metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.162D6, C6.1012- (1+4), C4⋊C4.213D6, C42⋊2S3⋊7C2, C42⋊2C2⋊5S3, D6⋊Q8⋊44C2, (C4×Dic6)⋊15C2, C22⋊C4.80D6, D6.27(C4○D4), Dic3.Q8⋊38C2, (C4×C12).34C22, (C2×C6).252C24, C2.65(Q8○D12), C12.3Q8⋊40C2, C23.9D6.4C2, (C2×C12).604C23, D6⋊C4.140C22, Dic3⋊4D4.5C2, C23.8D6⋊46C2, C23.68(C22×S3), (C22×C6).66C23, Dic3.32(C4○D4), Dic3.D4⋊46C2, C23.16D6⋊22C2, C4⋊Dic3.247C22, C22.273(S3×C23), Dic3⋊C4.146C22, (C22×S3).226C23, (C4×Dic3).218C22, (C2×Dic3).130C23, (C2×Dic6).255C22, C6.D4.68C22, C3⋊11(C22.46C24), (C22×Dic3).152C22, (S3×C4⋊C4)⋊42C2, C4⋊C4⋊7S3⋊41C2, C2.99(S3×C4○D4), C6.210(C2×C4○D4), (C3×C42⋊2C2)⋊7C2, (S3×C2×C4).220C22, (C2×C4).88(C22×S3), (C3×C4⋊C4).204C22, (C2×C3⋊D4).72C22, (C3×C22⋊C4).77C22, SmallGroup(192,1267)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 480 in 214 conjugacy classes, 95 normal (91 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×14], C22, C22 [×7], S3 [×2], C6 [×3], C6, C2×C4 [×6], C2×C4 [×15], D4 [×2], Q8 [×2], C23, C23, Dic3 [×2], Dic3 [×6], C12 [×6], D6 [×2], D6 [×2], C2×C6, C2×C6 [×3], C42, C42 [×4], C22⋊C4 [×3], C22⋊C4 [×5], C4⋊C4 [×3], C4⋊C4 [×13], C22×C4 [×4], C2×D4, C2×Q8, Dic6 [×2], C4×S3 [×6], C2×Dic3 [×7], C2×Dic3 [×2], C3⋊D4 [×2], C2×C12 [×6], C22×S3, C22×C6, C2×C4⋊C4, C42⋊C2 [×3], C4×D4, C4×Q8, C22⋊Q8 [×2], C22.D4 [×2], C42.C2 [×3], C42⋊2C2, C42⋊2C2, C4×Dic3 [×4], Dic3⋊C4 [×9], C4⋊Dic3 [×4], D6⋊C4 [×3], C6.D4 [×2], C4×C12, C3×C22⋊C4 [×3], C3×C4⋊C4 [×3], C2×Dic6, S3×C2×C4 [×3], C22×Dic3, C2×C3⋊D4, C22.46C24, C4×Dic6, C42⋊2S3, C23.16D6, Dic3.D4, C23.8D6, Dic3⋊4D4, C23.9D6 [×2], Dic3.Q8 [×2], C12.3Q8, S3×C4⋊C4, C4⋊C4⋊7S3, D6⋊Q8, C3×C42⋊2C2, C42.162D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×4], C24, C22×S3 [×7], C2×C4○D4 [×2], 2- (1+4), S3×C23, C22.46C24, S3×C4○D4 [×2], Q8○D12, C42.162D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=d2=b2, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b-1, dbd-1=b-1, dcd-1=c5 >
►On 96 points
(1 85 73 29)(2 92 74 36)(3 87 75 31)(4 94 76 26)(5 89 77 33)(6 96 78 28)(7 91 79 35)(8 86 80 30)(9 93 81 25)(10 88 82 32)(11 95 83 27)(12 90 84 34)(13 70 54 38)(14 65 55 45)(15 72 56 40)(16 67 57 47)(17 62 58 42)(18 69 59 37)(19 64 60 44)(20 71 49 39)(21 66 50 46)(22 61 51 41)(23 68 52 48)(24 63 53 43)
(1 61 7 67)(2 48 8 42)(3 63 9 69)(4 38 10 44)(5 65 11 71)(6 40 12 46)(13 88 19 94)(14 27 20 33)(15 90 21 96)(16 29 22 35)(17 92 23 86)(18 31 24 25)(26 54 32 60)(28 56 34 50)(30 58 36 52)(37 75 43 81)(39 77 45 83)(41 79 47 73)(49 89 55 95)(51 91 57 85)(53 93 59 87)(62 74 68 80)(64 76 70 82)(66 78 72 84)
(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 59 19 53)(14 52 20 58)(15 57 21 51)(16 50 22 56)(17 55 23 49)(18 60 24 54)(25 26 31 32)(27 36 33 30)(28 29 34 35)(37 70 43 64)(38 63 44 69)(39 68 45 62)(40 61 46 67)(41 66 47 72)(42 71 48 65)(73 78 79 84)(74 83 80 77)(75 76 81 82)(85 90 91 96)(86 95 92 89)(87 88 93 94)
G:=sub<Sym(96)| (1,85,73,29)(2,92,74,36)(3,87,75,31)(4,94,76,26)(5,89,77,33)(6,96,78,28)(7,91,79,35)(8,86,80,30)(9,93,81,25)(10,88,82,32)(11,95,83,27)(12,90,84,34)(13,70,54,38)(14,65,55,45)(15,72,56,40)(16,67,57,47)(17,62,58,42)(18,69,59,37)(19,64,60,44)(20,71,49,39)(21,66,50,46)(22,61,51,41)(23,68,52,48)(24,63,53,43), (1,61,7,67)(2,48,8,42)(3,63,9,69)(4,38,10,44)(5,65,11,71)(6,40,12,46)(13,88,19,94)(14,27,20,33)(15,90,21,96)(16,29,22,35)(17,92,23,86)(18,31,24,25)(26,54,32,60)(28,56,34,50)(30,58,36,52)(37,75,43,81)(39,77,45,83)(41,79,47,73)(49,89,55,95)(51,91,57,85)(53,93,59,87)(62,74,68,80)(64,76,70,82)(66,78,72,84), (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)(13,59,19,53)(14,52,20,58)(15,57,21,51)(16,50,22,56)(17,55,23,49)(18,60,24,54)(25,26,31,32)(27,36,33,30)(28,29,34,35)(37,70,43,64)(38,63,44,69)(39,68,45,62)(40,61,46,67)(41,66,47,72)(42,71,48,65)(73,78,79,84)(74,83,80,77)(75,76,81,82)(85,90,91,96)(86,95,92,89)(87,88,93,94)>;
G:=Group( (1,85,73,29)(2,92,74,36)(3,87,75,31)(4,94,76,26)(5,89,77,33)(6,96,78,28)(7,91,79,35)(8,86,80,30)(9,93,81,25)(10,88,82,32)(11,95,83,27)(12,90,84,34)(13,70,54,38)(14,65,55,45)(15,72,56,40)(16,67,57,47)(17,62,58,42)(18,69,59,37)(19,64,60,44)(20,71,49,39)(21,66,50,46)(22,61,51,41)(23,68,52,48)(24,63,53,43), (1,61,7,67)(2,48,8,42)(3,63,9,69)(4,38,10,44)(5,65,11,71)(6,40,12,46)(13,88,19,94)(14,27,20,33)(15,90,21,96)(16,29,22,35)(17,92,23,86)(18,31,24,25)(26,54,32,60)(28,56,34,50)(30,58,36,52)(37,75,43,81)(39,77,45,83)(41,79,47,73)(49,89,55,95)(51,91,57,85)(53,93,59,87)(62,74,68,80)(64,76,70,82)(66,78,72,84), (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)(13,59,19,53)(14,52,20,58)(15,57,21,51)(16,50,22,56)(17,55,23,49)(18,60,24,54)(25,26,31,32)(27,36,33,30)(28,29,34,35)(37,70,43,64)(38,63,44,69)(39,68,45,62)(40,61,46,67)(41,66,47,72)(42,71,48,65)(73,78,79,84)(74,83,80,77)(75,76,81,82)(85,90,91,96)(86,95,92,89)(87,88,93,94) );
G=PermutationGroup([(1,85,73,29),(2,92,74,36),(3,87,75,31),(4,94,76,26),(5,89,77,33),(6,96,78,28),(7,91,79,35),(8,86,80,30),(9,93,81,25),(10,88,82,32),(11,95,83,27),(12,90,84,34),(13,70,54,38),(14,65,55,45),(15,72,56,40),(16,67,57,47),(17,62,58,42),(18,69,59,37),(19,64,60,44),(20,71,49,39),(21,66,50,46),(22,61,51,41),(23,68,52,48),(24,63,53,43)], [(1,61,7,67),(2,48,8,42),(3,63,9,69),(4,38,10,44),(5,65,11,71),(6,40,12,46),(13,88,19,94),(14,27,20,33),(15,90,21,96),(16,29,22,35),(17,92,23,86),(18,31,24,25),(26,54,32,60),(28,56,34,50),(30,58,36,52),(37,75,43,81),(39,77,45,83),(41,79,47,73),(49,89,55,95),(51,91,57,85),(53,93,59,87),(62,74,68,80),(64,76,70,82),(66,78,72,84)], [(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),(13,59,19,53),(14,52,20,58),(15,57,21,51),(16,50,22,56),(17,55,23,49),(18,60,24,54),(25,26,31,32),(27,36,33,30),(28,29,34,35),(37,70,43,64),(38,63,44,69),(39,68,45,62),(40,61,46,67),(41,66,47,72),(42,71,48,65),(73,78,79,84),(74,83,80,77),(75,76,81,82),(85,90,91,96),(86,95,92,89),(87,88,93,94)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 0 | 5 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(13))| [0,8,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[5,0,0,0,0,0,0,8,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1],[5,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 4O | 4P | 4Q | 4R | 6A | 6B | 6C | 6D | 12A | ··· | 12F | 12G | 12H | 12I |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 8 | 4 | ··· | 4 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | S3 | D6 | D6 | D6 | C4○D4 | C4○D4 | 2- (1+4) | S3×C4○D4 | Q8○D12 |
| kernel | C42.162D6 | C4×Dic6 | C42⋊2S3 | C23.16D6 | Dic3.D4 | C23.8D6 | Dic3⋊4D4 | C23.9D6 | Dic3.Q8 | C12.3Q8 | S3×C4⋊C4 | C4⋊C4⋊7S3 | D6⋊Q8 | C3×C42⋊2C2 | C42⋊2C2 | C42 | C22⋊C4 | C4⋊C4 | Dic3 | D6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 4 | 4 | 1 | 4 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{162}D_6 % in TeX
G:=Group("C4^2.162D6"); // GroupNames label
G:=SmallGroup(192,1267);
// by ID
G=gap.SmallGroup(192,1267);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,268,1571,570,80,6278]);
// 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=d*a*d^-1=a*b^2,c*b*c^-1=a^2*b^-1,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations