metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.134D6, C6.142- (1+4), (C4×Q8)⋊20S3, C4⋊C4.301D6, (Q8×C12)⋊18C2, (C4×Dic6)⋊40C2, (C2×Q8).206D6, C4.50(C4○D12), Dic3.Q8⋊10C2, C42⋊3S3.2C2, C42⋊2S3.5C2, (C2×C6).127C24, C4.D12.10C2, D6⋊3Q8.10C2, C2.24(Q8○D12), C12.6Q8⋊27C2, C12.3Q8⋊17C2, Dic3⋊Q8⋊10C2, C12.121(C4○D4), (C4×C12).179C22, (C2×C12).624C23, D6⋊C4.126C22, (C6×Q8).227C22, Dic3⋊C4.78C22, (C22×S3).49C23, C4⋊Dic3.370C22, C22.148(S3×C23), (C2×Dic3).58C23, (C4×Dic3).87C22, C2.15(Q8.15D6), C3⋊2(C22.35C24), (C2×Dic6).292C22, C6.57(C2×C4○D4), C4⋊C4⋊S3.1C2, C2.66(C2×C4○D12), (S3×C2×C4).77C22, (C3×C4⋊C4).355C22, (C2×C4).290(C22×S3), SmallGroup(192,1142)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 408 in 192 conjugacy classes, 95 normal (43 characteristic)
C1, C2 [×3], C2, C3, C4 [×2], C4 [×13], C22, C22 [×3], S3, C6 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×9], Q8 [×4], C23, Dic3 [×7], C12 [×2], C12 [×6], D6 [×3], C2×C6, C42, C42 [×2], C42 [×3], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×17], C22×C4, C2×Q8, C2×Q8, Dic6 [×2], C4×S3 [×2], C2×Dic3 [×3], C2×Dic3 [×4], C2×C12 [×3], C2×C12 [×4], C3×Q8 [×2], C22×S3, C42⋊C2, C4×Q8, C4×Q8, C22⋊Q8 [×2], C42.C2 [×5], C42⋊2C2 [×4], C4⋊Q8, C4×Dic3, C4×Dic3 [×2], Dic3⋊C4 [×2], Dic3⋊C4 [×10], C4⋊Dic3, C4⋊Dic3 [×4], D6⋊C4 [×2], D6⋊C4 [×4], C4×C12, C4×C12 [×2], C3×C4⋊C4, C3×C4⋊C4 [×2], C2×Dic6, S3×C2×C4, C6×Q8, C22.35C24, C4×Dic6, C12.6Q8 [×2], C42⋊2S3, C42⋊3S3 [×2], Dic3.Q8 [×2], C12.3Q8, C4.D12, C4⋊C4⋊S3 [×2], Dic3⋊Q8, D6⋊3Q8, Q8×C12, C42.134D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×2], C24, C22×S3 [×7], C2×C4○D4, 2- (1+4) [×2], C4○D12 [×2], S3×C23, C22.35C24, C2×C4○D12, Q8.15D6, Q8○D12, C42.134D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=a2b2, d2=a2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=b2c5 >
►On 96 points
(1 19 34 42)(2 43 35 20)(3 21 36 44)(4 45 25 22)(5 23 26 46)(6 47 27 24)(7 13 28 48)(8 37 29 14)(9 15 30 38)(10 39 31 16)(11 17 32 40)(12 41 33 18)(49 96 63 74)(50 75 64 85)(51 86 65 76)(52 77 66 87)(53 88 67 78)(54 79 68 89)(55 90 69 80)(56 81 70 91)(57 92 71 82)(58 83 72 93)(59 94 61 84)(60 73 62 95)
(1 75 28 91)(2 76 29 92)(3 77 30 93)(4 78 31 94)(5 79 32 95)(6 80 33 96)(7 81 34 85)(8 82 35 86)(9 83 36 87)(10 84 25 88)(11 73 26 89)(12 74 27 90)(13 70 42 50)(14 71 43 51)(15 72 44 52)(16 61 45 53)(17 62 46 54)(18 63 47 55)(19 64 48 56)(20 65 37 57)(21 66 38 58)(22 67 39 59)(23 68 40 60)(24 69 41 49)
(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 34 27)(2 26 35 5)(3 4 36 25)(7 12 28 33)(8 32 29 11)(9 10 30 31)(13 18 48 41)(14 40 37 17)(15 16 38 39)(19 24 42 47)(20 46 43 23)(21 22 44 45)(49 56 63 70)(50 69 64 55)(51 54 65 68)(52 67 66 53)(57 60 71 62)(58 61 72 59)(73 92 95 82)(74 81 96 91)(75 90 85 80)(76 79 86 89)(77 88 87 78)(83 94 93 84)
G:=sub<Sym(96)| (1,19,34,42)(2,43,35,20)(3,21,36,44)(4,45,25,22)(5,23,26,46)(6,47,27,24)(7,13,28,48)(8,37,29,14)(9,15,30,38)(10,39,31,16)(11,17,32,40)(12,41,33,18)(49,96,63,74)(50,75,64,85)(51,86,65,76)(52,77,66,87)(53,88,67,78)(54,79,68,89)(55,90,69,80)(56,81,70,91)(57,92,71,82)(58,83,72,93)(59,94,61,84)(60,73,62,95), (1,75,28,91)(2,76,29,92)(3,77,30,93)(4,78,31,94)(5,79,32,95)(6,80,33,96)(7,81,34,85)(8,82,35,86)(9,83,36,87)(10,84,25,88)(11,73,26,89)(12,74,27,90)(13,70,42,50)(14,71,43,51)(15,72,44,52)(16,61,45,53)(17,62,46,54)(18,63,47,55)(19,64,48,56)(20,65,37,57)(21,66,38,58)(22,67,39,59)(23,68,40,60)(24,69,41,49), (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), (1,6,34,27)(2,26,35,5)(3,4,36,25)(7,12,28,33)(8,32,29,11)(9,10,30,31)(13,18,48,41)(14,40,37,17)(15,16,38,39)(19,24,42,47)(20,46,43,23)(21,22,44,45)(49,56,63,70)(50,69,64,55)(51,54,65,68)(52,67,66,53)(57,60,71,62)(58,61,72,59)(73,92,95,82)(74,81,96,91)(75,90,85,80)(76,79,86,89)(77,88,87,78)(83,94,93,84)>;
G:=Group( (1,19,34,42)(2,43,35,20)(3,21,36,44)(4,45,25,22)(5,23,26,46)(6,47,27,24)(7,13,28,48)(8,37,29,14)(9,15,30,38)(10,39,31,16)(11,17,32,40)(12,41,33,18)(49,96,63,74)(50,75,64,85)(51,86,65,76)(52,77,66,87)(53,88,67,78)(54,79,68,89)(55,90,69,80)(56,81,70,91)(57,92,71,82)(58,83,72,93)(59,94,61,84)(60,73,62,95), (1,75,28,91)(2,76,29,92)(3,77,30,93)(4,78,31,94)(5,79,32,95)(6,80,33,96)(7,81,34,85)(8,82,35,86)(9,83,36,87)(10,84,25,88)(11,73,26,89)(12,74,27,90)(13,70,42,50)(14,71,43,51)(15,72,44,52)(16,61,45,53)(17,62,46,54)(18,63,47,55)(19,64,48,56)(20,65,37,57)(21,66,38,58)(22,67,39,59)(23,68,40,60)(24,69,41,49), (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), (1,6,34,27)(2,26,35,5)(3,4,36,25)(7,12,28,33)(8,32,29,11)(9,10,30,31)(13,18,48,41)(14,40,37,17)(15,16,38,39)(19,24,42,47)(20,46,43,23)(21,22,44,45)(49,56,63,70)(50,69,64,55)(51,54,65,68)(52,67,66,53)(57,60,71,62)(58,61,72,59)(73,92,95,82)(74,81,96,91)(75,90,85,80)(76,79,86,89)(77,88,87,78)(83,94,93,84) );
G=PermutationGroup([(1,19,34,42),(2,43,35,20),(3,21,36,44),(4,45,25,22),(5,23,26,46),(6,47,27,24),(7,13,28,48),(8,37,29,14),(9,15,30,38),(10,39,31,16),(11,17,32,40),(12,41,33,18),(49,96,63,74),(50,75,64,85),(51,86,65,76),(52,77,66,87),(53,88,67,78),(54,79,68,89),(55,90,69,80),(56,81,70,91),(57,92,71,82),(58,83,72,93),(59,94,61,84),(60,73,62,95)], [(1,75,28,91),(2,76,29,92),(3,77,30,93),(4,78,31,94),(5,79,32,95),(6,80,33,96),(7,81,34,85),(8,82,35,86),(9,83,36,87),(10,84,25,88),(11,73,26,89),(12,74,27,90),(13,70,42,50),(14,71,43,51),(15,72,44,52),(16,61,45,53),(17,62,46,54),(18,63,47,55),(19,64,48,56),(20,65,37,57),(21,66,38,58),(22,67,39,59),(23,68,40,60),(24,69,41,49)], [(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)], [(1,6,34,27),(2,26,35,5),(3,4,36,25),(7,12,28,33),(8,32,29,11),(9,10,30,31),(13,18,48,41),(14,40,37,17),(15,16,38,39),(19,24,42,47),(20,46,43,23),(21,22,44,45),(49,56,63,70),(50,69,64,55),(51,54,65,68),(52,67,66,53),(57,60,71,62),(58,61,72,59),(73,92,95,82),(74,81,96,91),(75,90,85,80),(76,79,86,89),(77,88,87,78),(83,94,93,84)])
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 | 5 | 0 | 10 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 | 1 | 5 |
| 0 | 0 | 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 8 |
| 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 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 | 5 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 5 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 5 | 2 | 0 | 1 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 5 | 1 | 5 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 8 | 11 | 0 | 12 |
G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,10,1,8,0,0,0,0,0,0,5,0,8],[5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,1,0,0,0,0,0,0,2,8,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,12,0,1,5,0,0,0,0,0,12,0,2,0,0,0,0,11,5,1,0,0,0,0,0,0,12,0,1],[0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,12,5,1,8,0,0,0,0,0,1,0,11,0,0,0,0,11,5,1,0,0,0,0,0,0,1,0,12] >;
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4Q | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 12 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 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 | C2 | C2 | S3 | D6 | D6 | D6 | C4○D4 | C4○D12 | 2- (1+4) | Q8.15D6 | Q8○D12 |
| kernel | C42.134D6 | C4×Dic6 | C12.6Q8 | C42⋊2S3 | C42⋊3S3 | Dic3.Q8 | C12.3Q8 | C4.D12 | C4⋊C4⋊S3 | Dic3⋊Q8 | D6⋊3Q8 | Q8×C12 | C4×Q8 | C42 | C4⋊C4 | C2×Q8 | C12 | C4 | C6 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 3 | 3 | 1 | 4 | 8 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{134}D_6 % in TeX
G:=Group("C4^2.134D6"); // GroupNames label
G:=SmallGroup(192,1142);
// by ID
G=gap.SmallGroup(192,1142);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,758,219,268,675,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=a^2*b^2,d^2=a^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,d*c*d^-1=b^2*c^5>;
// generators/relations