metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.274D6, (C2×C12)⋊13Q8, (C4×Dic6)⋊3C2, (C2×C4)⋊10Dic6, C12.89(C2×Q8), C12⋊2Q8⋊37C2, C6.4(C22×Q8), (C2×C42).22S3, (C2×C6).14C24, C4.54(C2×Dic6), C12.6Q8⋊31C2, (C22×C4).452D6, C12.233(C4○D4), C4.117(C4○D12), C2.6(C22×Dic6), (C4×C12).314C22, (C2×C12).692C23, C22.61(S3×C23), (C2×Dic3).3C23, C22.10(C2×Dic6), C12.48D4.20C2, Dic3⋊C4.95C22, C4⋊Dic3.288C22, (C22×C6).376C23, C23.226(C22×S3), C23.26D6.6C2, (C22×C12).522C22, C3⋊1(C23.37C23), (C2×Dic6).223C22, (C4×Dic3).190C22, C6.D4.80C22, (C2×C4×C12).23C2, C6.3(C2×C4○D4), C2.8(C2×C4○D12), (C2×C6).48(C2×Q8), (C2×C4).728(C22×S3), SmallGroup(192,1029)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 440 in 222 conjugacy classes, 119 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×8], C4 [×10], C22, C22 [×2], C22 [×2], C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×12], Q8 [×8], C23, Dic3 [×8], C12 [×8], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×2], C42 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×16], C22×C4, C22×C4 [×2], C2×Q8 [×4], Dic6 [×8], C2×Dic3 [×8], C2×C12 [×2], C2×C12 [×8], C2×C12 [×4], C22×C6, C2×C42, C42⋊C2 [×2], C4×Q8 [×4], C22⋊Q8 [×4], C42.C2 [×2], C4⋊Q8 [×2], C4×Dic3 [×4], Dic3⋊C4 [×8], C4⋊Dic3 [×8], C6.D4 [×4], C4×C12 [×2], C4×C12 [×2], C2×Dic6 [×4], C22×C12, C22×C12 [×2], C23.37C23, C4×Dic6 [×4], C12⋊2Q8 [×2], C12.6Q8 [×2], C12.48D4 [×4], C23.26D6 [×2], C2×C4×C12, C42.274D6
Quotients:
C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D6 [×7], C2×Q8 [×6], C4○D4 [×4], C24, Dic6 [×4], C22×S3 [×7], C22×Q8, C2×C4○D4 [×2], C2×Dic6 [×6], C4○D12 [×4], S3×C23, C23.37C23, C22×Dic6, C2×C4○D12 [×2], C42.274D6
Generators and relations
G = < a,b,c,d | a4=b4=c6=1, d2=a2b2, ab=ba, ac=ca, dad-1=a-1, bc=cb, bd=db, dcd-1=b2c-1 >
►On 96 points
(1 28 4 25)(2 29 5 26)(3 30 6 27)(7 34 10 31)(8 35 11 32)(9 36 12 33)(13 40 16 37)(14 41 17 38)(15 42 18 39)(19 46 22 43)(20 47 23 44)(21 48 24 45)(49 84 64 56)(50 79 65 57)(51 80 66 58)(52 81 61 59)(53 82 62 60)(54 83 63 55)(67 93 77 87)(68 94 78 88)(69 95 73 89)(70 96 74 90)(71 91 75 85)(72 92 76 86)
(1 19 7 13)(2 20 8 14)(3 21 9 15)(4 22 10 16)(5 23 11 17)(6 24 12 18)(25 43 31 37)(26 44 32 38)(27 45 33 39)(28 46 34 40)(29 47 35 41)(30 48 36 42)(49 69 52 72)(50 70 53 67)(51 71 54 68)(55 88 58 85)(56 89 59 86)(57 90 60 87)(61 76 64 73)(62 77 65 74)(63 78 66 75)(79 96 82 93)(80 91 83 94)(81 92 84 95)
(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 63 10 51)(2 65 11 53)(3 61 12 49)(4 54 7 66)(5 50 8 62)(6 52 9 64)(13 75 22 68)(14 77 23 70)(15 73 24 72)(16 71 19 78)(17 67 20 74)(18 69 21 76)(25 55 34 80)(26 57 35 82)(27 59 36 84)(28 83 31 58)(29 79 32 60)(30 81 33 56)(37 85 46 94)(38 87 47 96)(39 89 48 92)(40 91 43 88)(41 93 44 90)(42 95 45 86)
G:=sub<Sym(96)| (1,28,4,25)(2,29,5,26)(3,30,6,27)(7,34,10,31)(8,35,11,32)(9,36,12,33)(13,40,16,37)(14,41,17,38)(15,42,18,39)(19,46,22,43)(20,47,23,44)(21,48,24,45)(49,84,64,56)(50,79,65,57)(51,80,66,58)(52,81,61,59)(53,82,62,60)(54,83,63,55)(67,93,77,87)(68,94,78,88)(69,95,73,89)(70,96,74,90)(71,91,75,85)(72,92,76,86), (1,19,7,13)(2,20,8,14)(3,21,9,15)(4,22,10,16)(5,23,11,17)(6,24,12,18)(25,43,31,37)(26,44,32,38)(27,45,33,39)(28,46,34,40)(29,47,35,41)(30,48,36,42)(49,69,52,72)(50,70,53,67)(51,71,54,68)(55,88,58,85)(56,89,59,86)(57,90,60,87)(61,76,64,73)(62,77,65,74)(63,78,66,75)(79,96,82,93)(80,91,83,94)(81,92,84,95), (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), (1,63,10,51)(2,65,11,53)(3,61,12,49)(4,54,7,66)(5,50,8,62)(6,52,9,64)(13,75,22,68)(14,77,23,70)(15,73,24,72)(16,71,19,78)(17,67,20,74)(18,69,21,76)(25,55,34,80)(26,57,35,82)(27,59,36,84)(28,83,31,58)(29,79,32,60)(30,81,33,56)(37,85,46,94)(38,87,47,96)(39,89,48,92)(40,91,43,88)(41,93,44,90)(42,95,45,86)>;
G:=Group( (1,28,4,25)(2,29,5,26)(3,30,6,27)(7,34,10,31)(8,35,11,32)(9,36,12,33)(13,40,16,37)(14,41,17,38)(15,42,18,39)(19,46,22,43)(20,47,23,44)(21,48,24,45)(49,84,64,56)(50,79,65,57)(51,80,66,58)(52,81,61,59)(53,82,62,60)(54,83,63,55)(67,93,77,87)(68,94,78,88)(69,95,73,89)(70,96,74,90)(71,91,75,85)(72,92,76,86), (1,19,7,13)(2,20,8,14)(3,21,9,15)(4,22,10,16)(5,23,11,17)(6,24,12,18)(25,43,31,37)(26,44,32,38)(27,45,33,39)(28,46,34,40)(29,47,35,41)(30,48,36,42)(49,69,52,72)(50,70,53,67)(51,71,54,68)(55,88,58,85)(56,89,59,86)(57,90,60,87)(61,76,64,73)(62,77,65,74)(63,78,66,75)(79,96,82,93)(80,91,83,94)(81,92,84,95), (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), (1,63,10,51)(2,65,11,53)(3,61,12,49)(4,54,7,66)(5,50,8,62)(6,52,9,64)(13,75,22,68)(14,77,23,70)(15,73,24,72)(16,71,19,78)(17,67,20,74)(18,69,21,76)(25,55,34,80)(26,57,35,82)(27,59,36,84)(28,83,31,58)(29,79,32,60)(30,81,33,56)(37,85,46,94)(38,87,47,96)(39,89,48,92)(40,91,43,88)(41,93,44,90)(42,95,45,86) );
G=PermutationGroup([(1,28,4,25),(2,29,5,26),(3,30,6,27),(7,34,10,31),(8,35,11,32),(9,36,12,33),(13,40,16,37),(14,41,17,38),(15,42,18,39),(19,46,22,43),(20,47,23,44),(21,48,24,45),(49,84,64,56),(50,79,65,57),(51,80,66,58),(52,81,61,59),(53,82,62,60),(54,83,63,55),(67,93,77,87),(68,94,78,88),(69,95,73,89),(70,96,74,90),(71,91,75,85),(72,92,76,86)], [(1,19,7,13),(2,20,8,14),(3,21,9,15),(4,22,10,16),(5,23,11,17),(6,24,12,18),(25,43,31,37),(26,44,32,38),(27,45,33,39),(28,46,34,40),(29,47,35,41),(30,48,36,42),(49,69,52,72),(50,70,53,67),(51,71,54,68),(55,88,58,85),(56,89,59,86),(57,90,60,87),(61,76,64,73),(62,77,65,74),(63,78,66,75),(79,96,82,93),(80,91,83,94),(81,92,84,95)], [(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)], [(1,63,10,51),(2,65,11,53),(3,61,12,49),(4,54,7,66),(5,50,8,62),(6,52,9,64),(13,75,22,68),(14,77,23,70),(15,73,24,72),(16,71,19,78),(17,67,20,74),(18,69,21,76),(25,55,34,80),(26,57,35,82),(27,59,36,84),(28,83,31,58),(29,79,32,60),(30,81,33,56),(37,85,46,94),(38,87,47,96),(39,89,48,92),(40,91,43,88),(41,93,44,90),(42,95,45,86)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 6 | 8 |
| 5 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 9 | 5 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 1 | 3 |
| 4 | 8 | 0 | 0 |
| 6 | 9 | 0 | 0 |
| 0 | 0 | 5 | 9 |
| 0 | 0 | 0 | 8 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,5,6,0,0,0,8],[5,0,0,0,0,5,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,5,10,0,0,0,0,9,1,0,0,0,3],[4,6,0,0,8,9,0,0,0,0,5,0,0,0,9,8] >;
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4V | 6A | ··· | 6G | 12A | ··· | 12X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | C4○D4 | Dic6 | C4○D12 |
| kernel | C42.274D6 | C4×Dic6 | C12⋊2Q8 | C12.6Q8 | C12.48D4 | C23.26D6 | C2×C4×C12 | C2×C42 | C2×C12 | C42 | C22×C4 | C12 | C2×C4 | C4 |
| # reps | 1 | 4 | 2 | 2 | 4 | 2 | 1 | 1 | 4 | 4 | 3 | 8 | 8 | 16 |
In GAP, Magma, Sage, TeX
C_4^2._{274}D_6 % in TeX
G:=Group("C4^2.274D6"); // GroupNames label
G:=SmallGroup(192,1029);
// by ID
G=gap.SmallGroup(192,1029);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,477,232,100,675,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=a^2*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=b^2*c^-1>;
// generators/relations