metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.171D6, C6.342- (1+4), C4⋊Q8⋊9S3, C4.37(S3×D4), C4⋊C4.122D6, D6.48(C2×D4), (C4×S3).13D4, C12.69(C2×D4), D6⋊Q8⋊47C2, (C2×Q8).167D6, C6.98(C22×D4), C42⋊2S3⋊25C2, C42⋊7S3⋊26C2, D6.D4⋊45C2, (C2×C6).268C24, D6⋊C4.49C22, Dic3.53(C2×D4), Dic3⋊Q8⋊26C2, C12.23D4⋊25C2, (C4×C12).209C22, (C2×C12).101C23, (C6×Q8).135C22, (C2×D12).171C22, C22.289(S3×C23), Dic3⋊C4.165C22, (C22×S3).230C23, C2.35(Q8.15D6), C3⋊5(C23.38C23), (C2×Dic3).140C23, (C4×Dic3).159C22, (C2×Dic6).188C22, (C2×S3×Q8)⋊12C2, C2.71(C2×S3×D4), (C3×C4⋊Q8)⋊10C2, (S3×C2×C4).142C22, (C2×Q8⋊3S3).7C2, (C3×C4⋊C4).211C22, (C2×C4).217(C22×S3), SmallGroup(192,1283)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 688 in 270 conjugacy classes, 103 normal (29 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×12], C22, C22 [×10], S3 [×4], C6, C6 [×2], C2×C4, C2×C4 [×6], C2×C4 [×17], D4 [×6], Q8 [×10], C23 [×3], Dic3 [×2], Dic3 [×4], C12 [×2], C12 [×6], D6 [×2], D6 [×8], C2×C6, C42, C42, C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×6], C22×C4 [×5], C2×D4 [×3], C2×Q8 [×2], C2×Q8 [×7], C4○D4 [×4], Dic6 [×6], C4×S3 [×4], C4×S3 [×8], D12 [×6], C2×Dic3, C2×Dic3 [×4], C2×C12, C2×C12 [×6], C3×Q8 [×4], C22×S3, C22×S3 [×2], C42⋊C2, C22⋊Q8 [×4], C22.D4 [×4], C4.4D4 [×2], C4⋊Q8, C4⋊Q8, C22×Q8, C2×C4○D4, C4×Dic3, Dic3⋊C4 [×6], D6⋊C4 [×10], C4×C12, C3×C4⋊C4 [×4], C2×Dic6, C2×Dic6 [×2], S3×C2×C4, S3×C2×C4 [×4], C2×D12, C2×D12 [×2], S3×Q8 [×4], Q8⋊3S3 [×4], C6×Q8 [×2], C23.38C23, C42⋊2S3, C42⋊7S3, D6.D4 [×4], D6⋊Q8 [×4], Dic3⋊Q8, C12.23D4, C3×C4⋊Q8, C2×S3×Q8, C2×Q8⋊3S3, C42.171D6
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C22×S3 [×7], C22×D4, 2- (1+4) [×2], S3×D4 [×2], S3×C23, C23.38C23, C2×S3×D4, Q8.15D6 [×2], C42.171D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=d2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c5 >
►On 96 points
(1 15 7 21)(2 22 8 16)(3 17 9 23)(4 24 10 18)(5 19 11 13)(6 14 12 20)(25 61 31 67)(26 68 32 62)(27 63 33 69)(28 70 34 64)(29 65 35 71)(30 72 36 66)(37 91 43 85)(38 86 44 92)(39 93 45 87)(40 88 46 94)(41 95 47 89)(42 90 48 96)(49 84 55 78)(50 79 56 73)(51 74 57 80)(52 81 58 75)(53 76 59 82)(54 83 60 77)
(1 28 76 86)(2 87 77 29)(3 30 78 88)(4 89 79 31)(5 32 80 90)(6 91 81 33)(7 34 82 92)(8 93 83 35)(9 36 84 94)(10 95 73 25)(11 26 74 96)(12 85 75 27)(13 68 57 42)(14 43 58 69)(15 70 59 44)(16 45 60 71)(17 72 49 46)(18 47 50 61)(19 62 51 48)(20 37 52 63)(21 64 53 38)(22 39 54 65)(23 66 55 40)(24 41 56 67)
(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 94 7 88)(2 87 8 93)(3 92 9 86)(4 85 10 91)(5 90 11 96)(6 95 12 89)(13 48 19 42)(14 41 20 47)(15 46 21 40)(16 39 22 45)(17 44 23 38)(18 37 24 43)(25 75 31 81)(26 80 32 74)(27 73 33 79)(28 78 34 84)(29 83 35 77)(30 76 36 82)(49 70 55 64)(50 63 56 69)(51 68 57 62)(52 61 58 67)(53 66 59 72)(54 71 60 65)
G:=sub<Sym(96)| (1,15,7,21)(2,22,8,16)(3,17,9,23)(4,24,10,18)(5,19,11,13)(6,14,12,20)(25,61,31,67)(26,68,32,62)(27,63,33,69)(28,70,34,64)(29,65,35,71)(30,72,36,66)(37,91,43,85)(38,86,44,92)(39,93,45,87)(40,88,46,94)(41,95,47,89)(42,90,48,96)(49,84,55,78)(50,79,56,73)(51,74,57,80)(52,81,58,75)(53,76,59,82)(54,83,60,77), (1,28,76,86)(2,87,77,29)(3,30,78,88)(4,89,79,31)(5,32,80,90)(6,91,81,33)(7,34,82,92)(8,93,83,35)(9,36,84,94)(10,95,73,25)(11,26,74,96)(12,85,75,27)(13,68,57,42)(14,43,58,69)(15,70,59,44)(16,45,60,71)(17,72,49,46)(18,47,50,61)(19,62,51,48)(20,37,52,63)(21,64,53,38)(22,39,54,65)(23,66,55,40)(24,41,56,67), (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), (1,94,7,88)(2,87,8,93)(3,92,9,86)(4,85,10,91)(5,90,11,96)(6,95,12,89)(13,48,19,42)(14,41,20,47)(15,46,21,40)(16,39,22,45)(17,44,23,38)(18,37,24,43)(25,75,31,81)(26,80,32,74)(27,73,33,79)(28,78,34,84)(29,83,35,77)(30,76,36,82)(49,70,55,64)(50,63,56,69)(51,68,57,62)(52,61,58,67)(53,66,59,72)(54,71,60,65)>;
G:=Group( (1,15,7,21)(2,22,8,16)(3,17,9,23)(4,24,10,18)(5,19,11,13)(6,14,12,20)(25,61,31,67)(26,68,32,62)(27,63,33,69)(28,70,34,64)(29,65,35,71)(30,72,36,66)(37,91,43,85)(38,86,44,92)(39,93,45,87)(40,88,46,94)(41,95,47,89)(42,90,48,96)(49,84,55,78)(50,79,56,73)(51,74,57,80)(52,81,58,75)(53,76,59,82)(54,83,60,77), (1,28,76,86)(2,87,77,29)(3,30,78,88)(4,89,79,31)(5,32,80,90)(6,91,81,33)(7,34,82,92)(8,93,83,35)(9,36,84,94)(10,95,73,25)(11,26,74,96)(12,85,75,27)(13,68,57,42)(14,43,58,69)(15,70,59,44)(16,45,60,71)(17,72,49,46)(18,47,50,61)(19,62,51,48)(20,37,52,63)(21,64,53,38)(22,39,54,65)(23,66,55,40)(24,41,56,67), (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), (1,94,7,88)(2,87,8,93)(3,92,9,86)(4,85,10,91)(5,90,11,96)(6,95,12,89)(13,48,19,42)(14,41,20,47)(15,46,21,40)(16,39,22,45)(17,44,23,38)(18,37,24,43)(25,75,31,81)(26,80,32,74)(27,73,33,79)(28,78,34,84)(29,83,35,77)(30,76,36,82)(49,70,55,64)(50,63,56,69)(51,68,57,62)(52,61,58,67)(53,66,59,72)(54,71,60,65) );
G=PermutationGroup([(1,15,7,21),(2,22,8,16),(3,17,9,23),(4,24,10,18),(5,19,11,13),(6,14,12,20),(25,61,31,67),(26,68,32,62),(27,63,33,69),(28,70,34,64),(29,65,35,71),(30,72,36,66),(37,91,43,85),(38,86,44,92),(39,93,45,87),(40,88,46,94),(41,95,47,89),(42,90,48,96),(49,84,55,78),(50,79,56,73),(51,74,57,80),(52,81,58,75),(53,76,59,82),(54,83,60,77)], [(1,28,76,86),(2,87,77,29),(3,30,78,88),(4,89,79,31),(5,32,80,90),(6,91,81,33),(7,34,82,92),(8,93,83,35),(9,36,84,94),(10,95,73,25),(11,26,74,96),(12,85,75,27),(13,68,57,42),(14,43,58,69),(15,70,59,44),(16,45,60,71),(17,72,49,46),(18,47,50,61),(19,62,51,48),(20,37,52,63),(21,64,53,38),(22,39,54,65),(23,66,55,40),(24,41,56,67)], [(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)], [(1,94,7,88),(2,87,8,93),(3,92,9,86),(4,85,10,91),(5,90,11,96),(6,95,12,89),(13,48,19,42),(14,41,20,47),(15,46,21,40),(16,39,22,45),(17,44,23,38),(18,37,24,43),(25,75,31,81),(26,80,32,74),(27,73,33,79),(28,78,34,84),(29,83,35,77),(30,76,36,82),(49,70,55,64),(50,63,56,69),(51,68,57,62),(52,61,58,67),(53,66,59,72),(54,71,60,65)])
Matrix representation ►G ⊆ GL8(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 4 | 8 | 6 |
| 0 | 0 | 0 | 0 | 9 | 3 | 6 | 5 |
| 0 | 0 | 0 | 0 | 4 | 0 | 10 | 4 |
| 0 | 0 | 0 | 0 | 0 | 9 | 9 | 10 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 11 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 11 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
| 12 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,3,9,4,0,0,0,0,0,4,3,0,9,0,0,0,0,8,6,10,9,0,0,0,0,6,5,4,10],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0],[0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,12,0,0,1,0,0,0,0,11,0,0,1,0,0,0,0,0,11,1,0],[12,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0] >;
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | 2- (1+4) | S3×D4 | Q8.15D6 |
| kernel | C42.171D6 | C42⋊2S3 | C42⋊7S3 | D6.D4 | D6⋊Q8 | Dic3⋊Q8 | C12.23D4 | C3×C4⋊Q8 | C2×S3×Q8 | C2×Q8⋊3S3 | C4⋊Q8 | C4×S3 | C42 | C4⋊C4 | C2×Q8 | C6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 4 | 2 | 2 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{171}D_6 % in TeX
G:=Group("C4^2.171D6"); // GroupNames label
G:=SmallGroup(192,1283);
// by ID
G=gap.SmallGroup(192,1283);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,100,675,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^5>;
// generators/relations