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, C4.Dic6⋊22C2, (C22×C4).386D6, Dic3⋊4D4⋊13C2, C12.208(C4○D4), C4.71(D4⋊2S3), C12.23D4⋊11C2, (C2×C12).502C23, (C6×Q8).103C22, Dic3.42(C4○D4), (C2×D12).147C22, C23.21D6⋊15C2, Dic3⋊C4.23C22, (C22×S3).73C23, C4⋊Dic3.212C22, C23.196(C22×S3), (C22×C6).196C23, 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
Generators and relations for C4⋊C4.187D6
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 >
Subgroups: 544 in 234 conjugacy classes, 101 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4×S3, D12, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×Q8, C22×S3, C22×C6, C2×C42, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4×Dic3, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C22×Dic3, C2×C3⋊D4, C22×C12, C6×Q8, C23.36C23, Dic3⋊4D4, C23.21D6, C4.Dic6, C4⋊C4⋊7S3, Dic3⋊5D4, C4⋊C4⋊S3, C2×C4×Dic3, C12⋊7D4, Q8×Dic3, C12.23D4, C3×C22⋊Q8, C4⋊C4.187D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, D4⋊2S3, Q8⋊3S3, S3×C23, C23.36C23, C2×D4⋊2S3, C2×Q8⋊3S3, S3×C4○D4, C4⋊C4.187D6
►On 96 points
(1 44 55 88)(2 45 56 89)(3 46 57 90)(4 47 58 85)(5 48 59 86)(6 43 60 87)(7 38 29 82)(8 39 30 83)(9 40 25 84)(10 41 26 79)(11 42 27 80)(12 37 28 81)(13 50 63 96)(14 51 64 91)(15 52 65 92)(16 53 66 93)(17 54 61 94)(18 49 62 95)(19 78 69 34)(20 73 70 35)(21 74 71 36)(22 75 72 31)(23 76 67 32)(24 77 68 33)
(1 79 73 17)(2 62 74 42)(3 81 75 13)(4 64 76 38)(5 83 77 15)(6 66 78 40)(7 47 51 67)(8 24 52 86)(9 43 53 69)(10 20 54 88)(11 45 49 71)(12 22 50 90)(14 32 82 58)(16 34 84 60)(18 36 80 56)(19 25 87 93)(21 27 89 95)(23 29 85 91)(26 70 94 44)(28 72 96 46)(30 68 92 48)(31 63 57 37)(33 65 59 39)(35 61 55 41)
(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 16 73 84)(2 15 74 83)(3 14 75 82)(4 13 76 81)(5 18 77 80)(6 17 78 79)(7 46 51 72)(8 45 52 71)(9 44 53 70)(10 43 54 69)(11 48 49 68)(12 47 50 67)(19 26 87 94)(20 25 88 93)(21 30 89 92)(22 29 90 91)(23 28 85 96)(24 27 86 95)(31 38 57 64)(32 37 58 63)(33 42 59 62)(34 41 60 61)(35 40 55 66)(36 39 56 65)
G:=sub<Sym(96)| (1,44,55,88)(2,45,56,89)(3,46,57,90)(4,47,58,85)(5,48,59,86)(6,43,60,87)(7,38,29,82)(8,39,30,83)(9,40,25,84)(10,41,26,79)(11,42,27,80)(12,37,28,81)(13,50,63,96)(14,51,64,91)(15,52,65,92)(16,53,66,93)(17,54,61,94)(18,49,62,95)(19,78,69,34)(20,73,70,35)(21,74,71,36)(22,75,72,31)(23,76,67,32)(24,77,68,33), (1,79,73,17)(2,62,74,42)(3,81,75,13)(4,64,76,38)(5,83,77,15)(6,66,78,40)(7,47,51,67)(8,24,52,86)(9,43,53,69)(10,20,54,88)(11,45,49,71)(12,22,50,90)(14,32,82,58)(16,34,84,60)(18,36,80,56)(19,25,87,93)(21,27,89,95)(23,29,85,91)(26,70,94,44)(28,72,96,46)(30,68,92,48)(31,63,57,37)(33,65,59,39)(35,61,55,41), (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), (1,16,73,84)(2,15,74,83)(3,14,75,82)(4,13,76,81)(5,18,77,80)(6,17,78,79)(7,46,51,72)(8,45,52,71)(9,44,53,70)(10,43,54,69)(11,48,49,68)(12,47,50,67)(19,26,87,94)(20,25,88,93)(21,30,89,92)(22,29,90,91)(23,28,85,96)(24,27,86,95)(31,38,57,64)(32,37,58,63)(33,42,59,62)(34,41,60,61)(35,40,55,66)(36,39,56,65)>;
G:=Group( (1,44,55,88)(2,45,56,89)(3,46,57,90)(4,47,58,85)(5,48,59,86)(6,43,60,87)(7,38,29,82)(8,39,30,83)(9,40,25,84)(10,41,26,79)(11,42,27,80)(12,37,28,81)(13,50,63,96)(14,51,64,91)(15,52,65,92)(16,53,66,93)(17,54,61,94)(18,49,62,95)(19,78,69,34)(20,73,70,35)(21,74,71,36)(22,75,72,31)(23,76,67,32)(24,77,68,33), (1,79,73,17)(2,62,74,42)(3,81,75,13)(4,64,76,38)(5,83,77,15)(6,66,78,40)(7,47,51,67)(8,24,52,86)(9,43,53,69)(10,20,54,88)(11,45,49,71)(12,22,50,90)(14,32,82,58)(16,34,84,60)(18,36,80,56)(19,25,87,93)(21,27,89,95)(23,29,85,91)(26,70,94,44)(28,72,96,46)(30,68,92,48)(31,63,57,37)(33,65,59,39)(35,61,55,41), (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), (1,16,73,84)(2,15,74,83)(3,14,75,82)(4,13,76,81)(5,18,77,80)(6,17,78,79)(7,46,51,72)(8,45,52,71)(9,44,53,70)(10,43,54,69)(11,48,49,68)(12,47,50,67)(19,26,87,94)(20,25,88,93)(21,30,89,92)(22,29,90,91)(23,28,85,96)(24,27,86,95)(31,38,57,64)(32,37,58,63)(33,42,59,62)(34,41,60,61)(35,40,55,66)(36,39,56,65) );
G=PermutationGroup([[(1,44,55,88),(2,45,56,89),(3,46,57,90),(4,47,58,85),(5,48,59,86),(6,43,60,87),(7,38,29,82),(8,39,30,83),(9,40,25,84),(10,41,26,79),(11,42,27,80),(12,37,28,81),(13,50,63,96),(14,51,64,91),(15,52,65,92),(16,53,66,93),(17,54,61,94),(18,49,62,95),(19,78,69,34),(20,73,70,35),(21,74,71,36),(22,75,72,31),(23,76,67,32),(24,77,68,33)], [(1,79,73,17),(2,62,74,42),(3,81,75,13),(4,64,76,38),(5,83,77,15),(6,66,78,40),(7,47,51,67),(8,24,52,86),(9,43,53,69),(10,20,54,88),(11,45,49,71),(12,22,50,90),(14,32,82,58),(16,34,84,60),(18,36,80,56),(19,25,87,93),(21,27,89,95),(23,29,85,91),(26,70,94,44),(28,72,96,46),(30,68,92,48),(31,63,57,37),(33,65,59,39),(35,61,55,41)], [(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)], [(1,16,73,84),(2,15,74,83),(3,14,75,82),(4,13,76,81),(5,18,77,80),(6,17,78,79),(7,46,51,72),(8,45,52,71),(9,44,53,70),(10,43,54,69),(11,48,49,68),(12,47,50,67),(19,26,87,94),(20,25,88,93),(21,30,89,92),(22,29,90,91),(23,28,85,96),(24,27,86,95),(31,38,57,64),(32,37,58,63),(33,42,59,62),(34,41,60,61),(35,40,55,66),(36,39,56,65)]])
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 | C23.21D6 | C4.Dic6 | 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 |
Matrix representation of C4⋊C4.187D6 ►in 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] >;
C4⋊C4.187D6 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