metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.177D6, C6.392- (1+4), C4⋊Q8⋊15S3, C4⋊C4.221D6, C4.D12⋊45C2, (C4×Dic6)⋊54C2, (Q8×Dic3)⋊24C2, (C4×D12).28C2, (C2×Q8).173D6, (C2×C6).276C24, C12.139(C4○D4), C4.19(D4⋊2S3), (C2×C12).109C23, (C4×C12).217C22, D6⋊C4.155C22, C4.41(Q8⋊3S3), C12.23D4.8C2, (C6×Q8).143C22, (C2×D12).274C22, C4⋊Dic3.386C22, C22.297(S3×C23), Dic3⋊C4.168C22, (C22×S3).121C23, C2.40(Q8.15D6), C3⋊8(C22.50C24), (C4×Dic3).165C22, (C2×Dic6).304C22, (C2×Dic3).146C23, (C3×C4⋊Q8)⋊18C2, C4⋊C4⋊7S3⋊43C2, C4⋊C4⋊S3⋊46C2, C6.123(C2×C4○D4), C2.66(C2×D4⋊2S3), (S3×C2×C4).149C22, C2.31(C2×Q8⋊3S3), (C3×C4⋊C4).219C22, (C2×C4).601(C22×S3), SmallGroup(192,1291)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 480 in 212 conjugacy classes, 99 normal (27 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×4], C4 [×11], C22, C22 [×6], S3 [×2], C6 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×10], D4 [×2], Q8 [×6], C23 [×2], Dic3 [×6], C12 [×4], C12 [×5], D6 [×6], C2×C6, C42, C42 [×6], C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×8], C22×C4 [×2], C2×D4, C2×Q8 [×2], C2×Q8, Dic6 [×2], C4×S3 [×4], D12 [×2], C2×Dic3 [×6], C2×C12 [×3], C2×C12 [×4], C3×Q8 [×4], C22×S3 [×2], C42⋊C2 [×2], C4×D4, C4×Q8 [×3], C22⋊Q8 [×2], C4.4D4 [×2], C42⋊2C2 [×4], C4⋊Q8, C4×Dic3 [×6], Dic3⋊C4 [×2], C4⋊Dic3 [×2], C4⋊Dic3 [×4], D6⋊C4 [×10], C4×C12, C3×C4⋊C4 [×4], C2×Dic6, S3×C2×C4 [×2], C2×D12, C6×Q8 [×2], C22.50C24, C4×Dic6, C4×D12, C4⋊C4⋊7S3 [×2], C4.D12 [×2], C4⋊C4⋊S3 [×4], Q8×Dic3 [×2], C12.23D4 [×2], C3×C4⋊Q8, C42.177D6
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), D4⋊2S3 [×2], Q8⋊3S3 [×2], S3×C23, C22.50C24, C2×D4⋊2S3, C2×Q8⋊3S3, Q8.15D6, C42.177D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=a2b2, d2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, dbd-1=a2b, dcd-1=b2c5 >
►On 96 points
(1 68 91 18)(2 19 92 69)(3 70 93 20)(4 21 94 71)(5 72 95 22)(6 23 96 61)(7 62 85 24)(8 13 86 63)(9 64 87 14)(10 15 88 65)(11 66 89 16)(12 17 90 67)(25 57 48 75)(26 76 37 58)(27 59 38 77)(28 78 39 60)(29 49 40 79)(30 80 41 50)(31 51 42 81)(32 82 43 52)(33 53 44 83)(34 84 45 54)(35 55 46 73)(36 74 47 56)
(1 52 85 76)(2 77 86 53)(3 54 87 78)(4 79 88 55)(5 56 89 80)(6 81 90 57)(7 58 91 82)(8 83 92 59)(9 60 93 84)(10 73 94 49)(11 50 95 74)(12 75 96 51)(13 33 69 38)(14 39 70 34)(15 35 71 40)(16 41 72 36)(17 25 61 42)(18 43 62 26)(19 27 63 44)(20 45 64 28)(21 29 65 46)(22 47 66 30)(23 31 67 48)(24 37 68 32)
(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 91 96)(2 95 92 5)(3 4 93 94)(7 12 85 90)(8 89 86 11)(9 10 87 88)(13 66 63 16)(14 15 64 65)(17 62 67 24)(18 23 68 61)(19 72 69 22)(20 21 70 71)(25 37 48 26)(27 47 38 36)(28 35 39 46)(29 45 40 34)(30 33 41 44)(31 43 42 32)(49 84 79 54)(50 53 80 83)(51 82 81 52)(55 78 73 60)(56 59 74 77)(57 76 75 58)
G:=sub<Sym(96)| (1,68,91,18)(2,19,92,69)(3,70,93,20)(4,21,94,71)(5,72,95,22)(6,23,96,61)(7,62,85,24)(8,13,86,63)(9,64,87,14)(10,15,88,65)(11,66,89,16)(12,17,90,67)(25,57,48,75)(26,76,37,58)(27,59,38,77)(28,78,39,60)(29,49,40,79)(30,80,41,50)(31,51,42,81)(32,82,43,52)(33,53,44,83)(34,84,45,54)(35,55,46,73)(36,74,47,56), (1,52,85,76)(2,77,86,53)(3,54,87,78)(4,79,88,55)(5,56,89,80)(6,81,90,57)(7,58,91,82)(8,83,92,59)(9,60,93,84)(10,73,94,49)(11,50,95,74)(12,75,96,51)(13,33,69,38)(14,39,70,34)(15,35,71,40)(16,41,72,36)(17,25,61,42)(18,43,62,26)(19,27,63,44)(20,45,64,28)(21,29,65,46)(22,47,66,30)(23,31,67,48)(24,37,68,32), (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), (1,6,91,96)(2,95,92,5)(3,4,93,94)(7,12,85,90)(8,89,86,11)(9,10,87,88)(13,66,63,16)(14,15,64,65)(17,62,67,24)(18,23,68,61)(19,72,69,22)(20,21,70,71)(25,37,48,26)(27,47,38,36)(28,35,39,46)(29,45,40,34)(30,33,41,44)(31,43,42,32)(49,84,79,54)(50,53,80,83)(51,82,81,52)(55,78,73,60)(56,59,74,77)(57,76,75,58)>;
G:=Group( (1,68,91,18)(2,19,92,69)(3,70,93,20)(4,21,94,71)(5,72,95,22)(6,23,96,61)(7,62,85,24)(8,13,86,63)(9,64,87,14)(10,15,88,65)(11,66,89,16)(12,17,90,67)(25,57,48,75)(26,76,37,58)(27,59,38,77)(28,78,39,60)(29,49,40,79)(30,80,41,50)(31,51,42,81)(32,82,43,52)(33,53,44,83)(34,84,45,54)(35,55,46,73)(36,74,47,56), (1,52,85,76)(2,77,86,53)(3,54,87,78)(4,79,88,55)(5,56,89,80)(6,81,90,57)(7,58,91,82)(8,83,92,59)(9,60,93,84)(10,73,94,49)(11,50,95,74)(12,75,96,51)(13,33,69,38)(14,39,70,34)(15,35,71,40)(16,41,72,36)(17,25,61,42)(18,43,62,26)(19,27,63,44)(20,45,64,28)(21,29,65,46)(22,47,66,30)(23,31,67,48)(24,37,68,32), (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), (1,6,91,96)(2,95,92,5)(3,4,93,94)(7,12,85,90)(8,89,86,11)(9,10,87,88)(13,66,63,16)(14,15,64,65)(17,62,67,24)(18,23,68,61)(19,72,69,22)(20,21,70,71)(25,37,48,26)(27,47,38,36)(28,35,39,46)(29,45,40,34)(30,33,41,44)(31,43,42,32)(49,84,79,54)(50,53,80,83)(51,82,81,52)(55,78,73,60)(56,59,74,77)(57,76,75,58) );
G=PermutationGroup([(1,68,91,18),(2,19,92,69),(3,70,93,20),(4,21,94,71),(5,72,95,22),(6,23,96,61),(7,62,85,24),(8,13,86,63),(9,64,87,14),(10,15,88,65),(11,66,89,16),(12,17,90,67),(25,57,48,75),(26,76,37,58),(27,59,38,77),(28,78,39,60),(29,49,40,79),(30,80,41,50),(31,51,42,81),(32,82,43,52),(33,53,44,83),(34,84,45,54),(35,55,46,73),(36,74,47,56)], [(1,52,85,76),(2,77,86,53),(3,54,87,78),(4,79,88,55),(5,56,89,80),(6,81,90,57),(7,58,91,82),(8,83,92,59),(9,60,93,84),(10,73,94,49),(11,50,95,74),(12,75,96,51),(13,33,69,38),(14,39,70,34),(15,35,71,40),(16,41,72,36),(17,25,61,42),(18,43,62,26),(19,27,63,44),(20,45,64,28),(21,29,65,46),(22,47,66,30),(23,31,67,48),(24,37,68,32)], [(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)], [(1,6,91,96),(2,95,92,5),(3,4,93,94),(7,12,85,90),(8,89,86,11),(9,10,87,88),(13,66,63,16),(14,15,64,65),(17,62,67,24),(18,23,68,61),(19,72,69,22),(20,21,70,71),(25,37,48,26),(27,47,38,36),(28,35,39,46),(29,45,40,34),(30,33,41,44),(31,43,42,32),(49,84,79,54),(50,53,80,83),(51,82,81,52),(55,78,73,60),(56,59,74,77),(57,76,75,58)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 10 | 5 |
| 4 | 3 | 0 | 0 | 0 | 0 |
| 3 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 10 | 5 |
| 7 | 2 | 0 | 0 | 0 | 0 |
| 1 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 6 | 11 | 0 | 0 | 0 | 0 |
| 11 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 2 | 1 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,10,0,0,0,0,0,5],[4,3,0,0,0,0,3,9,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,10,0,0,0,0,0,5],[7,1,0,0,0,0,2,6,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,1,12],[6,11,0,0,0,0,11,7,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,0,0,0,0,12,2,0,0,0,0,12,1] >;
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | ··· | 4Q | 4R | 4S | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | C4○D4 | 2- (1+4) | D4⋊2S3 | Q8⋊3S3 | Q8.15D6 |
| kernel | C42.177D6 | C4×Dic6 | C4×D12 | C4⋊C4⋊7S3 | C4.D12 | C4⋊C4⋊S3 | Q8×Dic3 | C12.23D4 | C3×C4⋊Q8 | C4⋊Q8 | C42 | C4⋊C4 | C2×Q8 | C12 | C6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 8 | 1 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{177}D_6 % in TeX
G:=Group("C4^2.177D6"); // GroupNames label
G:=SmallGroup(192,1291);
// by ID
G=gap.SmallGroup(192,1291);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,758,219,268,1571,297,136,6278]);
// 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,c*b*c^-1=b^-1,d*b*d^-1=a^2*b,d*c*d^-1=b^2*c^5>;
// generators/relations