metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.102D6, (C4×D4)⋊5S3, (D4×C12)⋊6C2, C4⋊C4.277D6, (C4×Dic6)⋊25C2, (C2×D4).207D6, (C2×C6).82C24, C42⋊2S3⋊10C2, Dic3.Q8⋊45C2, D6⋊C4.97C22, C22⋊C4.105D6, (C22×C4).336D6, C12.309(C4○D4), (C4×C12).145C22, (C2×C12).584C23, Dic3.3(C4○D4), C23.14D6.6C2, (C6×D4).301C22, C22.5(C4○D12), C23.26D6⋊5C2, C23.8D6⋊49C2, C4.136(D4⋊2S3), C23.11D6⋊49C2, Dic3.D4⋊49C2, C23.21D6⋊33C2, C23.23D6⋊33C2, (C22×S3).28C23, C4⋊Dic3.295C22, C22.110(S3×C23), (C22×C6).152C23, C23.101(C22×S3), Dic3⋊C4.107C22, (C22×C12).104C22, C3⋊3(C23.36C23), (C2×Dic3).200C23, (C4×Dic3).292C22, (C2×Dic6).235C22, C6.D4.101C22, (C22×Dic3).219C22, (C4×C3⋊D4)⋊3C2, C4⋊C4⋊S3⋊50C2, (C2×C4×Dic3)⋊35C2, C2.17(S3×C4○D4), C6.136(C2×C4○D4), C2.39(C2×C4○D12), (C2×C6).13(C4○D4), C2.18(C2×D4⋊2S3), (S3×C2×C4).197C22, (C3×C4⋊C4).318C22, (C2×C4).153(C22×S3), (C2×C3⋊D4).108C22, (C3×C22⋊C4).119C22, SmallGroup(192,1097)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.102D6
G = < a,b,c,d | a4=b4=c6=1, d2=a2b2, ab=ba, cac-1=a-1, dad-1=ab2, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 520 in 234 conjugacy classes, 101 normal (91 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, C23.36C23, C4×Dic6, C42⋊2S3, Dic3.D4, C23.8D6, C23.11D6, C23.21D6, Dic3.Q8, C4⋊C4⋊S3, C2×C4×Dic3, C23.26D6, C4×C3⋊D4, C23.23D6, C23.14D6, D4×C12, C42.102D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, C4○D12, D4⋊2S3, S3×C23, C23.36C23, C2×C4○D12, C2×D4⋊2S3, S3×C4○D4, C42.102D6
►On 96 points
(1 38 35 16)(2 17 36 39)(3 40 31 18)(4 13 32 41)(5 42 33 14)(6 15 34 37)(7 88 91 72)(8 67 92 89)(9 90 93 68)(10 69 94 85)(11 86 95 70)(12 71 96 87)(19 54 47 28)(20 29 48 49)(21 50 43 30)(22 25 44 51)(23 52 45 26)(24 27 46 53)(55 80 77 64)(56 65 78 81)(57 82 73 66)(58 61 74 83)(59 84 75 62)(60 63 76 79)
(1 69 57 19)(2 70 58 20)(3 71 59 21)(4 72 60 22)(5 67 55 23)(6 68 56 24)(7 63 25 13)(8 64 26 14)(9 65 27 15)(10 66 28 16)(11 61 29 17)(12 62 30 18)(31 87 75 43)(32 88 76 44)(33 89 77 45)(34 90 78 46)(35 85 73 47)(36 86 74 48)(37 93 81 53)(38 94 82 54)(39 95 83 49)(40 96 84 50)(41 91 79 51)(42 92 80 52)
(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 18 73 84)(2 17 74 83)(3 16 75 82)(4 15 76 81)(5 14 77 80)(6 13 78 79)(7 46 51 68)(8 45 52 67)(9 44 53 72)(10 43 54 71)(11 48 49 70)(12 47 50 69)(19 30 85 96)(20 29 86 95)(21 28 87 94)(22 27 88 93)(23 26 89 92)(24 25 90 91)(31 38 59 66)(32 37 60 65)(33 42 55 64)(34 41 56 63)(35 40 57 62)(36 39 58 61)
G:=sub<Sym(96)| (1,38,35,16)(2,17,36,39)(3,40,31,18)(4,13,32,41)(5,42,33,14)(6,15,34,37)(7,88,91,72)(8,67,92,89)(9,90,93,68)(10,69,94,85)(11,86,95,70)(12,71,96,87)(19,54,47,28)(20,29,48,49)(21,50,43,30)(22,25,44,51)(23,52,45,26)(24,27,46,53)(55,80,77,64)(56,65,78,81)(57,82,73,66)(58,61,74,83)(59,84,75,62)(60,63,76,79), (1,69,57,19)(2,70,58,20)(3,71,59,21)(4,72,60,22)(5,67,55,23)(6,68,56,24)(7,63,25,13)(8,64,26,14)(9,65,27,15)(10,66,28,16)(11,61,29,17)(12,62,30,18)(31,87,75,43)(32,88,76,44)(33,89,77,45)(34,90,78,46)(35,85,73,47)(36,86,74,48)(37,93,81,53)(38,94,82,54)(39,95,83,49)(40,96,84,50)(41,91,79,51)(42,92,80,52), (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), (1,18,73,84)(2,17,74,83)(3,16,75,82)(4,15,76,81)(5,14,77,80)(6,13,78,79)(7,46,51,68)(8,45,52,67)(9,44,53,72)(10,43,54,71)(11,48,49,70)(12,47,50,69)(19,30,85,96)(20,29,86,95)(21,28,87,94)(22,27,88,93)(23,26,89,92)(24,25,90,91)(31,38,59,66)(32,37,60,65)(33,42,55,64)(34,41,56,63)(35,40,57,62)(36,39,58,61)>;
G:=Group( (1,38,35,16)(2,17,36,39)(3,40,31,18)(4,13,32,41)(5,42,33,14)(6,15,34,37)(7,88,91,72)(8,67,92,89)(9,90,93,68)(10,69,94,85)(11,86,95,70)(12,71,96,87)(19,54,47,28)(20,29,48,49)(21,50,43,30)(22,25,44,51)(23,52,45,26)(24,27,46,53)(55,80,77,64)(56,65,78,81)(57,82,73,66)(58,61,74,83)(59,84,75,62)(60,63,76,79), (1,69,57,19)(2,70,58,20)(3,71,59,21)(4,72,60,22)(5,67,55,23)(6,68,56,24)(7,63,25,13)(8,64,26,14)(9,65,27,15)(10,66,28,16)(11,61,29,17)(12,62,30,18)(31,87,75,43)(32,88,76,44)(33,89,77,45)(34,90,78,46)(35,85,73,47)(36,86,74,48)(37,93,81,53)(38,94,82,54)(39,95,83,49)(40,96,84,50)(41,91,79,51)(42,92,80,52), (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), (1,18,73,84)(2,17,74,83)(3,16,75,82)(4,15,76,81)(5,14,77,80)(6,13,78,79)(7,46,51,68)(8,45,52,67)(9,44,53,72)(10,43,54,71)(11,48,49,70)(12,47,50,69)(19,30,85,96)(20,29,86,95)(21,28,87,94)(22,27,88,93)(23,26,89,92)(24,25,90,91)(31,38,59,66)(32,37,60,65)(33,42,55,64)(34,41,56,63)(35,40,57,62)(36,39,58,61) );
G=PermutationGroup([[(1,38,35,16),(2,17,36,39),(3,40,31,18),(4,13,32,41),(5,42,33,14),(6,15,34,37),(7,88,91,72),(8,67,92,89),(9,90,93,68),(10,69,94,85),(11,86,95,70),(12,71,96,87),(19,54,47,28),(20,29,48,49),(21,50,43,30),(22,25,44,51),(23,52,45,26),(24,27,46,53),(55,80,77,64),(56,65,78,81),(57,82,73,66),(58,61,74,83),(59,84,75,62),(60,63,76,79)], [(1,69,57,19),(2,70,58,20),(3,71,59,21),(4,72,60,22),(5,67,55,23),(6,68,56,24),(7,63,25,13),(8,64,26,14),(9,65,27,15),(10,66,28,16),(11,61,29,17),(12,62,30,18),(31,87,75,43),(32,88,76,44),(33,89,77,45),(34,90,78,46),(35,85,73,47),(36,86,74,48),(37,93,81,53),(38,94,82,54),(39,95,83,49),(40,96,84,50),(41,91,79,51),(42,92,80,52)], [(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)], [(1,18,73,84),(2,17,74,83),(3,16,75,82),(4,15,76,81),(5,14,77,80),(6,13,78,79),(7,46,51,68),(8,45,52,67),(9,44,53,72),(10,43,54,71),(11,48,49,70),(12,47,50,69),(19,30,85,96),(20,29,86,95),(21,28,87,94),(22,27,88,93),(23,26,89,92),(24,25,90,91),(31,38,59,66),(32,37,60,65),(33,42,55,64),(34,41,56,63),(35,40,57,62),(36,39,58,61)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | ··· | 4Q | 4R | 4S | 4T | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 12 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | D6 | C4○D4 | C4○D4 | C4○D4 | C4○D12 | D4⋊2S3 | S3×C4○D4 |
| kernel | C42.102D6 | C4×Dic6 | C42⋊2S3 | Dic3.D4 | C23.8D6 | C23.11D6 | C23.21D6 | Dic3.Q8 | C4⋊C4⋊S3 | C2×C4×Dic3 | C23.26D6 | C4×C3⋊D4 | C23.23D6 | C23.14D6 | D4×C12 | C4×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | Dic3 | C12 | C2×C6 | C22 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 4 | 4 | 8 | 2 | 2 |
Matrix representation of C42.102D6 ►in GL6(𝔽13)
| 1 | 5 | 0 | 0 | 0 | 0 |
| 0 | 12 | 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 | 0 | 5 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 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 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 12 | 8 | 0 | 0 | 0 | 0 |
| 3 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,5,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,5],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,0,8,0,0,0,0,5,0],[12,3,0,0,0,0,8,1,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,5,0,0,0,0,0,0,5] >;
C42.102D6 in GAP, Magma, Sage, TeX
C_4^2._{102}D_6 % in TeX
G:=Group("C4^2.102D6"); // GroupNames label
G:=SmallGroup(192,1097);
// by ID
G=gap.SmallGroup(192,1097);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,100,1123,794,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations