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, C4.66(C4○D12), C2.19(D4○D12), (C2×C6).107C24, D6⋊C4.55C22, C22⋊C4.119D6, (C22×C4).231D6, C12.114(C4○D4), (C4×C12).161C22, (C2×C12).165C23, (C6×D4).266C22, C23.26D6⋊9C2, C4.118(D4⋊2S3), C23.11D6⋊11C2, (C2×D12).214C22, (C22×S3).41C23, C4⋊Dic3.302C22, (C22×C6).177C23, C22.132(S3×C23), C23.114(C22×S3), (C2×Dic6).28C22, (C22×C12).111C22, C3⋊2(C22.49C24), (C4×Dic3).206C22, (C2×Dic3).209C23, 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
Subgroups: 600 in 236 conjugacy classes, 99 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C4 [×9], C22, C22 [×12], S3 [×2], C6 [×3], C6 [×2], C2×C4 [×3], C2×C4 [×2], C2×C4 [×14], D4 [×8], Q8 [×2], C23 [×2], C23 [×2], Dic3 [×6], C12 [×4], C12 [×3], D6 [×6], C2×C6, C2×C6 [×6], C42, C42 [×4], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×5], C22×C4 [×2], C22×C4 [×2], C2×D4, C2×D4 [×5], C2×Q8 [×2], Dic6 [×2], C4×S3 [×4], D12 [×2], C2×Dic3 [×6], C3⋊D4 [×4], C2×C12 [×3], C2×C12 [×2], C2×C12 [×4], C3×D4 [×2], C22×S3 [×2], C22×C6 [×2], C42⋊C2 [×4], C4×D4, C4×D4, C4⋊D4 [×4], C4.4D4 [×4], C4⋊Q8, C4×Dic3 [×4], C4⋊Dic3, C4⋊Dic3 [×4], D6⋊C4 [×6], C6.D4 [×4], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6 [×2], S3×C2×C4 [×2], C2×D12, C2×C3⋊D4 [×4], C22×C12 [×2], C6×D4, C22.49C24, C12⋊2Q8, C4×D12, C23.11D6 [×4], C4⋊C4⋊7S3 [×2], C23.26D6 [×2], C12⋊7D4 [×2], D6⋊3D4 [×2], D4×C12, C42.117D6
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), C4○D12 [×2], D4⋊2S3 [×2], S3×C23, C22.49C24, C2×C4○D12, C2×D4⋊2S3, D4○D12, C42.117D6
Generators and relations
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 >
►On 96 points
(1 72 57 22)(2 23 58 67)(3 68 59 24)(4 19 60 69)(5 70 55 20)(6 21 56 71)(7 16 25 66)(8 61 26 17)(9 18 27 62)(10 63 28 13)(11 14 29 64)(12 65 30 15)(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 13)(2 42 36 14)(3 37 31 15)(4 38 32 16)(5 39 33 17)(6 40 34 18)(7 69 91 85)(8 70 92 86)(9 71 93 87)(10 72 94 88)(11 67 95 89)(12 68 96 90)(19 51 47 25)(20 52 48 26)(21 53 43 27)(22 54 44 28)(23 49 45 29)(24 50 46 30)(55 83 77 61)(56 84 78 62)(57 79 73 63)(58 80 74 64)(59 81 75 65)(60 82 76 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 30 51 96)(8 95 52 29)(9 28 53 94)(10 93 54 27)(11 26 49 92)(12 91 50 25)(13 40 79 62)(14 61 80 39)(15 38 81 66)(16 65 82 37)(17 42 83 64)(18 63 84 41)(19 90 85 24)(20 23 86 89)(21 88 87 22)(31 32 59 60)(33 36 55 58)(34 57 56 35)(43 72 71 44)(45 70 67 48)(46 47 68 69)
G:=sub<Sym(96)| (1,72,57,22)(2,23,58,67)(3,68,59,24)(4,19,60,69)(5,70,55,20)(6,21,56,71)(7,16,25,66)(8,61,26,17)(9,18,27,62)(10,63,28,13)(11,14,29,64)(12,65,30,15)(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), (1,41,35,13)(2,42,36,14)(3,37,31,15)(4,38,32,16)(5,39,33,17)(6,40,34,18)(7,69,91,85)(8,70,92,86)(9,71,93,87)(10,72,94,88)(11,67,95,89)(12,68,96,90)(19,51,47,25)(20,52,48,26)(21,53,43,27)(22,54,44,28)(23,49,45,29)(24,50,46,30)(55,83,77,61)(56,84,78,62)(57,79,73,63)(58,80,74,64)(59,81,75,65)(60,82,76,66), (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)(7,30,51,96)(8,95,52,29)(9,28,53,94)(10,93,54,27)(11,26,49,92)(12,91,50,25)(13,40,79,62)(14,61,80,39)(15,38,81,66)(16,65,82,37)(17,42,83,64)(18,63,84,41)(19,90,85,24)(20,23,86,89)(21,88,87,22)(31,32,59,60)(33,36,55,58)(34,57,56,35)(43,72,71,44)(45,70,67,48)(46,47,68,69)>;
G:=Group( (1,72,57,22)(2,23,58,67)(3,68,59,24)(4,19,60,69)(5,70,55,20)(6,21,56,71)(7,16,25,66)(8,61,26,17)(9,18,27,62)(10,63,28,13)(11,14,29,64)(12,65,30,15)(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), (1,41,35,13)(2,42,36,14)(3,37,31,15)(4,38,32,16)(5,39,33,17)(6,40,34,18)(7,69,91,85)(8,70,92,86)(9,71,93,87)(10,72,94,88)(11,67,95,89)(12,68,96,90)(19,51,47,25)(20,52,48,26)(21,53,43,27)(22,54,44,28)(23,49,45,29)(24,50,46,30)(55,83,77,61)(56,84,78,62)(57,79,73,63)(58,80,74,64)(59,81,75,65)(60,82,76,66), (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)(7,30,51,96)(8,95,52,29)(9,28,53,94)(10,93,54,27)(11,26,49,92)(12,91,50,25)(13,40,79,62)(14,61,80,39)(15,38,81,66)(16,65,82,37)(17,42,83,64)(18,63,84,41)(19,90,85,24)(20,23,86,89)(21,88,87,22)(31,32,59,60)(33,36,55,58)(34,57,56,35)(43,72,71,44)(45,70,67,48)(46,47,68,69) );
G=PermutationGroup([(1,72,57,22),(2,23,58,67),(3,68,59,24),(4,19,60,69),(5,70,55,20),(6,21,56,71),(7,16,25,66),(8,61,26,17),(9,18,27,62),(10,63,28,13),(11,14,29,64),(12,65,30,15),(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)], [(1,41,35,13),(2,42,36,14),(3,37,31,15),(4,38,32,16),(5,39,33,17),(6,40,34,18),(7,69,91,85),(8,70,92,86),(9,71,93,87),(10,72,94,88),(11,67,95,89),(12,68,96,90),(19,51,47,25),(20,52,48,26),(21,53,43,27),(22,54,44,28),(23,49,45,29),(24,50,46,30),(55,83,77,61),(56,84,78,62),(57,79,73,63),(58,80,74,64),(59,81,75,65),(60,82,76,66)], [(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),(7,30,51,96),(8,95,52,29),(9,28,53,94),(10,93,54,27),(11,26,49,92),(12,91,50,25),(13,40,79,62),(14,61,80,39),(15,38,81,66),(16,65,82,37),(17,42,83,64),(18,63,84,41),(19,90,85,24),(20,23,86,89),(21,88,87,22),(31,32,59,60),(33,36,55,58),(34,57,56,35),(43,72,71,44),(45,70,67,48),(46,47,68,69)])
Matrix representation ►G ⊆ 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] >;
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 |
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