metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.187D6, C22⋊Q8⋊25S3, (Q8×Dic3)⋊11C2, (C2×Q8).146D6, C22⋊C4.54D6, Dic3⋊5D4⋊23C2, C12⋊7D4.15C2, (C2×C6).168C24, D6⋊C4.19C22, C12.3Q8⋊22C2, (C22×C4).386D6, Dic3⋊4D4⋊13C2, C12.208(C4○D4), C12.23D4⋊11C2, C4.71(D4⋊2S3), (C2×C12).502C23, (C6×Q8).103C22, Dic3.42(C4○D4), (C2×D12).147C22, C22.D12⋊15C2, Dic3⋊C4.23C22, (C22×S3).73C23, C4⋊Dic3.212C22, (C22×C6).196C23, C23.196(C22×S3), C22.189(S3×C23), C22.3(Q8⋊3S3), (C22×C12).248C22, C3⋊7(C23.36C23), (C2×Dic3).232C23, (C4×Dic3).102C22, (C22×Dic3).223C22, (C2×C4×Dic3)⋊9C2, C4⋊C4⋊7S3⋊23C2, C4⋊C4⋊S3⋊16C2, C2.46(S3×C4○D4), (C3×C22⋊Q8)⋊5C2, C6.158(C2×C4○D4), (S3×C2×C4).91C22, (C2×C6).25(C4○D4), C2.44(C2×D4⋊2S3), (C2×C4).44(C22×S3), C2.15(C2×Q8⋊3S3), (C3×C4⋊C4).154C22, (C2×C3⋊D4).37C22, (C3×C22⋊C4).23C22, SmallGroup(192,1183)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 544 in 234 conjugacy classes, 101 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×12], C22, C22 [×2], C22 [×8], S3 [×2], C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×16], D4 [×6], Q8 [×2], C23, C23 [×2], Dic3 [×2], Dic3 [×5], C12 [×2], C12 [×5], D6 [×6], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×6], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×7], C22×C4, C22×C4 [×4], C2×D4 [×3], C2×Q8, C4×S3 [×4], D12 [×2], C2×Dic3 [×4], C2×Dic3 [×2], C2×Dic3 [×4], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×4], C2×C12 [×2], C3×Q8 [×2], C22×S3 [×2], C22×C6, C2×C42, C42⋊C2 [×2], C4×D4 [×3], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C42⋊2C2 [×2], C4×Dic3 [×4], C4×Dic3 [×2], Dic3⋊C4 [×2], C4⋊Dic3, C4⋊Dic3 [×4], D6⋊C4 [×8], C3×C22⋊C4 [×2], C3×C4⋊C4, C3×C4⋊C4 [×2], S3×C2×C4 [×2], C2×D12, C22×Dic3 [×2], C2×C3⋊D4 [×2], C22×C12, C6×Q8, C23.36C23, Dic3⋊4D4 [×2], C22.D12 [×2], C12.3Q8, C4⋊C4⋊7S3 [×2], Dic3⋊5D4, C4⋊C4⋊S3 [×2], C2×C4×Dic3, C12⋊7D4, Q8×Dic3, C12.23D4, C3×C22⋊Q8, C4⋊C4.187D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×6], C24, C22×S3 [×7], C2×C4○D4 [×3], D4⋊2S3 [×2], Q8⋊3S3 [×2], S3×C23, C23.36C23, C2×D4⋊2S3, C2×Q8⋊3S3, S3×C4○D4, C4⋊C4.187D6
Generators and relations
G = < a,b,c,d | a4=b4=c6=1, d2=b2, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd-1=a2b-1, dcd-1=c-1 >
►On 96 points
(1 44 15 88)(2 45 16 89)(3 46 17 90)(4 47 18 85)(5 48 13 86)(6 43 14 87)(7 39 56 83)(8 40 57 84)(9 41 58 79)(10 42 59 80)(11 37 60 81)(12 38 55 82)(19 92 63 52)(20 93 64 53)(21 94 65 54)(22 95 66 49)(23 96 61 50)(24 91 62 51)(25 36 69 74)(26 31 70 75)(27 32 71 76)(28 33 72 77)(29 34 67 78)(30 35 68 73)
(1 79 73 65)(2 22 74 42)(3 81 75 61)(4 24 76 38)(5 83 77 63)(6 20 78 40)(7 72 52 86)(8 43 53 29)(9 68 54 88)(10 45 49 25)(11 70 50 90)(12 47 51 27)(13 39 33 19)(14 64 34 84)(15 41 35 21)(16 66 36 80)(17 37 31 23)(18 62 32 82)(26 96 46 60)(28 92 48 56)(30 94 44 58)(55 85 91 71)(57 87 93 67)(59 89 95 69)
(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 64 73 84)(2 63 74 83)(3 62 75 82)(4 61 76 81)(5 66 77 80)(6 65 78 79)(7 45 52 25)(8 44 53 30)(9 43 54 29)(10 48 49 28)(11 47 50 27)(12 46 51 26)(13 22 33 42)(14 21 34 41)(15 20 35 40)(16 19 36 39)(17 24 31 38)(18 23 32 37)(55 90 91 70)(56 89 92 69)(57 88 93 68)(58 87 94 67)(59 86 95 72)(60 85 96 71)
G:=sub<Sym(96)| (1,44,15,88)(2,45,16,89)(3,46,17,90)(4,47,18,85)(5,48,13,86)(6,43,14,87)(7,39,56,83)(8,40,57,84)(9,41,58,79)(10,42,59,80)(11,37,60,81)(12,38,55,82)(19,92,63,52)(20,93,64,53)(21,94,65,54)(22,95,66,49)(23,96,61,50)(24,91,62,51)(25,36,69,74)(26,31,70,75)(27,32,71,76)(28,33,72,77)(29,34,67,78)(30,35,68,73), (1,79,73,65)(2,22,74,42)(3,81,75,61)(4,24,76,38)(5,83,77,63)(6,20,78,40)(7,72,52,86)(8,43,53,29)(9,68,54,88)(10,45,49,25)(11,70,50,90)(12,47,51,27)(13,39,33,19)(14,64,34,84)(15,41,35,21)(16,66,36,80)(17,37,31,23)(18,62,32,82)(26,96,46,60)(28,92,48,56)(30,94,44,58)(55,85,91,71)(57,87,93,67)(59,89,95,69), (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), (1,64,73,84)(2,63,74,83)(3,62,75,82)(4,61,76,81)(5,66,77,80)(6,65,78,79)(7,45,52,25)(8,44,53,30)(9,43,54,29)(10,48,49,28)(11,47,50,27)(12,46,51,26)(13,22,33,42)(14,21,34,41)(15,20,35,40)(16,19,36,39)(17,24,31,38)(18,23,32,37)(55,90,91,70)(56,89,92,69)(57,88,93,68)(58,87,94,67)(59,86,95,72)(60,85,96,71)>;
G:=Group( (1,44,15,88)(2,45,16,89)(3,46,17,90)(4,47,18,85)(5,48,13,86)(6,43,14,87)(7,39,56,83)(8,40,57,84)(9,41,58,79)(10,42,59,80)(11,37,60,81)(12,38,55,82)(19,92,63,52)(20,93,64,53)(21,94,65,54)(22,95,66,49)(23,96,61,50)(24,91,62,51)(25,36,69,74)(26,31,70,75)(27,32,71,76)(28,33,72,77)(29,34,67,78)(30,35,68,73), (1,79,73,65)(2,22,74,42)(3,81,75,61)(4,24,76,38)(5,83,77,63)(6,20,78,40)(7,72,52,86)(8,43,53,29)(9,68,54,88)(10,45,49,25)(11,70,50,90)(12,47,51,27)(13,39,33,19)(14,64,34,84)(15,41,35,21)(16,66,36,80)(17,37,31,23)(18,62,32,82)(26,96,46,60)(28,92,48,56)(30,94,44,58)(55,85,91,71)(57,87,93,67)(59,89,95,69), (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), (1,64,73,84)(2,63,74,83)(3,62,75,82)(4,61,76,81)(5,66,77,80)(6,65,78,79)(7,45,52,25)(8,44,53,30)(9,43,54,29)(10,48,49,28)(11,47,50,27)(12,46,51,26)(13,22,33,42)(14,21,34,41)(15,20,35,40)(16,19,36,39)(17,24,31,38)(18,23,32,37)(55,90,91,70)(56,89,92,69)(57,88,93,68)(58,87,94,67)(59,86,95,72)(60,85,96,71) );
G=PermutationGroup([(1,44,15,88),(2,45,16,89),(3,46,17,90),(4,47,18,85),(5,48,13,86),(6,43,14,87),(7,39,56,83),(8,40,57,84),(9,41,58,79),(10,42,59,80),(11,37,60,81),(12,38,55,82),(19,92,63,52),(20,93,64,53),(21,94,65,54),(22,95,66,49),(23,96,61,50),(24,91,62,51),(25,36,69,74),(26,31,70,75),(27,32,71,76),(28,33,72,77),(29,34,67,78),(30,35,68,73)], [(1,79,73,65),(2,22,74,42),(3,81,75,61),(4,24,76,38),(5,83,77,63),(6,20,78,40),(7,72,52,86),(8,43,53,29),(9,68,54,88),(10,45,49,25),(11,70,50,90),(12,47,51,27),(13,39,33,19),(14,64,34,84),(15,41,35,21),(16,66,36,80),(17,37,31,23),(18,62,32,82),(26,96,46,60),(28,92,48,56),(30,94,44,58),(55,85,91,71),(57,87,93,67),(59,89,95,69)], [(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)], [(1,64,73,84),(2,63,74,83),(3,62,75,82),(4,61,76,81),(5,66,77,80),(6,65,78,79),(7,45,52,25),(8,44,53,30),(9,43,54,29),(10,48,49,28),(11,47,50,27),(12,46,51,26),(13,22,33,42),(14,21,34,41),(15,20,35,40),(16,19,36,39),(17,24,31,38),(18,23,32,37),(55,90,91,70),(56,89,92,69),(57,88,93,68),(58,87,94,67),(59,86,95,72),(60,85,96,71)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 9 | 0 | 0 |
| 0 | 0 | 9 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 11 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 5 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 8 | 8 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 7 | 0 | 0 |
| 0 | 0 | 7 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 7 | 0 | 0 |
| 0 | 0 | 7 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,9,0,0,0,0,9,3,0,0,0,0,0,0,12,1,0,0,0,0,11,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,5,0,0,0,0,0,12,0,0,0,0,0,0,5,8,0,0,0,0,0,8],[0,1,0,0,0,0,12,12,0,0,0,0,0,0,2,7,0,0,0,0,7,11,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,12,12,0,0,0,0,0,0,2,7,0,0,0,0,7,11,0,0,0,0,0,0,5,0,0,0,0,0,0,5] >;
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4R | 4S | 4T | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 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 | D6 | C4○D4 | C4○D4 | C4○D4 | D4⋊2S3 | Q8⋊3S3 | S3×C4○D4 |
| kernel | C4⋊C4.187D6 | Dic3⋊4D4 | C22.D12 | C12.3Q8 | C4⋊C4⋊7S3 | Dic3⋊5D4 | C4⋊C4⋊S3 | C2×C4×Dic3 | C12⋊7D4 | Q8×Dic3 | C12.23D4 | C3×C22⋊Q8 | C22⋊Q8 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | Dic3 | C12 | C2×C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 1 | 1 | 4 | 4 | 4 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4\rtimes C_4._{187}D_6 % in TeX
G:=Group("C4:C4.187D6"); // GroupNames label
G:=SmallGroup(192,1183);
// by ID
G=gap.SmallGroup(192,1183);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,100,794,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=b^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a^2*b^-1,d*c*d^-1=c^-1>;
// generators/relations