metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.141D6, C6.902- (1+4), C4.33(S3×D4), (C4×S3).12D4, D6.46(C2×D4), C12.62(C2×D4), C4.4D4⋊9S3, C12⋊2Q8⋊30C2, (C2×D4).172D6, (C2×Q8).160D6, C22⋊C4.35D6, C6.89(C22×D4), C42⋊2S3⋊20C2, C23.9D6⋊41C2, (C2×C12).80C23, (C2×C6).219C24, C2.51(Q8○D12), Dic3.51(C2×D4), Dic3⋊Q8⋊24C2, C23.12D6⋊24C2, (C4×C12).185C22, D6⋊C4.110C22, (C6×D4).154C22, C4⋊Dic3.51C22, C23.51(C22×S3), (C22×C6).49C23, (C6×Q8).126C22, Dic3.D4⋊40C2, C22.240(S3×C23), Dic3⋊C4.120C22, (C22×S3).214C23, C3⋊4(C23.38C23), (C2×Dic6).177C22, (C2×Dic3).114C23, (C4×Dic3).133C22, C6.D4.54C22, (C22×Dic3).142C22, (C2×S3×Q8)⋊10C2, C2.62(C2×S3×D4), (C2×D4⋊2S3).9C2, (C3×C4.4D4)⋊11C2, (S3×C2×C4).120C22, (C2×C4).194(C22×S3), (C2×C3⋊D4).59C22, (C3×C22⋊C4).64C22, SmallGroup(192,1234)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 656 in 270 conjugacy classes, 103 normal (29 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×12], C22, C22 [×10], S3 [×2], C6, C6 [×2], C6 [×2], C2×C4, C2×C4 [×4], C2×C4 [×19], D4 [×6], Q8 [×10], C23 [×2], C23, Dic3 [×2], Dic3 [×6], C12 [×2], C12 [×4], D6 [×2], D6 [×2], C2×C6, C2×C6 [×6], C42, C42, C22⋊C4 [×4], C22⋊C4 [×6], C4⋊C4 [×10], C22×C4 [×5], C2×D4, C2×D4 [×2], C2×Q8, C2×Q8 [×8], C4○D4 [×4], Dic6 [×8], C4×S3 [×4], C4×S3 [×4], C2×Dic3, C2×Dic3 [×6], C2×Dic3 [×4], C3⋊D4 [×4], C2×C12, C2×C12 [×4], C3×D4 [×2], C3×Q8 [×2], C22×S3, C22×C6 [×2], C42⋊C2, C22⋊Q8 [×4], C22.D4 [×4], C4.4D4, C4.4D4, C4⋊Q8 [×2], C22×Q8, C2×C4○D4, C4×Dic3, Dic3⋊C4 [×6], C4⋊Dic3 [×4], D6⋊C4 [×2], C6.D4 [×4], C4×C12, C3×C22⋊C4 [×4], C2×Dic6 [×2], C2×Dic6 [×2], S3×C2×C4, S3×C2×C4 [×2], D4⋊2S3 [×4], S3×Q8 [×4], C22×Dic3 [×2], C2×C3⋊D4 [×2], C6×D4, C6×Q8, C23.38C23, C12⋊2Q8, C42⋊2S3, Dic3.D4 [×4], C23.9D6 [×4], C23.12D6, Dic3⋊Q8, C3×C4.4D4, C2×D4⋊2S3, C2×S3×Q8, C42.141D6
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C22×S3 [×7], C22×D4, 2- (1+4) [×2], S3×D4 [×2], S3×C23, C23.38C23, C2×S3×D4, Q8○D12 [×2], C42.141D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=d2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, dbd-1=b-1, dcd-1=c5 >
►On 96 points
(1 91 7 85)(2 86 8 92)(3 93 9 87)(4 88 10 94)(5 95 11 89)(6 90 12 96)(13 51 19 57)(14 58 20 52)(15 53 21 59)(16 60 22 54)(17 55 23 49)(18 50 24 56)(25 61 31 67)(26 68 32 62)(27 63 33 69)(28 70 34 64)(29 65 35 71)(30 72 36 66)(37 78 43 84)(38 73 44 79)(39 80 45 74)(40 75 46 81)(41 82 47 76)(42 77 48 83)
(1 60 63 47)(2 42 64 55)(3 50 65 37)(4 44 66 57)(5 52 67 39)(6 46 68 59)(7 54 69 41)(8 48 70 49)(9 56 71 43)(10 38 72 51)(11 58 61 45)(12 40 62 53)(13 88 79 30)(14 25 80 95)(15 90 81 32)(16 27 82 85)(17 92 83 34)(18 29 84 87)(19 94 73 36)(20 31 74 89)(21 96 75 26)(22 33 76 91)(23 86 77 28)(24 35 78 93)
(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 24 19 18)(14 17 20 23)(15 22 21 16)(25 34 31 28)(26 27 32 33)(29 30 35 36)(37 44 43 38)(39 42 45 48)(40 47 46 41)(49 52 55 58)(50 57 56 51)(53 60 59 54)(61 70 67 64)(62 63 68 69)(65 66 71 72)(73 84 79 78)(74 77 80 83)(75 82 81 76)(85 90 91 96)(86 95 92 89)(87 88 93 94)
G:=sub<Sym(96)| (1,91,7,85)(2,86,8,92)(3,93,9,87)(4,88,10,94)(5,95,11,89)(6,90,12,96)(13,51,19,57)(14,58,20,52)(15,53,21,59)(16,60,22,54)(17,55,23,49)(18,50,24,56)(25,61,31,67)(26,68,32,62)(27,63,33,69)(28,70,34,64)(29,65,35,71)(30,72,36,66)(37,78,43,84)(38,73,44,79)(39,80,45,74)(40,75,46,81)(41,82,47,76)(42,77,48,83), (1,60,63,47)(2,42,64,55)(3,50,65,37)(4,44,66,57)(5,52,67,39)(6,46,68,59)(7,54,69,41)(8,48,70,49)(9,56,71,43)(10,38,72,51)(11,58,61,45)(12,40,62,53)(13,88,79,30)(14,25,80,95)(15,90,81,32)(16,27,82,85)(17,92,83,34)(18,29,84,87)(19,94,73,36)(20,31,74,89)(21,96,75,26)(22,33,76,91)(23,86,77,28)(24,35,78,93), (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)(13,24,19,18)(14,17,20,23)(15,22,21,16)(25,34,31,28)(26,27,32,33)(29,30,35,36)(37,44,43,38)(39,42,45,48)(40,47,46,41)(49,52,55,58)(50,57,56,51)(53,60,59,54)(61,70,67,64)(62,63,68,69)(65,66,71,72)(73,84,79,78)(74,77,80,83)(75,82,81,76)(85,90,91,96)(86,95,92,89)(87,88,93,94)>;
G:=Group( (1,91,7,85)(2,86,8,92)(3,93,9,87)(4,88,10,94)(5,95,11,89)(6,90,12,96)(13,51,19,57)(14,58,20,52)(15,53,21,59)(16,60,22,54)(17,55,23,49)(18,50,24,56)(25,61,31,67)(26,68,32,62)(27,63,33,69)(28,70,34,64)(29,65,35,71)(30,72,36,66)(37,78,43,84)(38,73,44,79)(39,80,45,74)(40,75,46,81)(41,82,47,76)(42,77,48,83), (1,60,63,47)(2,42,64,55)(3,50,65,37)(4,44,66,57)(5,52,67,39)(6,46,68,59)(7,54,69,41)(8,48,70,49)(9,56,71,43)(10,38,72,51)(11,58,61,45)(12,40,62,53)(13,88,79,30)(14,25,80,95)(15,90,81,32)(16,27,82,85)(17,92,83,34)(18,29,84,87)(19,94,73,36)(20,31,74,89)(21,96,75,26)(22,33,76,91)(23,86,77,28)(24,35,78,93), (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)(13,24,19,18)(14,17,20,23)(15,22,21,16)(25,34,31,28)(26,27,32,33)(29,30,35,36)(37,44,43,38)(39,42,45,48)(40,47,46,41)(49,52,55,58)(50,57,56,51)(53,60,59,54)(61,70,67,64)(62,63,68,69)(65,66,71,72)(73,84,79,78)(74,77,80,83)(75,82,81,76)(85,90,91,96)(86,95,92,89)(87,88,93,94) );
G=PermutationGroup([(1,91,7,85),(2,86,8,92),(3,93,9,87),(4,88,10,94),(5,95,11,89),(6,90,12,96),(13,51,19,57),(14,58,20,52),(15,53,21,59),(16,60,22,54),(17,55,23,49),(18,50,24,56),(25,61,31,67),(26,68,32,62),(27,63,33,69),(28,70,34,64),(29,65,35,71),(30,72,36,66),(37,78,43,84),(38,73,44,79),(39,80,45,74),(40,75,46,81),(41,82,47,76),(42,77,48,83)], [(1,60,63,47),(2,42,64,55),(3,50,65,37),(4,44,66,57),(5,52,67,39),(6,46,68,59),(7,54,69,41),(8,48,70,49),(9,56,71,43),(10,38,72,51),(11,58,61,45),(12,40,62,53),(13,88,79,30),(14,25,80,95),(15,90,81,32),(16,27,82,85),(17,92,83,34),(18,29,84,87),(19,94,73,36),(20,31,74,89),(21,96,75,26),(22,33,76,91),(23,86,77,28),(24,35,78,93)], [(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),(13,24,19,18),(14,17,20,23),(15,22,21,16),(25,34,31,28),(26,27,32,33),(29,30,35,36),(37,44,43,38),(39,42,45,48),(40,47,46,41),(49,52,55,58),(50,57,56,51),(53,60,59,54),(61,70,67,64),(62,63,68,69),(65,66,71,72),(73,84,79,78),(74,77,80,83),(75,82,81,76),(85,90,91,96),(86,95,92,89),(87,88,93,94)])
Matrix representation ►G ⊆ GL8(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 2 | 2 |
| 0 | 0 | 0 | 0 | 0 | 12 | 9 | 2 |
| 0 | 0 | 0 | 0 | 4 | 9 | 1 | 0 |
| 0 | 0 | 0 | 0 | 8 | 4 | 0 | 1 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 10 | 6 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 3 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 | 5 | 3 |
| 0 | 0 | 0 | 0 | 1 | 0 | 5 | 5 |
| 0 | 0 | 0 | 0 | 2 | 1 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 2 | 12 | 1 |
| 1 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 | 5 | 3 |
| 0 | 0 | 0 | 0 | 0 | 1 | 3 | 5 |
| 0 | 0 | 0 | 0 | 1 | 2 | 12 | 1 |
| 0 | 0 | 0 | 0 | 2 | 1 | 0 | 1 |
G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,4,8,0,0,0,0,0,12,9,4,0,0,0,0,2,9,1,0,0,0,0,0,2,2,0,1],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,3,7,0,0,0,0,0,0,6,10,0,0,0,0,0,0,0,0,10,7,0,0,0,0,0,0,6,3],[0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,12,1,2,1,0,0,0,0,12,0,1,2,0,0,0,0,5,5,0,12,0,0,0,0,3,5,1,1],[1,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,12,0,1,2,0,0,0,0,12,1,2,1,0,0,0,0,5,3,12,0,0,0,0,0,3,5,1,1] >;
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 6A | 6B | 6C | 6D | 6E | 12A | ··· | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 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 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 8 | 8 | 4 | ··· | 4 | 8 | 8 |
36 irreducible representations
| dim | 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 | S3 | D4 | D6 | D6 | D6 | D6 | 2- (1+4) | S3×D4 | Q8○D12 |
| kernel | C42.141D6 | C12⋊2Q8 | C42⋊2S3 | Dic3.D4 | C23.9D6 | C23.12D6 | Dic3⋊Q8 | C3×C4.4D4 | C2×D4⋊2S3 | C2×S3×Q8 | C4.4D4 | C4×S3 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 4 | 1 | 1 | 2 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{141}D_6 % in TeX
G:=Group("C4^2.141D6"); // GroupNames label
G:=SmallGroup(192,1234);
// by ID
G=gap.SmallGroup(192,1234);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,100,675,297,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations