metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.90D6, C6.482- (1+4), C6.922+ (1+4), (C2×C12)⋊5Q8, C12⋊Q8⋊11C2, (C2×C4)⋊4Dic6, C4⋊C4.268D6, C12⋊2Q8⋊6C2, C12.78(C2×Q8), C2.6(D4○D12), (C2×C6).63C24, (C4×C12).7C22, C22⋊C4.91D6, C2.7(Q8○D12), C12.6Q8⋊4C2, C4.34(C2×Dic6), C6.11(C22×Q8), C12.3Q8⋊11C2, (C22×C4).203D6, C22.7(C2×Dic6), (C2×C12).142C23, C42⋊C2.13S3, Dic3⋊C4.2C22, C4⋊Dic3.32C22, C22.96(S3×C23), C2.13(C22×Dic6), C23.161(C22×S3), (C22×C6).133C23, Dic3.D4.1C2, (C4×Dic3).68C22, (C2×Dic6).23C22, (C2×Dic3).22C23, (C22×C12).223C22, C3⋊2(C23.41C23), C6.D4.92C22, C23.26D6.23C2, (C22×Dic3).85C22, (C2×C6).13(C2×Q8), (C2×C4⋊Dic3).44C2, (C3×C4⋊C4).304C22, (C2×C4).148(C22×S3), (C3×C42⋊C2).14C2, (C3×C22⋊C4).99C22, SmallGroup(192,1078)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 456 in 206 conjugacy classes, 111 normal (29 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×4], C4 [×12], C22, C22 [×2], C22 [×2], C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×10], Q8 [×4], C23, Dic3 [×8], C12 [×4], C12 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×18], C22×C4, C22×C4 [×2], C2×Q8 [×4], Dic6 [×4], C2×Dic3 [×8], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×8], C22×C6, C2×C4⋊C4, C42⋊C2, C42⋊C2, C22⋊Q8 [×4], C42.C2 [×4], C4⋊Q8 [×4], C4×Dic3 [×2], Dic3⋊C4 [×8], C4⋊Dic3 [×2], C4⋊Dic3 [×8], C6.D4 [×2], C4×C12 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], C2×Dic6 [×4], C22×Dic3 [×2], C22×C12, C23.41C23, C12⋊2Q8 [×2], C12.6Q8 [×2], Dic3.D4 [×4], C12⋊Q8 [×2], C12.3Q8 [×2], C2×C4⋊Dic3, C23.26D6, C3×C42⋊C2, C42.90D6
Quotients:
C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D6 [×7], C2×Q8 [×6], C24, Dic6 [×4], C22×S3 [×7], C22×Q8, 2+ (1+4), 2- (1+4), C2×Dic6 [×6], S3×C23, C23.41C23, C22×Dic6, D4○D12, Q8○D12, C42.90D6
Generators and relations
G = < a,b,c,d | a4=b4=c6=1, d2=a2b2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >
►On 96 points
(1 57 37 78)(2 55 38 76)(3 59 39 74)(4 52 7 66)(5 50 8 64)(6 54 9 62)(10 63 13 49)(11 61 14 53)(12 65 15 51)(16 68 44 89)(17 72 45 87)(18 70 43 85)(19 60 24 75)(20 58 22 73)(21 56 23 77)(25 94 34 79)(26 92 35 83)(27 96 36 81)(28 91 31 82)(29 95 32 80)(30 93 33 84)(40 67 47 88)(41 71 48 86)(42 69 46 90)
(1 42 19 17)(2 40 20 18)(3 41 21 16)(4 25 13 28)(5 26 14 29)(6 27 15 30)(7 34 10 31)(8 35 11 32)(9 36 12 33)(22 43 38 47)(23 44 39 48)(24 45 37 46)(49 91 52 94)(50 92 53 95)(51 93 54 96)(55 67 58 70)(56 68 59 71)(57 69 60 72)(61 80 64 83)(62 81 65 84)(63 82 66 79)(73 85 76 88)(74 86 77 89)(75 87 78 90)
(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 14 24 8)(2 13 22 7)(3 15 23 9)(4 38 10 20)(5 37 11 19)(6 39 12 21)(16 30 48 36)(17 29 46 35)(18 28 47 34)(25 43 31 40)(26 45 32 42)(27 44 33 41)(49 58 66 76)(50 57 61 75)(51 56 62 74)(52 55 63 73)(53 60 64 78)(54 59 65 77)(67 79 85 91)(68 84 86 96)(69 83 87 95)(70 82 88 94)(71 81 89 93)(72 80 90 92)
G:=sub<Sym(96)| (1,57,37,78)(2,55,38,76)(3,59,39,74)(4,52,7,66)(5,50,8,64)(6,54,9,62)(10,63,13,49)(11,61,14,53)(12,65,15,51)(16,68,44,89)(17,72,45,87)(18,70,43,85)(19,60,24,75)(20,58,22,73)(21,56,23,77)(25,94,34,79)(26,92,35,83)(27,96,36,81)(28,91,31,82)(29,95,32,80)(30,93,33,84)(40,67,47,88)(41,71,48,86)(42,69,46,90), (1,42,19,17)(2,40,20,18)(3,41,21,16)(4,25,13,28)(5,26,14,29)(6,27,15,30)(7,34,10,31)(8,35,11,32)(9,36,12,33)(22,43,38,47)(23,44,39,48)(24,45,37,46)(49,91,52,94)(50,92,53,95)(51,93,54,96)(55,67,58,70)(56,68,59,71)(57,69,60,72)(61,80,64,83)(62,81,65,84)(63,82,66,79)(73,85,76,88)(74,86,77,89)(75,87,78,90), (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), (1,14,24,8)(2,13,22,7)(3,15,23,9)(4,38,10,20)(5,37,11,19)(6,39,12,21)(16,30,48,36)(17,29,46,35)(18,28,47,34)(25,43,31,40)(26,45,32,42)(27,44,33,41)(49,58,66,76)(50,57,61,75)(51,56,62,74)(52,55,63,73)(53,60,64,78)(54,59,65,77)(67,79,85,91)(68,84,86,96)(69,83,87,95)(70,82,88,94)(71,81,89,93)(72,80,90,92)>;
G:=Group( (1,57,37,78)(2,55,38,76)(3,59,39,74)(4,52,7,66)(5,50,8,64)(6,54,9,62)(10,63,13,49)(11,61,14,53)(12,65,15,51)(16,68,44,89)(17,72,45,87)(18,70,43,85)(19,60,24,75)(20,58,22,73)(21,56,23,77)(25,94,34,79)(26,92,35,83)(27,96,36,81)(28,91,31,82)(29,95,32,80)(30,93,33,84)(40,67,47,88)(41,71,48,86)(42,69,46,90), (1,42,19,17)(2,40,20,18)(3,41,21,16)(4,25,13,28)(5,26,14,29)(6,27,15,30)(7,34,10,31)(8,35,11,32)(9,36,12,33)(22,43,38,47)(23,44,39,48)(24,45,37,46)(49,91,52,94)(50,92,53,95)(51,93,54,96)(55,67,58,70)(56,68,59,71)(57,69,60,72)(61,80,64,83)(62,81,65,84)(63,82,66,79)(73,85,76,88)(74,86,77,89)(75,87,78,90), (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), (1,14,24,8)(2,13,22,7)(3,15,23,9)(4,38,10,20)(5,37,11,19)(6,39,12,21)(16,30,48,36)(17,29,46,35)(18,28,47,34)(25,43,31,40)(26,45,32,42)(27,44,33,41)(49,58,66,76)(50,57,61,75)(51,56,62,74)(52,55,63,73)(53,60,64,78)(54,59,65,77)(67,79,85,91)(68,84,86,96)(69,83,87,95)(70,82,88,94)(71,81,89,93)(72,80,90,92) );
G=PermutationGroup([(1,57,37,78),(2,55,38,76),(3,59,39,74),(4,52,7,66),(5,50,8,64),(6,54,9,62),(10,63,13,49),(11,61,14,53),(12,65,15,51),(16,68,44,89),(17,72,45,87),(18,70,43,85),(19,60,24,75),(20,58,22,73),(21,56,23,77),(25,94,34,79),(26,92,35,83),(27,96,36,81),(28,91,31,82),(29,95,32,80),(30,93,33,84),(40,67,47,88),(41,71,48,86),(42,69,46,90)], [(1,42,19,17),(2,40,20,18),(3,41,21,16),(4,25,13,28),(5,26,14,29),(6,27,15,30),(7,34,10,31),(8,35,11,32),(9,36,12,33),(22,43,38,47),(23,44,39,48),(24,45,37,46),(49,91,52,94),(50,92,53,95),(51,93,54,96),(55,67,58,70),(56,68,59,71),(57,69,60,72),(61,80,64,83),(62,81,65,84),(63,82,66,79),(73,85,76,88),(74,86,77,89),(75,87,78,90)], [(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)], [(1,14,24,8),(2,13,22,7),(3,15,23,9),(4,38,10,20),(5,37,11,19),(6,39,12,21),(16,30,48,36),(17,29,46,35),(18,28,47,34),(25,43,31,40),(26,45,32,42),(27,44,33,41),(49,58,66,76),(50,57,61,75),(51,56,62,74),(52,55,63,73),(53,60,64,78),(54,59,65,77),(67,79,85,91),(68,84,86,96),(69,83,87,95),(70,82,88,94),(71,81,89,93),(72,80,90,92)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 3 | 7 | 0 | 0 | 0 | 0 |
| 6 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 1 |
| 0 | 0 | 0 | 8 | 1 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 12 | 8 | 0 |
| 0 | 0 | 12 | 0 | 0 | 8 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 8 | 1 | 0 |
| 0 | 0 | 5 | 0 | 0 | 12 |
| 8 | 5 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 11 | 0 |
| 0 | 0 | 8 | 0 | 0 | 2 |
| 0 | 0 | 1 | 0 | 0 | 5 |
| 0 | 0 | 0 | 12 | 8 | 0 |
G:=sub<GL(6,GF(13))| [3,6,0,0,0,0,7,10,0,0,0,0,0,0,0,12,8,0,0,0,12,0,0,8,0,0,0,0,0,1,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,12,0,0,0,5,12,0,0,0,0,0,8,0,0,0,0,0,0,8],[0,12,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,5,0,0,0,12,8,0,0,0,0,0,1,0,0,0,0,0,0,12],[8,0,0,0,0,0,5,5,0,0,0,0,0,0,0,8,1,0,0,0,5,0,0,12,0,0,11,0,0,8,0,0,0,2,5,0] >;
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | + | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | D6 | D6 | Dic6 | 2+ (1+4) | 2- (1+4) | D4○D12 | Q8○D12 |
| kernel | C42.90D6 | C12⋊2Q8 | C12.6Q8 | Dic3.D4 | C12⋊Q8 | C12.3Q8 | C2×C4⋊Dic3 | C23.26D6 | C3×C42⋊C2 | C42⋊C2 | C2×C12 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 2 | 4 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 1 | 8 | 1 | 1 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{90}D_6 % in TeX
G:=Group("C4^2.90D6"); // GroupNames label
G:=SmallGroup(192,1078);
// by ID
G=gap.SmallGroup(192,1078);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,758,184,675,570,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations