metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.151D6, C6.292- (1+4), C4⋊C4.112D6, C42.C2⋊7S3, D6⋊Q8⋊36C2, C42⋊2S3⋊37C2, D6.24(C4○D4), Dic3.Q8⋊34C2, (C2×C12).89C23, (C2×C6).237C24, D6.D4.2C2, Dic6⋊C4⋊36C2, Dic3⋊5D4.12C2, (C4×C12).240C22, D6⋊C4.137C22, Dic3.30(C4○D4), (C2×D12).165C22, Dic3⋊C4.53C22, C4⋊Dic3.242C22, C22.258(S3×C23), (C22×S3).222C23, C2.30(Q8.15D6), C3⋊9(C22.46C24), (C2×Dic3).259C23, (C2×Dic6).181C22, (C4×Dic3).215C22, (S3×C4⋊C4)⋊37C2, C4⋊C4⋊S3⋊35C2, C4⋊C4⋊7S3⋊36C2, C2.88(S3×C4○D4), C6.199(C2×C4○D4), (S3×C2×C4).127C22, (C3×C42.C2)⋊10C2, (C3×C4⋊C4).192C22, (C2×C4).204(C22×S3), SmallGroup(192,1252)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 496 in 214 conjugacy classes, 95 normal (91 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×14], C22, C22 [×7], S3 [×3], C6 [×3], C2×C4 [×7], C2×C4 [×14], D4 [×2], Q8 [×2], C23 [×2], Dic3 [×2], Dic3 [×5], C12 [×7], D6 [×2], D6 [×5], C2×C6, C42, C42 [×4], C22⋊C4 [×8], C4⋊C4 [×6], C4⋊C4 [×10], C22×C4 [×4], C2×D4, C2×Q8, Dic6 [×2], C4×S3 [×8], D12 [×2], C2×Dic3 [×6], C2×C12 [×7], C22×S3 [×2], C2×C4⋊C4, C42⋊C2 [×3], C4×D4, C4×Q8, C22⋊Q8 [×2], C22.D4 [×2], C42.C2, C42.C2 [×2], C42⋊2C2 [×2], C4×Dic3 [×4], Dic3⋊C4 [×8], C4⋊Dic3 [×2], D6⋊C4 [×8], C4×C12, C3×C4⋊C4 [×6], C2×Dic6, S3×C2×C4 [×4], C2×D12, C22.46C24, C42⋊2S3 [×2], Dic6⋊C4, Dic3.Q8 [×2], S3×C4⋊C4, C4⋊C4⋊7S3, Dic3⋊5D4, D6.D4 [×2], D6⋊Q8 [×2], C4⋊C4⋊S3 [×2], C3×C42.C2, C42.151D6
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, Q8.15D6, S3×C4○D4 [×2], C42.151D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=d2=a2, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b, bd=db, dcd-1=c5 >
►On 96 points
(1 74 7 80)(2 50 8 56)(3 76 9 82)(4 52 10 58)(5 78 11 84)(6 54 12 60)(13 73 19 79)(14 49 20 55)(15 75 21 81)(16 51 22 57)(17 77 23 83)(18 53 24 59)(25 43 31 37)(26 65 32 71)(27 45 33 39)(28 67 34 61)(29 47 35 41)(30 69 36 63)(38 88 44 94)(40 90 46 96)(42 92 48 86)(62 91 68 85)(64 93 70 87)(66 95 72 89)
(1 70 14 37)(2 65 15 44)(3 72 16 39)(4 67 17 46)(5 62 18 41)(6 69 19 48)(7 64 20 43)(8 71 21 38)(9 66 22 45)(10 61 23 40)(11 68 24 47)(12 63 13 42)(25 74 87 49)(26 81 88 56)(27 76 89 51)(28 83 90 58)(29 78 91 53)(30 73 92 60)(31 80 93 55)(32 75 94 50)(33 82 95 57)(34 77 96 52)(35 84 85 59)(36 79 86 54)
(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 95 7 89)(2 88 8 94)(3 93 9 87)(4 86 10 92)(5 91 11 85)(6 96 12 90)(13 28 19 34)(14 33 20 27)(15 26 21 32)(16 31 22 25)(17 36 23 30)(18 29 24 35)(37 82 43 76)(38 75 44 81)(39 80 45 74)(40 73 46 79)(41 78 47 84)(42 83 48 77)(49 72 55 66)(50 65 56 71)(51 70 57 64)(52 63 58 69)(53 68 59 62)(54 61 60 67)
G:=sub<Sym(96)| (1,74,7,80)(2,50,8,56)(3,76,9,82)(4,52,10,58)(5,78,11,84)(6,54,12,60)(13,73,19,79)(14,49,20,55)(15,75,21,81)(16,51,22,57)(17,77,23,83)(18,53,24,59)(25,43,31,37)(26,65,32,71)(27,45,33,39)(28,67,34,61)(29,47,35,41)(30,69,36,63)(38,88,44,94)(40,90,46,96)(42,92,48,86)(62,91,68,85)(64,93,70,87)(66,95,72,89), (1,70,14,37)(2,65,15,44)(3,72,16,39)(4,67,17,46)(5,62,18,41)(6,69,19,48)(7,64,20,43)(8,71,21,38)(9,66,22,45)(10,61,23,40)(11,68,24,47)(12,63,13,42)(25,74,87,49)(26,81,88,56)(27,76,89,51)(28,83,90,58)(29,78,91,53)(30,73,92,60)(31,80,93,55)(32,75,94,50)(33,82,95,57)(34,77,96,52)(35,84,85,59)(36,79,86,54), (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), (1,95,7,89)(2,88,8,94)(3,93,9,87)(4,86,10,92)(5,91,11,85)(6,96,12,90)(13,28,19,34)(14,33,20,27)(15,26,21,32)(16,31,22,25)(17,36,23,30)(18,29,24,35)(37,82,43,76)(38,75,44,81)(39,80,45,74)(40,73,46,79)(41,78,47,84)(42,83,48,77)(49,72,55,66)(50,65,56,71)(51,70,57,64)(52,63,58,69)(53,68,59,62)(54,61,60,67)>;
G:=Group( (1,74,7,80)(2,50,8,56)(3,76,9,82)(4,52,10,58)(5,78,11,84)(6,54,12,60)(13,73,19,79)(14,49,20,55)(15,75,21,81)(16,51,22,57)(17,77,23,83)(18,53,24,59)(25,43,31,37)(26,65,32,71)(27,45,33,39)(28,67,34,61)(29,47,35,41)(30,69,36,63)(38,88,44,94)(40,90,46,96)(42,92,48,86)(62,91,68,85)(64,93,70,87)(66,95,72,89), (1,70,14,37)(2,65,15,44)(3,72,16,39)(4,67,17,46)(5,62,18,41)(6,69,19,48)(7,64,20,43)(8,71,21,38)(9,66,22,45)(10,61,23,40)(11,68,24,47)(12,63,13,42)(25,74,87,49)(26,81,88,56)(27,76,89,51)(28,83,90,58)(29,78,91,53)(30,73,92,60)(31,80,93,55)(32,75,94,50)(33,82,95,57)(34,77,96,52)(35,84,85,59)(36,79,86,54), (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), (1,95,7,89)(2,88,8,94)(3,93,9,87)(4,86,10,92)(5,91,11,85)(6,96,12,90)(13,28,19,34)(14,33,20,27)(15,26,21,32)(16,31,22,25)(17,36,23,30)(18,29,24,35)(37,82,43,76)(38,75,44,81)(39,80,45,74)(40,73,46,79)(41,78,47,84)(42,83,48,77)(49,72,55,66)(50,65,56,71)(51,70,57,64)(52,63,58,69)(53,68,59,62)(54,61,60,67) );
G=PermutationGroup([(1,74,7,80),(2,50,8,56),(3,76,9,82),(4,52,10,58),(5,78,11,84),(6,54,12,60),(13,73,19,79),(14,49,20,55),(15,75,21,81),(16,51,22,57),(17,77,23,83),(18,53,24,59),(25,43,31,37),(26,65,32,71),(27,45,33,39),(28,67,34,61),(29,47,35,41),(30,69,36,63),(38,88,44,94),(40,90,46,96),(42,92,48,86),(62,91,68,85),(64,93,70,87),(66,95,72,89)], [(1,70,14,37),(2,65,15,44),(3,72,16,39),(4,67,17,46),(5,62,18,41),(6,69,19,48),(7,64,20,43),(8,71,21,38),(9,66,22,45),(10,61,23,40),(11,68,24,47),(12,63,13,42),(25,74,87,49),(26,81,88,56),(27,76,89,51),(28,83,90,58),(29,78,91,53),(30,73,92,60),(31,80,93,55),(32,75,94,50),(33,82,95,57),(34,77,96,52),(35,84,85,59),(36,79,86,54)], [(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)], [(1,95,7,89),(2,88,8,94),(3,93,9,87),(4,86,10,92),(5,91,11,85),(6,96,12,90),(13,28,19,34),(14,33,20,27),(15,26,21,32),(16,31,22,25),(17,36,23,30),(18,29,24,35),(37,82,43,76),(38,75,44,81),(39,80,45,74),(40,73,46,79),(41,78,47,84),(42,83,48,77),(49,72,55,66),(50,65,56,71),(51,70,57,64),(52,63,58,69),(53,68,59,62),(54,61,60,67)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 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 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
G:=sub<GL(6,GF(13))| [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],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,12,0,0,0,0,0,0,8,0,0,0,0,0,0,8] >;
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | ··· | 4O | 4P | 4Q | 4R | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | C4○D4 | C4○D4 | 2- (1+4) | Q8.15D6 | S3×C4○D4 |
| kernel | C42.151D6 | C42⋊2S3 | Dic6⋊C4 | Dic3.Q8 | S3×C4⋊C4 | C4⋊C4⋊7S3 | Dic3⋊5D4 | D6.D4 | D6⋊Q8 | C4⋊C4⋊S3 | C3×C42.C2 | C42.C2 | C42 | C4⋊C4 | Dic3 | D6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 6 | 4 | 4 | 1 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{151}D_6 % in TeX
G:=Group("C4^2.151D6"); // GroupNames label
G:=SmallGroup(192,1252);
// by ID
G=gap.SmallGroup(192,1252);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,100,346,297,136,6278]);
// 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*b^2,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations