metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.132D6, C6.122- (1+4), (C4×Q8)⋊18S3, C4⋊C4.299D6, D6⋊3Q8⋊9C2, (Q8×C12)⋊16C2, (C4×D12).22C2, Dic3.Q8⋊9C2, (C2×Q8).204D6, C42⋊2S3⋊34C2, C42⋊3S3⋊18C2, D6.18(C4○D4), C4.68(C4○D12), (C2×C6).125C24, D6⋊C4.89C22, C12.6Q8⋊19C2, D6.D4.1C2, C12.119(C4○D4), (C2×C12).623C23, (C4×C12).177C22, (C6×Q8).225C22, (C2×D12).218C22, Dic3⋊C4.76C22, (C22×S3).47C23, C4⋊Dic3.309C22, C22.146(S3×C23), (C2×Dic3).56C23, C2.13(Q8.15D6), C3⋊5(C22.46C24), (C4×Dic3).209C22, (S3×C4⋊C4)⋊19C2, C4⋊C4⋊7S3⋊17C2, C2.32(S3×C4○D4), C6.147(C2×C4○D4), C2.64(C2×C4○D12), (S3×C2×C4).75C22, (C3×C4⋊C4).353C22, (C2×C4).289(C22×S3), SmallGroup(192,1140)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 488 in 214 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×2], C4 [×12], C22, C22 [×7], S3 [×3], C6 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×14], D4 [×2], Q8 [×2], C23 [×2], Dic3 [×6], C12 [×2], C12 [×6], D6 [×2], D6 [×5], C2×C6, C42, C42 [×2], C42 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×13], C22×C4 [×4], C2×D4, C2×Q8, C4×S3 [×8], D12 [×2], C2×Dic3 [×2], C2×Dic3 [×4], C2×C12 [×3], C2×C12 [×4], C3×Q8 [×2], C22×S3 [×2], C2×C4⋊C4, C42⋊C2 [×3], C4×D4, C4×Q8, C22⋊Q8 [×2], C22.D4 [×2], C42.C2 [×3], C42⋊2C2 [×2], C4×Dic3 [×2], Dic3⋊C4 [×10], C4⋊Dic3 [×3], D6⋊C4 [×2], D6⋊C4 [×6], C4×C12, C4×C12 [×2], C3×C4⋊C4, C3×C4⋊C4 [×2], S3×C2×C4 [×2], S3×C2×C4 [×2], C2×D12, C6×Q8, C22.46C24, C12.6Q8, C42⋊2S3 [×2], C4×D12, C42⋊3S3 [×2], Dic3.Q8 [×2], S3×C4⋊C4, C4⋊C4⋊7S3, D6.D4 [×2], D6⋊3Q8 [×2], Q8×C12, C42.132D6
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, Q8.15D6, S3×C4○D4, C42.132D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=d2=a2b2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b-1, dcd-1=c5 >
►On 96 points
(1 77 36 49)(2 50 25 78)(3 79 26 51)(4 52 27 80)(5 81 28 53)(6 54 29 82)(7 83 30 55)(8 56 31 84)(9 73 32 57)(10 58 33 74)(11 75 34 59)(12 60 35 76)(13 48 61 95)(14 96 62 37)(15 38 63 85)(16 86 64 39)(17 40 65 87)(18 88 66 41)(19 42 67 89)(20 90 68 43)(21 44 69 91)(22 92 70 45)(23 46 71 93)(24 94 72 47)
(1 96 30 43)(2 85 31 44)(3 86 32 45)(4 87 33 46)(5 88 34 47)(6 89 35 48)(7 90 36 37)(8 91 25 38)(9 92 26 39)(10 93 27 40)(11 94 28 41)(12 95 29 42)(13 82 67 60)(14 83 68 49)(15 84 69 50)(16 73 70 51)(17 74 71 52)(18 75 72 53)(19 76 61 54)(20 77 62 55)(21 78 63 56)(22 79 64 57)(23 80 65 58)(24 81 66 59)
(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 61 7 67)(2 66 8 72)(3 71 9 65)(4 64 10 70)(5 69 11 63)(6 62 12 68)(13 30 19 36)(14 35 20 29)(15 28 21 34)(16 33 22 27)(17 26 23 32)(18 31 24 25)(37 76 43 82)(38 81 44 75)(39 74 45 80)(40 79 46 73)(41 84 47 78)(42 77 48 83)(49 95 55 89)(50 88 56 94)(51 93 57 87)(52 86 58 92)(53 91 59 85)(54 96 60 90)
G:=sub<Sym(96)| (1,77,36,49)(2,50,25,78)(3,79,26,51)(4,52,27,80)(5,81,28,53)(6,54,29,82)(7,83,30,55)(8,56,31,84)(9,73,32,57)(10,58,33,74)(11,75,34,59)(12,60,35,76)(13,48,61,95)(14,96,62,37)(15,38,63,85)(16,86,64,39)(17,40,65,87)(18,88,66,41)(19,42,67,89)(20,90,68,43)(21,44,69,91)(22,92,70,45)(23,46,71,93)(24,94,72,47), (1,96,30,43)(2,85,31,44)(3,86,32,45)(4,87,33,46)(5,88,34,47)(6,89,35,48)(7,90,36,37)(8,91,25,38)(9,92,26,39)(10,93,27,40)(11,94,28,41)(12,95,29,42)(13,82,67,60)(14,83,68,49)(15,84,69,50)(16,73,70,51)(17,74,71,52)(18,75,72,53)(19,76,61,54)(20,77,62,55)(21,78,63,56)(22,79,64,57)(23,80,65,58)(24,81,66,59), (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), (1,61,7,67)(2,66,8,72)(3,71,9,65)(4,64,10,70)(5,69,11,63)(6,62,12,68)(13,30,19,36)(14,35,20,29)(15,28,21,34)(16,33,22,27)(17,26,23,32)(18,31,24,25)(37,76,43,82)(38,81,44,75)(39,74,45,80)(40,79,46,73)(41,84,47,78)(42,77,48,83)(49,95,55,89)(50,88,56,94)(51,93,57,87)(52,86,58,92)(53,91,59,85)(54,96,60,90)>;
G:=Group( (1,77,36,49)(2,50,25,78)(3,79,26,51)(4,52,27,80)(5,81,28,53)(6,54,29,82)(7,83,30,55)(8,56,31,84)(9,73,32,57)(10,58,33,74)(11,75,34,59)(12,60,35,76)(13,48,61,95)(14,96,62,37)(15,38,63,85)(16,86,64,39)(17,40,65,87)(18,88,66,41)(19,42,67,89)(20,90,68,43)(21,44,69,91)(22,92,70,45)(23,46,71,93)(24,94,72,47), (1,96,30,43)(2,85,31,44)(3,86,32,45)(4,87,33,46)(5,88,34,47)(6,89,35,48)(7,90,36,37)(8,91,25,38)(9,92,26,39)(10,93,27,40)(11,94,28,41)(12,95,29,42)(13,82,67,60)(14,83,68,49)(15,84,69,50)(16,73,70,51)(17,74,71,52)(18,75,72,53)(19,76,61,54)(20,77,62,55)(21,78,63,56)(22,79,64,57)(23,80,65,58)(24,81,66,59), (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), (1,61,7,67)(2,66,8,72)(3,71,9,65)(4,64,10,70)(5,69,11,63)(6,62,12,68)(13,30,19,36)(14,35,20,29)(15,28,21,34)(16,33,22,27)(17,26,23,32)(18,31,24,25)(37,76,43,82)(38,81,44,75)(39,74,45,80)(40,79,46,73)(41,84,47,78)(42,77,48,83)(49,95,55,89)(50,88,56,94)(51,93,57,87)(52,86,58,92)(53,91,59,85)(54,96,60,90) );
G=PermutationGroup([(1,77,36,49),(2,50,25,78),(3,79,26,51),(4,52,27,80),(5,81,28,53),(6,54,29,82),(7,83,30,55),(8,56,31,84),(9,73,32,57),(10,58,33,74),(11,75,34,59),(12,60,35,76),(13,48,61,95),(14,96,62,37),(15,38,63,85),(16,86,64,39),(17,40,65,87),(18,88,66,41),(19,42,67,89),(20,90,68,43),(21,44,69,91),(22,92,70,45),(23,46,71,93),(24,94,72,47)], [(1,96,30,43),(2,85,31,44),(3,86,32,45),(4,87,33,46),(5,88,34,47),(6,89,35,48),(7,90,36,37),(8,91,25,38),(9,92,26,39),(10,93,27,40),(11,94,28,41),(12,95,29,42),(13,82,67,60),(14,83,68,49),(15,84,69,50),(16,73,70,51),(17,74,71,52),(18,75,72,53),(19,76,61,54),(20,77,62,55),(21,78,63,56),(22,79,64,57),(23,80,65,58),(24,81,66,59)], [(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)], [(1,61,7,67),(2,66,8,72),(3,71,9,65),(4,64,10,70),(5,69,11,63),(6,62,12,68),(13,30,19,36),(14,35,20,29),(15,28,21,34),(16,33,22,27),(17,26,23,32),(18,31,24,25),(37,76,43,82),(38,81,44,75),(39,74,45,80),(40,79,46,73),(41,84,47,78),(42,77,48,83),(49,95,55,89),(50,88,56,94),(51,93,57,87),(52,86,58,92),(53,91,59,85),(54,96,60,90)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 5 | 8 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 10 | 7 |
| 0 | 0 | 6 | 3 |
| 1 | 0 | 0 | 0 |
| 2 | 12 | 0 | 0 |
| 0 | 0 | 8 | 8 |
| 0 | 0 | 5 | 0 |
| 1 | 0 | 0 | 0 |
| 2 | 12 | 0 | 0 |
| 0 | 0 | 2 | 11 |
| 0 | 0 | 9 | 11 |
G:=sub<GL(4,GF(13))| [5,0,0,0,8,8,0,0,0,0,12,0,0,0,0,12],[8,0,0,0,0,8,0,0,0,0,10,6,0,0,7,3],[1,2,0,0,0,12,0,0,0,0,8,5,0,0,8,0],[1,2,0,0,0,12,0,0,0,0,2,9,0,0,11,11] >;
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | ··· | 4R | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 4 | 4 | 4 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | C4○D4 | C4○D4 | C4○D12 | 2- (1+4) | Q8.15D6 | S3×C4○D4 |
| kernel | C42.132D6 | C12.6Q8 | C42⋊2S3 | C4×D12 | C42⋊3S3 | Dic3.Q8 | S3×C4⋊C4 | C4⋊C4⋊7S3 | D6.D4 | D6⋊3Q8 | Q8×C12 | C4×Q8 | C42 | C4⋊C4 | C2×Q8 | C12 | D6 | C4 | C6 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 3 | 3 | 1 | 4 | 4 | 8 | 1 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{132}D_6 % in TeX
G:=Group("C4^2.132D6"); // GroupNames label
G:=SmallGroup(192,1140);
// by ID
G=gap.SmallGroup(192,1140);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,268,1571,136,6278]);
// 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=d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^5>;
// generators/relations