metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.87D6, C6.462- (1+4), C4⋊C4.308D6, (C4×Dic6)⋊5C2, (C2×Dic6)⋊18C4, (C2×C6).60C24, C6.15(C23×C4), C2.1(Q8○D12), (C4×C12).20C22, C22⋊C4.123D6, Dic6.31(C2×C4), (C22×C4).201D6, Dic6⋊C4⋊11C2, (C2×C12).581C23, C12.119(C22×C4), C42⋊C2.10S3, C22.25(S3×C23), Dic3.6(C22×C4), C4⋊Dic3.396C22, (C22×C6).130C23, C23.159(C22×S3), (C22×Dic6).17C2, (C4×Dic3).67C22, C23.16D6.4C2, Dic3⋊C4.130C22, (C22×C12).221C22, C3⋊1(C23.32C23), (C2×Dic3).192C23, (C2×Dic6).283C22, C23.26D6.21C2, C6.D4.90C22, (C22×Dic3).84C22, C4.57(S3×C2×C4), (C2×C4).57(C4×S3), C22.25(S3×C2×C4), C2.17(S3×C22×C4), (C2×C12).129(C2×C4), (C2×C6).19(C22×C4), (C3×C4⋊C4).301C22, (C2×C4).268(C22×S3), (C2×Dic3).36(C2×C4), (C3×C42⋊C2).11C2, (C3×C22⋊C4).133C22, SmallGroup(192,1075)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 488 in 266 conjugacy classes, 151 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×4], C4 [×16], C22, C22 [×2], C22 [×2], C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×16], Q8 [×16], C23, Dic3 [×8], Dic3 [×4], C12 [×4], C12 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×2], C42 [×10], C22⋊C4 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×10], C22×C4, C22×C4 [×2], C2×Q8 [×12], Dic6 [×16], C2×Dic3 [×16], C2×C12 [×2], C2×C12 [×8], C22×C6, C42⋊C2, C42⋊C2 [×5], C4×Q8 [×8], C22×Q8, C4×Dic3 [×10], Dic3⋊C4 [×8], C4⋊Dic3 [×2], C6.D4 [×2], C4×C12 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], C2×Dic6 [×12], C22×Dic3 [×2], C22×C12, C23.32C23, C4×Dic6 [×4], C23.16D6 [×4], Dic6⋊C4 [×4], C23.26D6, C3×C42⋊C2, C22×Dic6, C42.87D6
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], D6 [×7], C22×C4 [×14], C24, C4×S3 [×4], C22×S3 [×7], C23×C4, 2- (1+4) [×2], S3×C2×C4 [×6], S3×C23, C23.32C23, S3×C22×C4, Q8○D12 [×2], C42.87D6
Generators and relations
G = < a,b,c,d | a4=b4=c6=1, d2=a2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=a2b, dcd-1=c-1 >
►On 96 points
(1 44 19 17)(2 45 20 18)(3 43 21 16)(4 25 42 28)(5 26 40 29)(6 27 41 30)(7 34 10 31)(8 35 11 32)(9 36 12 33)(13 37 46 24)(14 38 47 22)(15 39 48 23)(49 91 52 94)(50 92 53 95)(51 93 54 96)(55 67 58 70)(56 68 59 71)(57 69 60 72)(61 73 64 76)(62 74 65 77)(63 75 66 78)(79 85 82 88)(80 86 83 89)(81 87 84 90)
(1 81 24 57)(2 79 22 55)(3 83 23 59)(4 52 10 63)(5 50 11 61)(6 54 12 65)(7 66 42 49)(8 64 40 53)(9 62 41 51)(13 69 44 87)(14 67 45 85)(15 71 43 89)(16 86 48 68)(17 90 46 72)(18 88 47 70)(19 84 37 60)(20 82 38 58)(21 80 39 56)(25 94 31 75)(26 92 32 73)(27 96 33 77)(28 91 34 78)(29 95 35 76)(30 93 36 74)
(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 40 19 5)(2 42 20 4)(3 41 21 6)(7 38 10 22)(8 37 11 24)(9 39 12 23)(13 32 46 35)(14 31 47 34)(15 33 48 36)(16 30 43 27)(17 29 44 26)(18 28 45 25)(49 79 52 82)(50 84 53 81)(51 83 54 80)(55 63 58 66)(56 62 59 65)(57 61 60 64)(67 78 70 75)(68 77 71 74)(69 76 72 73)(85 91 88 94)(86 96 89 93)(87 95 90 92)
G:=sub<Sym(96)| (1,44,19,17)(2,45,20,18)(3,43,21,16)(4,25,42,28)(5,26,40,29)(6,27,41,30)(7,34,10,31)(8,35,11,32)(9,36,12,33)(13,37,46,24)(14,38,47,22)(15,39,48,23)(49,91,52,94)(50,92,53,95)(51,93,54,96)(55,67,58,70)(56,68,59,71)(57,69,60,72)(61,73,64,76)(62,74,65,77)(63,75,66,78)(79,85,82,88)(80,86,83,89)(81,87,84,90), (1,81,24,57)(2,79,22,55)(3,83,23,59)(4,52,10,63)(5,50,11,61)(6,54,12,65)(7,66,42,49)(8,64,40,53)(9,62,41,51)(13,69,44,87)(14,67,45,85)(15,71,43,89)(16,86,48,68)(17,90,46,72)(18,88,47,70)(19,84,37,60)(20,82,38,58)(21,80,39,56)(25,94,31,75)(26,92,32,73)(27,96,33,77)(28,91,34,78)(29,95,35,76)(30,93,36,74), (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), (1,40,19,5)(2,42,20,4)(3,41,21,6)(7,38,10,22)(8,37,11,24)(9,39,12,23)(13,32,46,35)(14,31,47,34)(15,33,48,36)(16,30,43,27)(17,29,44,26)(18,28,45,25)(49,79,52,82)(50,84,53,81)(51,83,54,80)(55,63,58,66)(56,62,59,65)(57,61,60,64)(67,78,70,75)(68,77,71,74)(69,76,72,73)(85,91,88,94)(86,96,89,93)(87,95,90,92)>;
G:=Group( (1,44,19,17)(2,45,20,18)(3,43,21,16)(4,25,42,28)(5,26,40,29)(6,27,41,30)(7,34,10,31)(8,35,11,32)(9,36,12,33)(13,37,46,24)(14,38,47,22)(15,39,48,23)(49,91,52,94)(50,92,53,95)(51,93,54,96)(55,67,58,70)(56,68,59,71)(57,69,60,72)(61,73,64,76)(62,74,65,77)(63,75,66,78)(79,85,82,88)(80,86,83,89)(81,87,84,90), (1,81,24,57)(2,79,22,55)(3,83,23,59)(4,52,10,63)(5,50,11,61)(6,54,12,65)(7,66,42,49)(8,64,40,53)(9,62,41,51)(13,69,44,87)(14,67,45,85)(15,71,43,89)(16,86,48,68)(17,90,46,72)(18,88,47,70)(19,84,37,60)(20,82,38,58)(21,80,39,56)(25,94,31,75)(26,92,32,73)(27,96,33,77)(28,91,34,78)(29,95,35,76)(30,93,36,74), (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), (1,40,19,5)(2,42,20,4)(3,41,21,6)(7,38,10,22)(8,37,11,24)(9,39,12,23)(13,32,46,35)(14,31,47,34)(15,33,48,36)(16,30,43,27)(17,29,44,26)(18,28,45,25)(49,79,52,82)(50,84,53,81)(51,83,54,80)(55,63,58,66)(56,62,59,65)(57,61,60,64)(67,78,70,75)(68,77,71,74)(69,76,72,73)(85,91,88,94)(86,96,89,93)(87,95,90,92) );
G=PermutationGroup([(1,44,19,17),(2,45,20,18),(3,43,21,16),(4,25,42,28),(5,26,40,29),(6,27,41,30),(7,34,10,31),(8,35,11,32),(9,36,12,33),(13,37,46,24),(14,38,47,22),(15,39,48,23),(49,91,52,94),(50,92,53,95),(51,93,54,96),(55,67,58,70),(56,68,59,71),(57,69,60,72),(61,73,64,76),(62,74,65,77),(63,75,66,78),(79,85,82,88),(80,86,83,89),(81,87,84,90)], [(1,81,24,57),(2,79,22,55),(3,83,23,59),(4,52,10,63),(5,50,11,61),(6,54,12,65),(7,66,42,49),(8,64,40,53),(9,62,41,51),(13,69,44,87),(14,67,45,85),(15,71,43,89),(16,86,48,68),(17,90,46,72),(18,88,47,70),(19,84,37,60),(20,82,38,58),(21,80,39,56),(25,94,31,75),(26,92,32,73),(27,96,33,77),(28,91,34,78),(29,95,35,76),(30,93,36,74)], [(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)], [(1,40,19,5),(2,42,20,4),(3,41,21,6),(7,38,10,22),(8,37,11,24),(9,39,12,23),(13,32,46,35),(14,31,47,34),(15,33,48,36),(16,30,43,27),(17,29,44,26),(18,28,45,25),(49,79,52,82),(50,84,53,81),(51,83,54,80),(55,63,58,66),(56,62,59,65),(57,61,60,64),(67,78,70,75),(68,77,71,74),(69,76,72,73),(85,91,88,94),(86,96,89,93),(87,95,90,92)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 8 | 12 | 8 | 3 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 8 | 0 | 1 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 8 | 12 | 8 | 3 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 12 | 0 | 12 | 12 |
| 3 | 3 | 0 | 0 | 0 | 0 |
| 6 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 9 | 0 |
| 0 | 0 | 6 | 6 | 6 | 12 |
| 0 | 0 | 9 | 0 | 10 | 0 |
| 0 | 0 | 4 | 11 | 7 | 7 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,8,12,1,0,0,0,12,0,8,0,0,1,8,0,0,0,0,0,3,0,1],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,12,8,0,0,0,1,0,12,0,0,0,0,0,8,0,0,0,0,0,3,5],[0,12,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,12,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12],[3,6,0,0,0,0,3,10,0,0,0,0,0,0,3,6,9,4,0,0,0,6,0,11,0,0,9,6,10,7,0,0,0,12,0,7] >;
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4L | 4M | ··· | 4AB | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D6 | D6 | D6 | D6 | C4×S3 | 2- (1+4) | Q8○D12 |
| kernel | C42.87D6 | C4×Dic6 | C23.16D6 | Dic6⋊C4 | C23.26D6 | C3×C42⋊C2 | C22×Dic6 | C2×Dic6 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C6 | C2 |
| # reps | 1 | 4 | 4 | 4 | 1 | 1 | 1 | 16 | 1 | 2 | 2 | 2 | 1 | 8 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{87}D_6 % in TeX
G:=Group("C4^2.87D6"); // GroupNames label
G:=SmallGroup(192,1075);
// by ID
G=gap.SmallGroup(192,1075);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,758,184,570,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations