metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.241D6, C4⋊Q8⋊20S3, (C4×S3)⋊5Q8, D6.5(C2×Q8), C4.40(S3×Q8), C4⋊C4.219D6, C12.54(C2×Q8), C12⋊2Q8⋊36C2, (Q8×Dic3)⋊22C2, (C2×Q8).171D6, C6.48(C22×Q8), (S3×C42).10C2, (C2×C6).272C24, D6⋊C4.51C22, D6⋊3Q8.11C2, C4.D12.13C2, C12.3Q8⋊42C2, Dic3.17(C2×Q8), Dic6⋊C4⋊42C2, C12.136(C4○D4), C4.41(D4⋊2S3), (C2×C12).105C23, (C4×C12).213C22, C4.22(Q8⋊3S3), (C6×Q8).139C22, Dic3⋊C4.61C22, C4⋊Dic3.251C22, C22.293(S3×C23), (C22×S3).233C23, C3⋊6(C23.37C23), (C2×Dic3).143C23, (C2×Dic6).190C22, (C4×Dic3).161C22, C2.31(C2×S3×Q8), (C3×C4⋊Q8)⋊14C2, C6.100(C2×C4○D4), C4⋊C4⋊7S3.14C2, C2.64(C2×D4⋊2S3), (S3×C2×C4).252C22, C2.29(C2×Q8⋊3S3), (C3×C4⋊C4).215C22, (C2×C4).600(C22×S3), SmallGroup(192,1287)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 464 in 222 conjugacy classes, 111 normal (33 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×6], C4 [×12], C22, C22 [×4], S3 [×2], C6 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×15], Q8 [×8], C23, Dic3 [×2], Dic3 [×6], C12 [×6], C12 [×4], D6 [×2], D6 [×2], C2×C6, C42, C42 [×7], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×12], C22×C4 [×3], C2×Q8 [×2], C2×Q8 [×2], Dic6 [×4], C4×S3 [×4], C4×S3 [×4], C2×Dic3 [×3], C2×Dic3 [×4], C2×C12 [×3], C2×C12 [×4], C3×Q8 [×4], C22×S3, C2×C42, C42⋊C2 [×2], C4×Q8 [×4], C22⋊Q8 [×4], C42.C2 [×2], C4⋊Q8, C4⋊Q8, C4×Dic3 [×3], C4×Dic3 [×4], Dic3⋊C4 [×4], C4⋊Dic3 [×8], D6⋊C4 [×4], C4×C12, C3×C4⋊C4 [×4], C2×Dic6 [×2], S3×C2×C4 [×3], C6×Q8 [×2], C23.37C23, C12⋊2Q8, S3×C42, Dic6⋊C4 [×2], C12.3Q8 [×2], C4⋊C4⋊7S3 [×2], C4.D12 [×2], Q8×Dic3 [×2], D6⋊3Q8 [×2], C3×C4⋊Q8, C42.241D6
Quotients:
C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D6 [×7], C2×Q8 [×6], C4○D4 [×4], C24, C22×S3 [×7], C22×Q8, C2×C4○D4 [×2], D4⋊2S3 [×2], S3×Q8 [×2], Q8⋊3S3 [×2], S3×C23, C23.37C23, C2×D4⋊2S3, C2×S3×Q8, C2×Q8⋊3S3, C42.241D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=b2, d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=a2c5 >
►On 96 points
(1 28 49 41)(2 42 50 29)(3 30 51 43)(4 44 52 31)(5 32 53 45)(6 46 54 33)(7 34 55 47)(8 48 56 35)(9 36 57 37)(10 38 58 25)(11 26 59 39)(12 40 60 27)(13 82 95 65)(14 66 96 83)(15 84 85 67)(16 68 86 73)(17 74 87 69)(18 70 88 75)(19 76 89 71)(20 72 90 77)(21 78 91 61)(22 62 92 79)(23 80 93 63)(24 64 94 81)
(1 66 7 72)(2 61 8 67)(3 68 9 62)(4 63 10 69)(5 70 11 64)(6 65 12 71)(13 40 19 46)(14 47 20 41)(15 42 21 48)(16 37 22 43)(17 44 23 38)(18 39 24 45)(25 87 31 93)(26 94 32 88)(27 89 33 95)(28 96 34 90)(29 91 35 85)(30 86 36 92)(49 83 55 77)(50 78 56 84)(51 73 57 79)(52 80 58 74)(53 75 59 81)(54 82 60 76)
(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 55 60)(2 59 56 5)(3 4 57 58)(7 12 49 54)(8 53 50 11)(9 10 51 52)(13 96 89 20)(14 19 90 95)(15 94 91 18)(16 17 92 93)(21 88 85 24)(22 23 86 87)(25 43 44 36)(26 35 45 42)(27 41 46 34)(28 33 47 40)(29 39 48 32)(30 31 37 38)(61 75 84 64)(62 63 73 74)(65 83 76 72)(66 71 77 82)(67 81 78 70)(68 69 79 80)
G:=sub<Sym(96)| (1,28,49,41)(2,42,50,29)(3,30,51,43)(4,44,52,31)(5,32,53,45)(6,46,54,33)(7,34,55,47)(8,48,56,35)(9,36,57,37)(10,38,58,25)(11,26,59,39)(12,40,60,27)(13,82,95,65)(14,66,96,83)(15,84,85,67)(16,68,86,73)(17,74,87,69)(18,70,88,75)(19,76,89,71)(20,72,90,77)(21,78,91,61)(22,62,92,79)(23,80,93,63)(24,64,94,81), (1,66,7,72)(2,61,8,67)(3,68,9,62)(4,63,10,69)(5,70,11,64)(6,65,12,71)(13,40,19,46)(14,47,20,41)(15,42,21,48)(16,37,22,43)(17,44,23,38)(18,39,24,45)(25,87,31,93)(26,94,32,88)(27,89,33,95)(28,96,34,90)(29,91,35,85)(30,86,36,92)(49,83,55,77)(50,78,56,84)(51,73,57,79)(52,80,58,74)(53,75,59,81)(54,82,60,76), (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), (1,6,55,60)(2,59,56,5)(3,4,57,58)(7,12,49,54)(8,53,50,11)(9,10,51,52)(13,96,89,20)(14,19,90,95)(15,94,91,18)(16,17,92,93)(21,88,85,24)(22,23,86,87)(25,43,44,36)(26,35,45,42)(27,41,46,34)(28,33,47,40)(29,39,48,32)(30,31,37,38)(61,75,84,64)(62,63,73,74)(65,83,76,72)(66,71,77,82)(67,81,78,70)(68,69,79,80)>;
G:=Group( (1,28,49,41)(2,42,50,29)(3,30,51,43)(4,44,52,31)(5,32,53,45)(6,46,54,33)(7,34,55,47)(8,48,56,35)(9,36,57,37)(10,38,58,25)(11,26,59,39)(12,40,60,27)(13,82,95,65)(14,66,96,83)(15,84,85,67)(16,68,86,73)(17,74,87,69)(18,70,88,75)(19,76,89,71)(20,72,90,77)(21,78,91,61)(22,62,92,79)(23,80,93,63)(24,64,94,81), (1,66,7,72)(2,61,8,67)(3,68,9,62)(4,63,10,69)(5,70,11,64)(6,65,12,71)(13,40,19,46)(14,47,20,41)(15,42,21,48)(16,37,22,43)(17,44,23,38)(18,39,24,45)(25,87,31,93)(26,94,32,88)(27,89,33,95)(28,96,34,90)(29,91,35,85)(30,86,36,92)(49,83,55,77)(50,78,56,84)(51,73,57,79)(52,80,58,74)(53,75,59,81)(54,82,60,76), (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), (1,6,55,60)(2,59,56,5)(3,4,57,58)(7,12,49,54)(8,53,50,11)(9,10,51,52)(13,96,89,20)(14,19,90,95)(15,94,91,18)(16,17,92,93)(21,88,85,24)(22,23,86,87)(25,43,44,36)(26,35,45,42)(27,41,46,34)(28,33,47,40)(29,39,48,32)(30,31,37,38)(61,75,84,64)(62,63,73,74)(65,83,76,72)(66,71,77,82)(67,81,78,70)(68,69,79,80) );
G=PermutationGroup([(1,28,49,41),(2,42,50,29),(3,30,51,43),(4,44,52,31),(5,32,53,45),(6,46,54,33),(7,34,55,47),(8,48,56,35),(9,36,57,37),(10,38,58,25),(11,26,59,39),(12,40,60,27),(13,82,95,65),(14,66,96,83),(15,84,85,67),(16,68,86,73),(17,74,87,69),(18,70,88,75),(19,76,89,71),(20,72,90,77),(21,78,91,61),(22,62,92,79),(23,80,93,63),(24,64,94,81)], [(1,66,7,72),(2,61,8,67),(3,68,9,62),(4,63,10,69),(5,70,11,64),(6,65,12,71),(13,40,19,46),(14,47,20,41),(15,42,21,48),(16,37,22,43),(17,44,23,38),(18,39,24,45),(25,87,31,93),(26,94,32,88),(27,89,33,95),(28,96,34,90),(29,91,35,85),(30,86,36,92),(49,83,55,77),(50,78,56,84),(51,73,57,79),(52,80,58,74),(53,75,59,81),(54,82,60,76)], [(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)], [(1,6,55,60),(2,59,56,5),(3,4,57,58),(7,12,49,54),(8,53,50,11),(9,10,51,52),(13,96,89,20),(14,19,90,95),(15,94,91,18),(16,17,92,93),(21,88,85,24),(22,23,86,87),(25,43,44,36),(26,35,45,42),(27,41,46,34),(28,33,47,40),(29,39,48,32),(30,31,37,38),(61,75,84,64),(62,63,73,74),(65,83,76,72),(66,71,77,82),(67,81,78,70),(68,69,79,80)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 5 | 0 | 0 | 0 | 0 | 0 |
| 5 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 5 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 11 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 12 | 2 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(13))| [5,5,0,0,0,0,0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[5,5,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,1,0,0,0,0,11,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[12,0,0,0,0,0,2,1,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,12,12] >;
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 4S | 4T | 4U | 4V | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 2 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | D6 | C4○D4 | D4⋊2S3 | S3×Q8 | Q8⋊3S3 |
| kernel | C42.241D6 | C12⋊2Q8 | S3×C42 | Dic6⋊C4 | C12.3Q8 | C4⋊C4⋊7S3 | C4.D12 | Q8×Dic3 | D6⋊3Q8 | C3×C4⋊Q8 | C4⋊Q8 | C4×S3 | C42 | C4⋊C4 | C2×Q8 | C12 | C4 | C4 | C4 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 4 | 1 | 4 | 2 | 8 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{241}D_6 % in TeX
G:=Group("C4^2.241D6"); // GroupNames label
G:=SmallGroup(192,1287);
// by ID
G=gap.SmallGroup(192,1287);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,100,1123,570,185,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=b^2,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*c^5>;
// generators/relations