metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.94D6, C6.512- (1+4), C4⋊C4.271D6, C42⋊2S3⋊2C2, C4.D12⋊12C2, (C2×C6).73C24, C2.9(Q8○D12), C22⋊C4.97D6, C12.6Q8⋊6C2, C42⋊C2⋊13S3, D6.16(C4○D4), C4.95(C4○D12), (C4×C12).24C22, D6⋊C4.96C22, C12.3Q8⋊12C2, C23.9D6.1C2, C23.8D6⋊3C2, (C22×C4).210D6, Dic6⋊C4⋊12C2, C12.197(C4○D4), C12.48D4⋊29C2, (C2×C12).148C23, C4⋊Dic3.34C22, C23.95(C22×S3), Dic3⋊C4.97C22, (C22×C6).143C23, C22.102(S3×C23), (C4×Dic3).69C22, (C2×Dic3).26C23, (C22×S3).166C23, (C22×C12).231C22, C3⋊2(C22.46C24), (C2×Dic6).143C22, C6.D4.96C22, (S3×C4⋊C4)⋊12C2, C2.12(S3×C4○D4), C6.30(C2×C4○D4), (C4×C3⋊D4).5C2, C2.32(C2×C4○D12), (S3×C2×C4).60C22, (C3×C42⋊C2)⋊15C2, (C3×C4⋊C4).309C22, (C2×C4).275(C22×S3), (C2×C3⋊D4).103C22, (C3×C22⋊C4).113C22, SmallGroup(192,1088)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 472 in 214 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×2], C4 [×12], C22, C22 [×7], S3 [×2], C6 [×3], C6, C2×C4 [×2], C2×C4 [×4], C2×C4 [×15], D4 [×2], Q8 [×2], C23, C23, Dic3 [×7], C12 [×2], C12 [×5], D6 [×2], D6 [×2], C2×C6, C2×C6 [×3], C42 [×2], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×14], C22×C4, C22×C4 [×3], C2×D4, C2×Q8, Dic6 [×2], C4×S3 [×6], C2×Dic3 [×3], C2×Dic3 [×4], C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×4], C2×C12 [×2], C22×S3, C22×C6, C2×C4⋊C4, C42⋊C2, C42⋊C2 [×2], C4×D4, C4×Q8, C22⋊Q8 [×2], C22.D4 [×2], C42.C2 [×3], C42⋊2C2 [×2], C4×Dic3, C4×Dic3 [×2], Dic3⋊C4, Dic3⋊C4 [×8], C4⋊Dic3, C4⋊Dic3 [×4], D6⋊C4, D6⋊C4 [×2], C6.D4, C6.D4 [×2], C4×C12 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], C2×Dic6, S3×C2×C4, S3×C2×C4 [×2], C2×C3⋊D4, C22×C12, C22.46C24, C12.6Q8 [×2], C42⋊2S3 [×2], C23.8D6 [×2], C23.9D6 [×2], Dic6⋊C4, C12.3Q8, S3×C4⋊C4, C4.D12, C12.48D4, C4×C3⋊D4, C3×C42⋊C2, C42.94D6
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), C4○D12 [×2], S3×C23, C22.46C24, C2×C4○D12, S3×C4○D4, Q8○D12, C42.94D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=d2=b2, ab=ba, cac-1=dad-1=ab2, bc=cb, dbd-1=a2b, dcd-1=c5 >
►On 96 points
(1 82 17 85)(2 77 18 92)(3 84 19 87)(4 79 20 94)(5 74 21 89)(6 81 22 96)(7 76 23 91)(8 83 24 86)(9 78 13 93)(10 73 14 88)(11 80 15 95)(12 75 16 90)(25 68 57 38)(26 63 58 45)(27 70 59 40)(28 65 60 47)(29 72 49 42)(30 67 50 37)(31 62 51 44)(32 69 52 39)(33 64 53 46)(34 71 54 41)(35 66 55 48)(36 61 56 43)
(1 70 7 64)(2 71 8 65)(3 72 9 66)(4 61 10 67)(5 62 11 68)(6 63 12 69)(13 48 19 42)(14 37 20 43)(15 38 21 44)(16 39 22 45)(17 40 23 46)(18 41 24 47)(25 89 31 95)(26 90 32 96)(27 91 33 85)(28 92 34 86)(29 93 35 87)(30 94 36 88)(49 78 55 84)(50 79 56 73)(51 80 57 74)(52 81 58 75)(53 82 59 76)(54 83 60 77)
(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 12 7 6)(2 5 8 11)(3 10 9 4)(13 20 19 14)(15 18 21 24)(16 23 22 17)(25 60 31 54)(26 53 32 59)(27 58 33 52)(28 51 34 57)(29 56 35 50)(30 49 36 55)(37 66 43 72)(38 71 44 65)(39 64 45 70)(40 69 46 63)(41 62 47 68)(42 67 48 61)(73 84 79 78)(74 77 80 83)(75 82 81 76)(85 96 91 90)(86 89 92 95)(87 94 93 88)
G:=sub<Sym(96)| (1,82,17,85)(2,77,18,92)(3,84,19,87)(4,79,20,94)(5,74,21,89)(6,81,22,96)(7,76,23,91)(8,83,24,86)(9,78,13,93)(10,73,14,88)(11,80,15,95)(12,75,16,90)(25,68,57,38)(26,63,58,45)(27,70,59,40)(28,65,60,47)(29,72,49,42)(30,67,50,37)(31,62,51,44)(32,69,52,39)(33,64,53,46)(34,71,54,41)(35,66,55,48)(36,61,56,43), (1,70,7,64)(2,71,8,65)(3,72,9,66)(4,61,10,67)(5,62,11,68)(6,63,12,69)(13,48,19,42)(14,37,20,43)(15,38,21,44)(16,39,22,45)(17,40,23,46)(18,41,24,47)(25,89,31,95)(26,90,32,96)(27,91,33,85)(28,92,34,86)(29,93,35,87)(30,94,36,88)(49,78,55,84)(50,79,56,73)(51,80,57,74)(52,81,58,75)(53,82,59,76)(54,83,60,77), (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), (1,12,7,6)(2,5,8,11)(3,10,9,4)(13,20,19,14)(15,18,21,24)(16,23,22,17)(25,60,31,54)(26,53,32,59)(27,58,33,52)(28,51,34,57)(29,56,35,50)(30,49,36,55)(37,66,43,72)(38,71,44,65)(39,64,45,70)(40,69,46,63)(41,62,47,68)(42,67,48,61)(73,84,79,78)(74,77,80,83)(75,82,81,76)(85,96,91,90)(86,89,92,95)(87,94,93,88)>;
G:=Group( (1,82,17,85)(2,77,18,92)(3,84,19,87)(4,79,20,94)(5,74,21,89)(6,81,22,96)(7,76,23,91)(8,83,24,86)(9,78,13,93)(10,73,14,88)(11,80,15,95)(12,75,16,90)(25,68,57,38)(26,63,58,45)(27,70,59,40)(28,65,60,47)(29,72,49,42)(30,67,50,37)(31,62,51,44)(32,69,52,39)(33,64,53,46)(34,71,54,41)(35,66,55,48)(36,61,56,43), (1,70,7,64)(2,71,8,65)(3,72,9,66)(4,61,10,67)(5,62,11,68)(6,63,12,69)(13,48,19,42)(14,37,20,43)(15,38,21,44)(16,39,22,45)(17,40,23,46)(18,41,24,47)(25,89,31,95)(26,90,32,96)(27,91,33,85)(28,92,34,86)(29,93,35,87)(30,94,36,88)(49,78,55,84)(50,79,56,73)(51,80,57,74)(52,81,58,75)(53,82,59,76)(54,83,60,77), (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), (1,12,7,6)(2,5,8,11)(3,10,9,4)(13,20,19,14)(15,18,21,24)(16,23,22,17)(25,60,31,54)(26,53,32,59)(27,58,33,52)(28,51,34,57)(29,56,35,50)(30,49,36,55)(37,66,43,72)(38,71,44,65)(39,64,45,70)(40,69,46,63)(41,62,47,68)(42,67,48,61)(73,84,79,78)(74,77,80,83)(75,82,81,76)(85,96,91,90)(86,89,92,95)(87,94,93,88) );
G=PermutationGroup([(1,82,17,85),(2,77,18,92),(3,84,19,87),(4,79,20,94),(5,74,21,89),(6,81,22,96),(7,76,23,91),(8,83,24,86),(9,78,13,93),(10,73,14,88),(11,80,15,95),(12,75,16,90),(25,68,57,38),(26,63,58,45),(27,70,59,40),(28,65,60,47),(29,72,49,42),(30,67,50,37),(31,62,51,44),(32,69,52,39),(33,64,53,46),(34,71,54,41),(35,66,55,48),(36,61,56,43)], [(1,70,7,64),(2,71,8,65),(3,72,9,66),(4,61,10,67),(5,62,11,68),(6,63,12,69),(13,48,19,42),(14,37,20,43),(15,38,21,44),(16,39,22,45),(17,40,23,46),(18,41,24,47),(25,89,31,95),(26,90,32,96),(27,91,33,85),(28,92,34,86),(29,93,35,87),(30,94,36,88),(49,78,55,84),(50,79,56,73),(51,80,57,74),(52,81,58,75),(53,82,59,76),(54,83,60,77)], [(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)], [(1,12,7,6),(2,5,8,11),(3,10,9,4),(13,20,19,14),(15,18,21,24),(16,23,22,17),(25,60,31,54),(26,53,32,59),(27,58,33,52),(28,51,34,57),(29,56,35,50),(30,49,36,55),(37,66,43,72),(38,71,44,65),(39,64,45,70),(40,69,46,63),(41,62,47,68),(42,67,48,61),(73,84,79,78),(74,77,80,83),(75,82,81,76),(85,96,91,90),(86,89,92,95),(87,94,93,88)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 5 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 12 |
| 11 | 9 | 0 | 0 |
| 4 | 2 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 12 | 12 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 12 | 5 |
| 12 | 12 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 1 | 8 |
G:=sub<GL(4,GF(13))| [5,0,0,0,0,5,0,0,0,0,1,0,0,0,3,12],[11,4,0,0,9,2,0,0,0,0,5,0,0,0,0,5],[12,1,0,0,12,0,0,0,0,0,8,12,0,0,0,5],[12,0,0,0,12,1,0,0,0,0,5,1,0,0,0,8] >;
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 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 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 2 | 2 | ··· | 2 | 4 | 4 | 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 | 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 | C4○D4 | C4○D4 | C4○D12 | 2- (1+4) | S3×C4○D4 | Q8○D12 |
| kernel | C42.94D6 | C12.6Q8 | C42⋊2S3 | C23.8D6 | C23.9D6 | Dic6⋊C4 | C12.3Q8 | S3×C4⋊C4 | C4.D12 | C12.48D4 | C4×C3⋊D4 | C3×C42⋊C2 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C12 | D6 | C4 | C6 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 8 | 1 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{94}D_6 % in TeX
G:=Group("C4^2.94D6"); // GroupNames label
G:=SmallGroup(192,1088);
// by ID
G=gap.SmallGroup(192,1088);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,100,675,136,6278]);
// 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=d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^5>;
// generators/relations