metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.117D6, C6.1062+ 1+4, (C4×D4)⋊25S3, (C4×D12)⋊34C2, (D4×C12)⋊27C2, C4⋊C4.320D6, D6⋊3D4⋊11C2, C12⋊7D4⋊20C2, C12⋊2Q8⋊26C2, (C2×D4).224D6, C2.19(D4○D12), C4.66(C4○D12), (C2×C6).107C24, D6⋊C4.55C22, C22⋊C4.119D6, (C22×C4).231D6, C12.114(C4○D4), (C2×C12).165C23, (C4×C12).161C22, (C6×D4).266C22, C23.26D6⋊9C2, C4.118(D4⋊2S3), C23.11D6⋊11C2, (C2×D12).214C22, (C22×S3).41C23, C4⋊Dic3.302C22, C23.114(C22×S3), (C22×C6).177C23, C22.132(S3×C23), (C2×Dic6).28C22, (C22×C12).111C22, C3⋊2(C22.49C24), (C2×Dic3).209C23, (C4×Dic3).206C22, C6.D4.108C22, C4⋊C4⋊7S3⋊16C2, C6.49(C2×C4○D4), C2.56(C2×C4○D12), (S3×C2×C4).68C22, C2.24(C2×D4⋊2S3), (C3×C4⋊C4).335C22, (C2×C4).163(C22×S3), (C2×C3⋊D4).20C22, (C3×C22⋊C4).130C22, SmallGroup(192,1122)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.117D6
G = < a,b,c,d | a4=b4=c6=1, d2=a2b2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=a2b2c-1 >
Subgroups: 600 in 236 conjugacy classes, 99 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, 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, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C42⋊C2, C4×D4, C4×D4, C4⋊D4, C4.4D4, C4⋊Q8, C4×Dic3, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C2×C3⋊D4, C22×C12, C6×D4, C22.49C24, C12⋊2Q8, C4×D12, C23.11D6, C4⋊C4⋊7S3, C23.26D6, C12⋊7D4, D6⋊3D4, D4×C12, C42.117D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, C4○D12, D4⋊2S3, S3×C23, C22.49C24, C2×C4○D12, C2×D4⋊2S3, D4○D12, C42.117D6
►On 96 points
(1 70 55 20)(2 21 56 71)(3 72 57 22)(4 23 58 67)(5 68 59 24)(6 19 60 69)(7 14 29 64)(8 65 30 15)(9 16 25 66)(10 61 26 17)(11 18 27 62)(12 63 28 13)(31 90 75 46)(32 47 76 85)(33 86 77 48)(34 43 78 87)(35 88 73 44)(36 45 74 89)(37 96 81 50)(38 51 82 91)(39 92 83 52)(40 53 84 93)(41 94 79 54)(42 49 80 95)
(1 41 35 17)(2 42 36 18)(3 37 31 13)(4 38 32 14)(5 39 33 15)(6 40 34 16)(7 67 91 85)(8 68 92 86)(9 69 93 87)(10 70 94 88)(11 71 95 89)(12 72 96 90)(19 53 43 25)(20 54 44 26)(21 49 45 27)(22 50 46 28)(23 51 47 29)(24 52 48 30)(55 79 73 61)(56 80 74 62)(57 81 75 63)(58 82 76 64)(59 83 77 65)(60 84 78 66)
(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 73 78)(2 77 74 5)(3 4 75 76)(7 28 51 96)(8 95 52 27)(9 26 53 94)(10 93 54 25)(11 30 49 92)(12 91 50 29)(13 38 81 64)(14 63 82 37)(15 42 83 62)(16 61 84 41)(17 40 79 66)(18 65 80 39)(19 88 87 20)(21 86 89 24)(22 23 90 85)(31 32 57 58)(33 36 59 56)(34 55 60 35)(43 70 69 44)(45 68 71 48)(46 47 72 67)
G:=sub<Sym(96)| (1,70,55,20)(2,21,56,71)(3,72,57,22)(4,23,58,67)(5,68,59,24)(6,19,60,69)(7,14,29,64)(8,65,30,15)(9,16,25,66)(10,61,26,17)(11,18,27,62)(12,63,28,13)(31,90,75,46)(32,47,76,85)(33,86,77,48)(34,43,78,87)(35,88,73,44)(36,45,74,89)(37,96,81,50)(38,51,82,91)(39,92,83,52)(40,53,84,93)(41,94,79,54)(42,49,80,95), (1,41,35,17)(2,42,36,18)(3,37,31,13)(4,38,32,14)(5,39,33,15)(6,40,34,16)(7,67,91,85)(8,68,92,86)(9,69,93,87)(10,70,94,88)(11,71,95,89)(12,72,96,90)(19,53,43,25)(20,54,44,26)(21,49,45,27)(22,50,46,28)(23,51,47,29)(24,52,48,30)(55,79,73,61)(56,80,74,62)(57,81,75,63)(58,82,76,64)(59,83,77,65)(60,84,78,66), (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), (1,6,73,78)(2,77,74,5)(3,4,75,76)(7,28,51,96)(8,95,52,27)(9,26,53,94)(10,93,54,25)(11,30,49,92)(12,91,50,29)(13,38,81,64)(14,63,82,37)(15,42,83,62)(16,61,84,41)(17,40,79,66)(18,65,80,39)(19,88,87,20)(21,86,89,24)(22,23,90,85)(31,32,57,58)(33,36,59,56)(34,55,60,35)(43,70,69,44)(45,68,71,48)(46,47,72,67)>;
G:=Group( (1,70,55,20)(2,21,56,71)(3,72,57,22)(4,23,58,67)(5,68,59,24)(6,19,60,69)(7,14,29,64)(8,65,30,15)(9,16,25,66)(10,61,26,17)(11,18,27,62)(12,63,28,13)(31,90,75,46)(32,47,76,85)(33,86,77,48)(34,43,78,87)(35,88,73,44)(36,45,74,89)(37,96,81,50)(38,51,82,91)(39,92,83,52)(40,53,84,93)(41,94,79,54)(42,49,80,95), (1,41,35,17)(2,42,36,18)(3,37,31,13)(4,38,32,14)(5,39,33,15)(6,40,34,16)(7,67,91,85)(8,68,92,86)(9,69,93,87)(10,70,94,88)(11,71,95,89)(12,72,96,90)(19,53,43,25)(20,54,44,26)(21,49,45,27)(22,50,46,28)(23,51,47,29)(24,52,48,30)(55,79,73,61)(56,80,74,62)(57,81,75,63)(58,82,76,64)(59,83,77,65)(60,84,78,66), (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), (1,6,73,78)(2,77,74,5)(3,4,75,76)(7,28,51,96)(8,95,52,27)(9,26,53,94)(10,93,54,25)(11,30,49,92)(12,91,50,29)(13,38,81,64)(14,63,82,37)(15,42,83,62)(16,61,84,41)(17,40,79,66)(18,65,80,39)(19,88,87,20)(21,86,89,24)(22,23,90,85)(31,32,57,58)(33,36,59,56)(34,55,60,35)(43,70,69,44)(45,68,71,48)(46,47,72,67) );
G=PermutationGroup([[(1,70,55,20),(2,21,56,71),(3,72,57,22),(4,23,58,67),(5,68,59,24),(6,19,60,69),(7,14,29,64),(8,65,30,15),(9,16,25,66),(10,61,26,17),(11,18,27,62),(12,63,28,13),(31,90,75,46),(32,47,76,85),(33,86,77,48),(34,43,78,87),(35,88,73,44),(36,45,74,89),(37,96,81,50),(38,51,82,91),(39,92,83,52),(40,53,84,93),(41,94,79,54),(42,49,80,95)], [(1,41,35,17),(2,42,36,18),(3,37,31,13),(4,38,32,14),(5,39,33,15),(6,40,34,16),(7,67,91,85),(8,68,92,86),(9,69,93,87),(10,70,94,88),(11,71,95,89),(12,72,96,90),(19,53,43,25),(20,54,44,26),(21,49,45,27),(22,50,46,28),(23,51,47,29),(24,52,48,30),(55,79,73,61),(56,80,74,62),(57,81,75,63),(58,82,76,64),(59,83,77,65),(60,84,78,66)], [(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)], [(1,6,73,78),(2,77,74,5),(3,4,75,76),(7,28,51,96),(8,95,52,27),(9,26,53,94),(10,93,54,25),(11,30,49,92),(12,91,50,29),(13,38,81,64),(14,63,82,37),(15,42,83,62),(16,61,84,41),(17,40,79,66),(18,65,80,39),(19,88,87,20),(21,86,89,24),(22,23,90,85),(31,32,57,58),(33,36,59,56),(34,55,60,35),(43,70,69,44),(45,68,71,48),(46,47,72,67)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 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 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 12 | 12 | 2 | 2 | ··· | 2 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 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 | S3 | D6 | D6 | D6 | D6 | D6 | C4○D4 | C4○D12 | 2+ 1+4 | D4⋊2S3 | D4○D12 |
| kernel | C42.117D6 | C12⋊2Q8 | C4×D12 | C23.11D6 | C4⋊C4⋊7S3 | C23.26D6 | C12⋊7D4 | D6⋊3D4 | D4×C12 | C4×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C12 | C4 | C6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 8 | 1 | 2 | 2 |
Matrix representation of C42.117D6 ►in GL4(𝔽13) generated by
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 9 | 5 |
| 3 | 6 | 0 | 0 |
| 7 | 10 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 11 | 2 | 0 | 0 |
| 11 | 9 | 0 | 0 |
| 0 | 0 | 4 | 3 |
| 0 | 0 | 8 | 9 |
| 2 | 11 | 0 | 0 |
| 9 | 11 | 0 | 0 |
| 0 | 0 | 9 | 10 |
| 0 | 0 | 10 | 4 |
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,8,9,0,0,0,5],[3,7,0,0,6,10,0,0,0,0,12,0,0,0,0,12],[11,11,0,0,2,9,0,0,0,0,4,8,0,0,3,9],[2,9,0,0,11,11,0,0,0,0,9,10,0,0,10,4] >;
C42.117D6 in GAP, Magma, Sage, TeX
C_4^2._{117}D_6 % in TeX
G:=Group("C4^2.117D6"); // GroupNames label
G:=SmallGroup(192,1122);
// by ID
G=gap.SmallGroup(192,1122);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,758,219,1571,192,6278]);
// 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=d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=a^2*b^2*c^-1>;
// generators/relations