metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.157D6, C6.312- (1+4), C6.1352+ (1+4), C12⋊Q8⋊38C2, C4⋊C4.114D6, C4.D12⋊39C2, D6⋊Q8⋊37C2, C42.C2⋊13S3, C2.60(D4○D12), (C2×C6).243C24, D6⋊C4.43C22, C2.61(Q8○D12), C12.6Q8⋊30C2, D6.D4.4C2, (C4×C12).224C22, (C2×C12).190C23, C42⋊7S3.12C2, (C2×D12).36C22, Dic3⋊C4.86C22, C4⋊Dic3.245C22, C22.264(S3×C23), (C2×Dic6).41C22, (C22×S3).108C23, C2.32(Q8.15D6), C3⋊4(C22.57C24), (C4×Dic3).148C22, (C2×Dic3).125C23, C4⋊C4⋊S3⋊38C2, (S3×C2×C4).133C22, (C3×C42.C2)⋊16C2, (C3×C4⋊C4).198C22, (C2×C4).207(C22×S3), SmallGroup(192,1258)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 496 in 196 conjugacy classes, 91 normal (31 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×13], C22, C22 [×6], S3 [×2], C6 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×8], D4, Q8 [×3], C23 [×2], Dic3 [×6], C12 [×7], D6 [×6], C2×C6, C42, C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×10], C22×C4 [×2], C2×D4, C2×Q8 [×3], Dic6 [×3], C4×S3 [×2], D12, C2×Dic3 [×6], C2×C12 [×3], C2×C12 [×4], C22×S3 [×2], C22⋊Q8 [×4], C22.D4 [×2], C4.4D4, C42.C2, C42.C2, C42⋊2C2 [×4], C4⋊Q8 [×2], C4×Dic3 [×2], Dic3⋊C4 [×6], C4⋊Dic3 [×2], C4⋊Dic3 [×2], D6⋊C4 [×10], C4×C12, C3×C4⋊C4 [×2], C3×C4⋊C4 [×4], C2×Dic6, C2×Dic6 [×2], S3×C2×C4 [×2], C2×D12, C22.57C24, C12.6Q8, C42⋊7S3, C12⋊Q8 [×2], D6.D4 [×2], D6⋊Q8 [×2], C4.D12 [×2], C4⋊C4⋊S3 [×4], C3×C42.C2, C42.157D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C24, C22×S3 [×7], 2+ (1+4), 2- (1+4) [×2], S3×C23, C22.57C24, Q8.15D6, D4○D12, Q8○D12, C42.157D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=d2=a2b2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c5 >
►On 96 points
(1 25 65 47)(2 32 66 42)(3 27 67 37)(4 34 68 44)(5 29 69 39)(6 36 70 46)(7 31 71 41)(8 26 72 48)(9 33 61 43)(10 28 62 38)(11 35 63 45)(12 30 64 40)(13 58 85 73)(14 53 86 80)(15 60 87 75)(16 55 88 82)(17 50 89 77)(18 57 90 84)(19 52 91 79)(20 59 92 74)(21 54 93 81)(22 49 94 76)(23 56 95 83)(24 51 96 78)
(1 74 71 53)(2 54 72 75)(3 76 61 55)(4 56 62 77)(5 78 63 57)(6 58 64 79)(7 80 65 59)(8 60 66 81)(9 82 67 49)(10 50 68 83)(11 84 69 51)(12 52 70 73)(13 30 91 46)(14 47 92 31)(15 32 93 48)(16 37 94 33)(17 34 95 38)(18 39 96 35)(19 36 85 40)(20 41 86 25)(21 26 87 42)(22 43 88 27)(23 28 89 44)(24 45 90 29)
(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 20 19 14)(15 18 21 24)(16 23 22 17)(25 46 31 40)(26 39 32 45)(27 44 33 38)(28 37 34 43)(29 42 35 48)(30 47 36 41)(49 77 55 83)(50 82 56 76)(51 75 57 81)(52 80 58 74)(53 73 59 79)(54 78 60 84)(61 62 67 68)(63 72 69 66)(64 65 70 71)(85 92 91 86)(87 90 93 96)(88 95 94 89)
G:=sub<Sym(96)| (1,25,65,47)(2,32,66,42)(3,27,67,37)(4,34,68,44)(5,29,69,39)(6,36,70,46)(7,31,71,41)(8,26,72,48)(9,33,61,43)(10,28,62,38)(11,35,63,45)(12,30,64,40)(13,58,85,73)(14,53,86,80)(15,60,87,75)(16,55,88,82)(17,50,89,77)(18,57,90,84)(19,52,91,79)(20,59,92,74)(21,54,93,81)(22,49,94,76)(23,56,95,83)(24,51,96,78), (1,74,71,53)(2,54,72,75)(3,76,61,55)(4,56,62,77)(5,78,63,57)(6,58,64,79)(7,80,65,59)(8,60,66,81)(9,82,67,49)(10,50,68,83)(11,84,69,51)(12,52,70,73)(13,30,91,46)(14,47,92,31)(15,32,93,48)(16,37,94,33)(17,34,95,38)(18,39,96,35)(19,36,85,40)(20,41,86,25)(21,26,87,42)(22,43,88,27)(23,28,89,44)(24,45,90,29), (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)(13,20,19,14)(15,18,21,24)(16,23,22,17)(25,46,31,40)(26,39,32,45)(27,44,33,38)(28,37,34,43)(29,42,35,48)(30,47,36,41)(49,77,55,83)(50,82,56,76)(51,75,57,81)(52,80,58,74)(53,73,59,79)(54,78,60,84)(61,62,67,68)(63,72,69,66)(64,65,70,71)(85,92,91,86)(87,90,93,96)(88,95,94,89)>;
G:=Group( (1,25,65,47)(2,32,66,42)(3,27,67,37)(4,34,68,44)(5,29,69,39)(6,36,70,46)(7,31,71,41)(8,26,72,48)(9,33,61,43)(10,28,62,38)(11,35,63,45)(12,30,64,40)(13,58,85,73)(14,53,86,80)(15,60,87,75)(16,55,88,82)(17,50,89,77)(18,57,90,84)(19,52,91,79)(20,59,92,74)(21,54,93,81)(22,49,94,76)(23,56,95,83)(24,51,96,78), (1,74,71,53)(2,54,72,75)(3,76,61,55)(4,56,62,77)(5,78,63,57)(6,58,64,79)(7,80,65,59)(8,60,66,81)(9,82,67,49)(10,50,68,83)(11,84,69,51)(12,52,70,73)(13,30,91,46)(14,47,92,31)(15,32,93,48)(16,37,94,33)(17,34,95,38)(18,39,96,35)(19,36,85,40)(20,41,86,25)(21,26,87,42)(22,43,88,27)(23,28,89,44)(24,45,90,29), (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)(13,20,19,14)(15,18,21,24)(16,23,22,17)(25,46,31,40)(26,39,32,45)(27,44,33,38)(28,37,34,43)(29,42,35,48)(30,47,36,41)(49,77,55,83)(50,82,56,76)(51,75,57,81)(52,80,58,74)(53,73,59,79)(54,78,60,84)(61,62,67,68)(63,72,69,66)(64,65,70,71)(85,92,91,86)(87,90,93,96)(88,95,94,89) );
G=PermutationGroup([(1,25,65,47),(2,32,66,42),(3,27,67,37),(4,34,68,44),(5,29,69,39),(6,36,70,46),(7,31,71,41),(8,26,72,48),(9,33,61,43),(10,28,62,38),(11,35,63,45),(12,30,64,40),(13,58,85,73),(14,53,86,80),(15,60,87,75),(16,55,88,82),(17,50,89,77),(18,57,90,84),(19,52,91,79),(20,59,92,74),(21,54,93,81),(22,49,94,76),(23,56,95,83),(24,51,96,78)], [(1,74,71,53),(2,54,72,75),(3,76,61,55),(4,56,62,77),(5,78,63,57),(6,58,64,79),(7,80,65,59),(8,60,66,81),(9,82,67,49),(10,50,68,83),(11,84,69,51),(12,52,70,73),(13,30,91,46),(14,47,92,31),(15,32,93,48),(16,37,94,33),(17,34,95,38),(18,39,96,35),(19,36,85,40),(20,41,86,25),(21,26,87,42),(22,43,88,27),(23,28,89,44),(24,45,90,29)], [(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),(13,20,19,14),(15,18,21,24),(16,23,22,17),(25,46,31,40),(26,39,32,45),(27,44,33,38),(28,37,34,43),(29,42,35,48),(30,47,36,41),(49,77,55,83),(50,82,56,76),(51,75,57,81),(52,80,58,74),(53,73,59,79),(54,78,60,84),(61,62,67,68),(63,72,69,66),(64,65,70,71),(85,92,91,86),(87,90,93,96),(88,95,94,89)])
Matrix representation ►G ⊆ GL8(𝔽13)
| 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 | 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 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 11 | 4 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 2 |
| 6 | 0 | 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 7 | 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 10 | 6 | 0 | 0 | 0 | 0 | 0 |
| 10 | 0 | 0 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 0 | 0 | 5 | 8 |
| 0 | 0 | 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 8 | 0 | 0 |
| 6 | 0 | 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 6 | 10 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 7 | 0 | 0 | 0 | 0 | 0 |
| 10 | 0 | 0 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 5 | 8 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 5 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 | 0 | 0 |
G:=sub<GL(8,GF(13))| [0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,11,9,0,0,0,0,0,0,4,2,0,0,0,0,0,0,0,0,11,9,0,0,0,0,0,0,4,2],[6,0,0,10,0,0,0,0,0,7,10,0,0,0,0,0,0,3,6,0,0,0,0,0,3,0,0,7,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,8,8,0,0,0,0,0,5,0,0,0,0,0,0,8,8,0,0],[6,0,0,10,0,0,0,0,0,6,3,0,0,0,0,0,0,10,7,0,0,0,0,0,3,0,0,7,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,8,8,0,0,0,0,5,0,0,0,0,0,0,0,8,8,0,0] >;
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4G | 4H | ··· | 4M | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 4 | ··· | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | 2+ (1+4) | 2- (1+4) | Q8.15D6 | D4○D12 | Q8○D12 |
| kernel | C42.157D6 | C12.6Q8 | C42⋊7S3 | C12⋊Q8 | D6.D4 | D6⋊Q8 | C4.D12 | C4⋊C4⋊S3 | C3×C42.C2 | C42.C2 | C42 | C4⋊C4 | C6 | C6 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 1 | 1 | 6 | 1 | 2 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{157}D_6 % in TeX
G:=Group("C4^2.157D6"); // GroupNames label
G:=SmallGroup(192,1258);
// by ID
G=gap.SmallGroup(192,1258);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,758,219,268,1571,570,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^2*b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^5>;
// generators/relations