,36,90)(37,64,43,70)(38,65,44,71)(39,66,45,72)(40,67,46,61)(41,68,47,62)(42,69,48,63), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,16,19,22)(14,21,20,15)(17,24,23,18)(25,53,31,59)(26,58,32,52)(27,51,33,57)(28,56,34,50)(29,49,35,55)(30,54,36,60)(37,46,43,40)(38,39,44,45)(41,42,47,48)(61,80,67,74)(62,73,68,79)(63,78,69,84)(64,83,70,77)(65,76,71,82)(66,81,72,75)(85,88,91,94)(86,93,92,87)(89,96,95,90)>;
G:=Group( (1,24,48,93)(2,88,37,19)(3,14,38,95)(4,90,39,21)(5,16,40,85)(6,92,41,23)(7,18,42,87)(8,94,43,13)(9,20,44,89)(10,96,45,15)(11,22,46,91)(12,86,47,17)(25,77,50,67)(26,62,51,84)(27,79,52,69)(28,64,53,74)(29,81,54,71)(30,66,55,76)(31,83,56,61)(32,68,57,78)(33,73,58,63)(34,70,59,80)(35,75,60,65)(36,72,49,82), (1,73,7,79)(2,74,8,80)(3,75,9,81)(4,76,10,82)(5,77,11,83)(6,78,12,84)(13,59,19,53)(14,60,20,54)(15,49,21,55)(16,50,22,56)(17,51,23,57)(18,52,24,58)(25,91,31,85)(26,92,32,86)(27,93,33,87)(28,94,34,88)(29,95,35,89)(30,96,36,90)(37,64,43,70)(38,65,44,71)(39,66,45,72)(40,67,46,61)(41,68,47,62)(42,69,48,63), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,16,19,22)(14,21,20,15)(17,24,23,18)(25,53,31,59)(26,58,32,52)(27,51,33,57)(28,56,34,50)(29,49,35,55)(30,54,36,60)(37,46,43,40)(38,39,44,45)(41,42,47,48)(61,80,67,74)(62,73,68,79)(63,78,69,84)(64,83,70,77)(65,76,71,82)(66,81,72,75)(85,88,91,94)(86,93,92,87)(89,96,95,90) );
G=PermutationGroup([[(1,24,48,93),(2,88,37,19),(3,14,38,95),(4,90,39,21),(5,16,40,85),(6,92,41,23),(7,18,42,87),(8,94,43,13),(9,20,44,89),(10,96,45,15),(11,22,46,91),(12,86,47,17),(25,77,50,67),(26,62,51,84),(27,79,52,69),(28,64,53,74),(29,81,54,71),(30,66,55,76),(31,83,56,61),(32,68,57,78),(33,73,58,63),(34,70,59,80),(35,75,60,65),(36,72,49,82)], [(1,73,7,79),(2,74,8,80),(3,75,9,81),(4,76,10,82),(5,77,11,83),(6,78,12,84),(13,59,19,53),(14,60,20,54),(15,49,21,55),(16,50,22,56),(17,51,23,57),(18,52,24,58),(25,91,31,85),(26,92,32,86),(27,93,33,87),(28,94,34,88),(29,95,35,89),(30,96,36,90),(37,64,43,70),(38,65,44,71),(39,66,45,72),(40,67,46,61),(41,68,47,62),(42,69,48,63)], [(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)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,16,19,22),(14,21,20,15),(17,24,23,18),(25,53,31,59),(26,58,32,52),(27,51,33,57),(28,56,34,50),(29,49,35,55),(30,54,36,60),(37,46,43,40),(38,39,44,45),(41,42,47,48),(61,80,67,74),(62,73,68,79),(63,78,69,84),(64,83,70,77),(65,76,71,82),(66,81,72,75),(85,88,91,94),(86,93,92,87),(89,96,95,90)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| 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 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 12 | 12 | 2 | 2 | ··· | 2 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | D6 | C4○D4 | C4○D4 | C4○D12 | 2+ 1+4 | D4⋊6D6 | S3×C4○D4 |
| kernel | C42.114D6 | C4×Dic6 | C42⋊7S3 | C23.9D6 | C23.11D6 | Dic6⋊C4 | Dic3⋊5D4 | C4×C3⋊D4 | C23.28D6 | C23.12D6 | C12⋊3D4 | D4×C12 | C4×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | Dic3 | C12 | C4 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 4 | 8 | 1 | 2 | 2 |
Matrix representation of C42.114D6 ►in GL4(𝔽13) generated by
| 0 | 8 | 0 | 0 |
| 5 | 0 | 0 | 0 |
| 0 | 0 | 3 | 6 |
| 0 | 0 | 7 | 10 |
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 3 | 6 |
| 0 | 0 | 7 | 10 |
| 5 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 5 | 5 |
| 0 | 0 | 8 | 0 |
| 5 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 5 | 5 |
| 0 | 0 | 0 | 8 |
G:=sub<GL(4,GF(13))| [0,5,0,0,8,0,0,0,0,0,3,7,0,0,6,10],[8,0,0,0,0,8,0,0,0,0,3,7,0,0,6,10],[5,0,0,0,0,8,0,0,0,0,5,8,0,0,5,0],[5,0,0,0,0,5,0,0,0,0,5,0,0,0,5,8] >;
C42.114D6 in GAP, Magma, Sage, TeX
C_4^2._{114}D_6 % in TeX
G:=Group("C4^2.114D6"); // GroupNames label
G:=SmallGroup(192,1118);
// by ID
G=gap.SmallGroup(192,1118);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,219,1571,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^5>;
// generators/relations