metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.168D6, C6.792+ 1+4, C4⋊1D4⋊9S3, D6⋊3D4⋊38C2, (D4×Dic3)⋊36C2, (C2×D4).179D6, C42⋊2S3⋊24C2, (C2×C6).263C24, C12.6Q8⋊23C2, C23.14D6⋊38C2, C12.134(C4○D4), C4.40(D4⋊2S3), C2.83(D4⋊6D6), (C2×C12).637C23, (C4×C12).205C22, D6⋊C4.150C22, (C6×D4).215C22, C23.79(C22×S3), (C22×C6).77C23, C23.23D6⋊27C2, Dic3⋊C4.87C22, C4⋊Dic3.249C22, C22.284(S3×C23), (C22×S3).117C23, C3⋊7(C22.34C24), (C2×Dic3).137C23, (C4×Dic3).156C22, C6.D4.74C22, (C22×Dic3).159C22, C6.98(C2×C4○D4), (C3×C4⋊1D4)⋊10C2, C2.62(C2×D4⋊2S3), (S3×C2×C4).140C22, (C2×C4).215(C22×S3), (C2×C3⋊D4).79C22, SmallGroup(192,1278)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.168D6
G = < a,b,c,d | a4=b4=c6=1, d2=b2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=b2c-1 >
Subgroups: 624 in 240 conjugacy classes, 95 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C4×S3, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C42.C2, C4⋊1D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C6×D4, C22.34C24, C12.6Q8, C42⋊2S3, D4×Dic3, C23.23D6, D6⋊3D4, C23.14D6, C3×C4⋊1D4, C42.168D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, D4⋊2S3, S3×C23, C22.34C24, C2×D4⋊2S3, D4⋊6D6, C42.168D6
►On 96 points
Generators in S96
(1 64 19 67)(2 68 20 65)(3 66 21 69)(4 70 22 61)(5 62 23 71)(6 72 24 63)(7 59 35 80)(8 81 36 60)(9 55 31 82)(10 83 32 56)(11 57 33 84)(12 79 34 58)(13 40 86 54)(14 49 87 41)(15 42 88 50)(16 51 89 37)(17 38 90 52)(18 53 85 39)(25 94 46 74)(26 75 47 95)(27 96 48 76)(28 77 43 91)(29 92 44 78)(30 73 45 93)
(1 28 7 42)(2 37 8 29)(3 30 9 38)(4 39 10 25)(5 26 11 40)(6 41 12 27)(13 71 95 84)(14 79 96 72)(15 67 91 80)(16 81 92 68)(17 69 93 82)(18 83 94 70)(19 43 35 50)(20 51 36 44)(21 45 31 52)(22 53 32 46)(23 47 33 54)(24 49 34 48)(55 90 66 73)(56 74 61 85)(57 86 62 75)(58 76 63 87)(59 88 64 77)(60 78 65 89)
(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 89 95 78)(14 77 96 88)(15 87 91 76)(16 75 92 86)(17 85 93 74)(18 73 94 90)(19 24 35 34)(20 33 36 23)(21 22 31 32)(25 52 39 45)(26 44 40 51)(27 50 41 43)(28 48 42 49)(29 54 37 47)(30 46 38 53)(55 56 66 61)(57 60 62 65)(58 64 63 59)(67 72 80 79)(68 84 81 71)(69 70 82 83)
G:=sub<Sym(96)| (1,64,19,67)(2,68,20,65)(3,66,21,69)(4,70,22,61)(5,62,23,71)(6,72,24,63)(7,59,35,80)(8,81,36,60)(9,55,31,82)(10,83,32,56)(11,57,33,84)(12,79,34,58)(13,40,86,54)(14,49,87,41)(15,42,88,50)(16,51,89,37)(17,38,90,52)(18,53,85,39)(25,94,46,74)(26,75,47,95)(27,96,48,76)(28,77,43,91)(29,92,44,78)(30,73,45,93), (1,28,7,42)(2,37,8,29)(3,30,9,38)(4,39,10,25)(5,26,11,40)(6,41,12,27)(13,71,95,84)(14,79,96,72)(15,67,91,80)(16,81,92,68)(17,69,93,82)(18,83,94,70)(19,43,35,50)(20,51,36,44)(21,45,31,52)(22,53,32,46)(23,47,33,54)(24,49,34,48)(55,90,66,73)(56,74,61,85)(57,86,62,75)(58,76,63,87)(59,88,64,77)(60,78,65,89), (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)(13,89,95,78)(14,77,96,88)(15,87,91,76)(16,75,92,86)(17,85,93,74)(18,73,94,90)(19,24,35,34)(20,33,36,23)(21,22,31,32)(25,52,39,45)(26,44,40,51)(27,50,41,43)(28,48,42,49)(29,54,37,47)(30,46,38,53)(55,56,66,61)(57,60,62,65)(58,64,63,59)(67,72,80,79)(68,84,81,71)(69,70,82,83)>;
G:=Group( (1,64,19,67)(2,68,20,65)(3,66,21,69)(4,70,22,61)(5,62,23,71)(6,72,24,63)(7,59,35,80)(8,81,36,60)(9,55,31,82)(10,83,32,56)(11,57,33,84)(12,79,34,58)(13,40,86,54)(14,49,87,41)(15,42,88,50)(16,51,89,37)(17,38,90,52)(18,53,85,39)(25,94,46,74)(26,75,47,95)(27,96,48,76)(28,77,43,91)(29,92,44,78)(30,73,45,93), (1,28,7,42)(2,37,8,29)(3,30,9,38)(4,39,10,25)(5,26,11,40)(6,41,12,27)(13,71,95,84)(14,79,96,72)(15,67,91,80)(16,81,92,68)(17,69,93,82)(18,83,94,70)(19,43,35,50)(20,51,36,44)(21,45,31,52)(22,53,32,46)(23,47,33,54)(24,49,34,48)(55,90,66,73)(56,74,61,85)(57,86,62,75)(58,76,63,87)(59,88,64,77)(60,78,65,89), (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)(13,89,95,78)(14,77,96,88)(15,87,91,76)(16,75,92,86)(17,85,93,74)(18,73,94,90)(19,24,35,34)(20,33,36,23)(21,22,31,32)(25,52,39,45)(26,44,40,51)(27,50,41,43)(28,48,42,49)(29,54,37,47)(30,46,38,53)(55,56,66,61)(57,60,62,65)(58,64,63,59)(67,72,80,79)(68,84,81,71)(69,70,82,83) );
G=PermutationGroup([[(1,64,19,67),(2,68,20,65),(3,66,21,69),(4,70,22,61),(5,62,23,71),(6,72,24,63),(7,59,35,80),(8,81,36,60),(9,55,31,82),(10,83,32,56),(11,57,33,84),(12,79,34,58),(13,40,86,54),(14,49,87,41),(15,42,88,50),(16,51,89,37),(17,38,90,52),(18,53,85,39),(25,94,46,74),(26,75,47,95),(27,96,48,76),(28,77,43,91),(29,92,44,78),(30,73,45,93)], [(1,28,7,42),(2,37,8,29),(3,30,9,38),(4,39,10,25),(5,26,11,40),(6,41,12,27),(13,71,95,84),(14,79,96,72),(15,67,91,80),(16,81,92,68),(17,69,93,82),(18,83,94,70),(19,43,35,50),(20,51,36,44),(21,45,31,52),(22,53,32,46),(23,47,33,54),(24,49,34,48),(55,90,66,73),(56,74,61,85),(57,86,62,75),(58,76,63,87),(59,88,64,77),(60,78,65,89)], [(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),(13,89,95,78),(14,77,96,88),(15,87,91,76),(16,75,92,86),(17,85,93,74),(18,73,94,90),(19,24,35,34),(20,33,36,23),(21,22,31,32),(25,52,39,45),(26,44,40,51),(27,50,41,43),(28,48,42,49),(29,54,37,47),(30,46,38,53),(55,56,66,61),(57,60,62,65),(58,64,63,59),(67,72,80,79),(68,84,81,71),(69,70,82,83)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4M | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | ··· | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 12 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | C4○D4 | 2+ 1+4 | D4⋊2S3 | D4⋊6D6 |
| kernel | C42.168D6 | C12.6Q8 | C42⋊2S3 | D4×Dic3 | C23.23D6 | D6⋊3D4 | C23.14D6 | C3×C4⋊1D4 | C4⋊1D4 | C42 | C2×D4 | C12 | C6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 4 | 2 | 4 | 1 | 1 | 1 | 6 | 4 | 2 | 2 | 4 |
Matrix representation of C42.168D6 ►in GL8(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 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 | 12 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 12 | 1 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 1 |
| 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 5 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 | 12 | 0 |
| 12 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 | 12 | 0 |
| 1 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 1 | 1 | 0 |
G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,2,1,1,1],[5,5,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,0,0,12,0,0,0,0,11,12,12,12,0,0,0,0,0,1,0,0],[12,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,1,1,0,0,0,0,11,12,12,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0],[1,1,0,0,0,0,0,0,11,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,12,12,0,12,0,0,0,0,2,1,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0] >;
C42.168D6 in GAP, Magma, Sage, TeX
C_4^2._{168}D_6 % in TeX
G:=Group("C4^2.168D6"); // GroupNames label
G:=SmallGroup(192,1278);
// by ID
G=gap.SmallGroup(192,1278);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,100,675,570,185,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations