metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.96D6, C6.522- 1+4, C4⋊C4.273D6, D6⋊Q8⋊5C2, C42⋊2S3⋊3C2, C42⋊3S3⋊5C2, (C4×Dic6)⋊10C2, Dic3.Q8⋊5C2, (C2×C6).75C24, C22⋊C4.99D6, C12.6Q8⋊7C2, C42⋊C2⋊15S3, (C4×C12).26C22, D6⋊C4.63C22, C2.10(Q8○D12), (C22×C4).212D6, (C2×C12).150C23, Dic3.D4⋊5C2, Dic3⋊4D4.6C2, C4⋊Dic3.35C22, Dic3.17(C4○D4), C22.19(C4○D12), C23.16D6⋊26C2, Dic3⋊C4.99C22, (C22×S3).23C23, C22.104(S3×C23), (C22×C6).145C23, C23.170(C22×S3), (C2×Dic3).28C23, C23.28D6.3C2, C23.21D6.1C2, C6.D4.5C22, (C22×C12).436C22, C3⋊3(C22.46C24), (C2×Dic6).231C22, (C4×Dic3).197C22, (C22×Dic3).89C22, C4⋊C4⋊S3⋊5C2, C2.14(S3×C4○D4), C6.31(C2×C4○D4), C2.34(C2×C4○D12), (C2×Dic3⋊C4)⋊46C2, (C2×C6).42(C4○D4), (S3×C2×C4).193C22, (C3×C42⋊C2)⋊17C2, (C3×C4⋊C4).311C22, (C2×C4).277(C22×S3), (C2×C3⋊D4).10C22, (C3×C22⋊C4).139C22, SmallGroup(192,1090)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.96D6
G = < a,b,c,d | a4=b4=c6=1, d2=b2, ab=ba, cac-1=dad-1=ab2, bc=cb, dbd-1=a2b, dcd-1=b2c-1 >
Subgroups: 472 in 214 conjugacy classes, 97 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C22×C12, C22.46C24, C4×Dic6, C12.6Q8, C42⋊2S3, C42⋊3S3, C23.16D6, Dic3.D4, Dic3⋊4D4, C23.21D6, Dic3.Q8, D6⋊Q8, C4⋊C4⋊S3, C2×Dic3⋊C4, C23.28D6, C3×C42⋊C2, C42.96D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2- 1+4, C4○D12, S3×C23, C22.46C24, C2×C4○D12, S3×C4○D4, Q8○D12, C42.96D6
►On 96 points
(1 67 55 23)(2 48 56 86)(3 69 57 19)(4 44 58 88)(5 71 59 21)(6 46 60 90)(7 61 29 17)(8 42 30 80)(9 63 25 13)(10 38 26 82)(11 65 27 15)(12 40 28 84)(14 54 64 94)(16 50 66 96)(18 52 62 92)(20 76 70 32)(22 78 72 34)(24 74 68 36)(31 87 75 43)(33 89 77 45)(35 85 73 47)(37 93 81 53)(39 95 83 49)(41 91 79 51)
(1 79 73 17)(2 80 74 18)(3 81 75 13)(4 82 76 14)(5 83 77 15)(6 84 78 16)(7 67 51 47)(8 68 52 48)(9 69 53 43)(10 70 54 44)(11 71 49 45)(12 72 50 46)(19 93 87 25)(20 94 88 26)(21 95 89 27)(22 96 90 28)(23 91 85 29)(24 92 86 30)(31 63 57 37)(32 64 58 38)(33 65 59 39)(34 66 60 40)(35 61 55 41)(36 62 56 42)
(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 73 78)(2 77 74 5)(3 4 75 76)(7 28 51 96)(8 95 52 27)(9 26 53 94)(10 93 54 25)(11 30 49 92)(12 91 50 29)(13 64 81 38)(14 37 82 63)(15 62 83 42)(16 41 84 61)(17 66 79 40)(18 39 80 65)(19 20 87 88)(21 24 89 86)(22 85 90 23)(31 32 57 58)(33 36 59 56)(34 55 60 35)(43 44 69 70)(45 48 71 68)(46 67 72 47)
G:=sub<Sym(96)| (1,67,55,23)(2,48,56,86)(3,69,57,19)(4,44,58,88)(5,71,59,21)(6,46,60,90)(7,61,29,17)(8,42,30,80)(9,63,25,13)(10,38,26,82)(11,65,27,15)(12,40,28,84)(14,54,64,94)(16,50,66,96)(18,52,62,92)(20,76,70,32)(22,78,72,34)(24,74,68,36)(31,87,75,43)(33,89,77,45)(35,85,73,47)(37,93,81,53)(39,95,83,49)(41,91,79,51), (1,79,73,17)(2,80,74,18)(3,81,75,13)(4,82,76,14)(5,83,77,15)(6,84,78,16)(7,67,51,47)(8,68,52,48)(9,69,53,43)(10,70,54,44)(11,71,49,45)(12,72,50,46)(19,93,87,25)(20,94,88,26)(21,95,89,27)(22,96,90,28)(23,91,85,29)(24,92,86,30)(31,63,57,37)(32,64,58,38)(33,65,59,39)(34,66,60,40)(35,61,55,41)(36,62,56,42), (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), (1,6,73,78)(2,77,74,5)(3,4,75,76)(7,28,51,96)(8,95,52,27)(9,26,53,94)(10,93,54,25)(11,30,49,92)(12,91,50,29)(13,64,81,38)(14,37,82,63)(15,62,83,42)(16,41,84,61)(17,66,79,40)(18,39,80,65)(19,20,87,88)(21,24,89,86)(22,85,90,23)(31,32,57,58)(33,36,59,56)(34,55,60,35)(43,44,69,70)(45,48,71,68)(46,67,72,47)>;
G:=Group( (1,67,55,23)(2,48,56,86)(3,69,57,19)(4,44,58,88)(5,71,59,21)(6,46,60,90)(7,61,29,17)(8,42,30,80)(9,63,25,13)(10,38,26,82)(11,65,27,15)(12,40,28,84)(14,54,64,94)(16,50,66,96)(18,52,62,92)(20,76,70,32)(22,78,72,34)(24,74,68,36)(31,87,75,43)(33,89,77,45)(35,85,73,47)(37,93,81,53)(39,95,83,49)(41,91,79,51), (1,79,73,17)(2,80,74,18)(3,81,75,13)(4,82,76,14)(5,83,77,15)(6,84,78,16)(7,67,51,47)(8,68,52,48)(9,69,53,43)(10,70,54,44)(11,71,49,45)(12,72,50,46)(19,93,87,25)(20,94,88,26)(21,95,89,27)(22,96,90,28)(23,91,85,29)(24,92,86,30)(31,63,57,37)(32,64,58,38)(33,65,59,39)(34,66,60,40)(35,61,55,41)(36,62,56,42), (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), (1,6,73,78)(2,77,74,5)(3,4,75,76)(7,28,51,96)(8,95,52,27)(9,26,53,94)(10,93,54,25)(11,30,49,92)(12,91,50,29)(13,64,81,38)(14,37,82,63)(15,62,83,42)(16,41,84,61)(17,66,79,40)(18,39,80,65)(19,20,87,88)(21,24,89,86)(22,85,90,23)(31,32,57,58)(33,36,59,56)(34,55,60,35)(43,44,69,70)(45,48,71,68)(46,67,72,47) );
G=PermutationGroup([[(1,67,55,23),(2,48,56,86),(3,69,57,19),(4,44,58,88),(5,71,59,21),(6,46,60,90),(7,61,29,17),(8,42,30,80),(9,63,25,13),(10,38,26,82),(11,65,27,15),(12,40,28,84),(14,54,64,94),(16,50,66,96),(18,52,62,92),(20,76,70,32),(22,78,72,34),(24,74,68,36),(31,87,75,43),(33,89,77,45),(35,85,73,47),(37,93,81,53),(39,95,83,49),(41,91,79,51)], [(1,79,73,17),(2,80,74,18),(3,81,75,13),(4,82,76,14),(5,83,77,15),(6,84,78,16),(7,67,51,47),(8,68,52,48),(9,69,53,43),(10,70,54,44),(11,71,49,45),(12,72,50,46),(19,93,87,25),(20,94,88,26),(21,95,89,27),(22,96,90,28),(23,91,85,29),(24,92,86,30),(31,63,57,37),(32,64,58,38),(33,65,59,39),(34,66,60,40),(35,61,55,41),(36,62,56,42)], [(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)], [(1,6,73,78),(2,77,74,5),(3,4,75,76),(7,28,51,96),(8,95,52,27),(9,26,53,94),(10,93,54,25),(11,30,49,92),(12,91,50,29),(13,64,81,38),(14,37,82,63),(15,62,83,42),(16,41,84,61),(17,66,79,40),(18,39,80,65),(19,20,87,88),(21,24,89,86),(22,85,90,23),(31,32,57,58),(33,36,59,56),(34,55,60,35),(43,44,69,70),(45,48,71,68),(46,67,72,47)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | ··· | 4R | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | C4○D4 | C4○D4 | C4○D12 | 2- 1+4 | S3×C4○D4 | Q8○D12 |
| kernel | C42.96D6 | C4×Dic6 | C12.6Q8 | C42⋊2S3 | C42⋊3S3 | C23.16D6 | Dic3.D4 | Dic3⋊4D4 | C23.21D6 | Dic3.Q8 | D6⋊Q8 | C4⋊C4⋊S3 | C2×Dic3⋊C4 | C23.28D6 | C3×C42⋊C2 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | Dic3 | C2×C6 | C22 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 8 | 1 | 2 | 2 |
Matrix representation of C42.96D6 ►in GL4(𝔽13) generated by
| 12 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 2 | 9 |
| 0 | 0 | 4 | 11 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 1 | 12 |
| 0 | 12 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 |
| 0 | 0 | 0 | 12 |
G:=sub<GL(4,GF(13))| [12,0,0,0,0,1,0,0,0,0,8,0,0,0,0,8],[8,0,0,0,0,8,0,0,0,0,2,4,0,0,9,11],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,12,12],[0,1,0,0,12,0,0,0,0,0,1,0,0,0,12,12] >;
C42.96D6 in GAP, Magma, Sage, TeX
C_4^2._{96}D_6 % in TeX
G:=Group("C4^2.96D6"); // GroupNames label
G:=SmallGroup(192,1090);
// by ID
G=gap.SmallGroup(192,1090);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,387,100,675,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=b^2*c^-1>;
// generators/relations