metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.174D6, C6.372- 1+4, C6.822+ 1+4, C4⋊Q8⋊12S3, (C4×S3)⋊2Q8, C12⋊Q8⋊45C2, D6.6(C2×Q8), C4.41(S3×Q8), C4⋊C4.124D6, C12.55(C2×Q8), D6⋊Q8.4C2, (C2×Q8).110D6, Dic3.7(C2×Q8), C6.49(C22×Q8), Dic3.Q8⋊41C2, C42⋊2S3.8C2, (C2×C6).273C24, D6⋊C4.52C22, D6⋊3Q8.12C2, C12.6Q8⋊24C2, C4.Dic6⋊43C2, Dic3⋊Q8⋊27C2, C2.86(D4⋊6D6), (C4×C12).214C22, (C2×C12).106C23, (C6×Q8).140C22, C4⋊Dic3.252C22, C22.294(S3×C23), Dic3⋊C4.167C22, (C22×S3).234C23, C2.38(Q8.15D6), C3⋊5(C23.41C23), (C2×Dic3).144C23, (C2×Dic6).191C22, (C4×Dic3).162C22, C2.32(C2×S3×Q8), (C3×C4⋊Q8)⋊15C2, (S3×C4⋊C4).13C2, C4⋊C4⋊7S3.15C2, (S3×C2×C4).146C22, (C3×C4⋊C4).216C22, (C2×C4).219(C22×S3), SmallGroup(192,1288)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.174D6
G = < a,b,c,d | a4=b4=1, c6=d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c5 >
Subgroups: 464 in 206 conjugacy classes, 103 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, C2×Q8, Dic6, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C2×C4⋊C4, C42⋊C2, C22⋊Q8, C42.C2, C4⋊Q8, C4⋊Q8, C4×Dic3, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, D6⋊C4, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, S3×C2×C4, C6×Q8, C23.41C23, C12.6Q8, C42⋊2S3, C12⋊Q8, Dic3.Q8, C4.Dic6, S3×C4⋊C4, C4⋊C4⋊7S3, D6⋊Q8, Dic3⋊Q8, D6⋊3Q8, C3×C4⋊Q8, C42.174D6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C24, C22×S3, C22×Q8, 2+ 1+4, 2- 1+4, S3×Q8, S3×C23, C23.41C23, D4⋊6D6, C2×S3×Q8, Q8.15D6, C42.174D6
►On 96 points
(1 87 41 73)(2 74 42 88)(3 89 43 75)(4 76 44 90)(5 91 45 77)(6 78 46 92)(7 93 47 79)(8 80 48 94)(9 95 37 81)(10 82 38 96)(11 85 39 83)(12 84 40 86)(13 71 50 33)(14 34 51 72)(15 61 52 35)(16 36 53 62)(17 63 54 25)(18 26 55 64)(19 65 56 27)(20 28 57 66)(21 67 58 29)(22 30 59 68)(23 69 60 31)(24 32 49 70)
(1 33 47 65)(2 66 48 34)(3 35 37 67)(4 68 38 36)(5 25 39 69)(6 70 40 26)(7 27 41 71)(8 72 42 28)(9 29 43 61)(10 62 44 30)(11 31 45 63)(12 64 46 32)(13 79 56 87)(14 88 57 80)(15 81 58 89)(16 90 59 82)(17 83 60 91)(18 92 49 84)(19 73 50 93)(20 94 51 74)(21 75 52 95)(22 96 53 76)(23 77 54 85)(24 86 55 78)
(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 7 12)(2 11 8 5)(3 4 9 10)(13 55 19 49)(14 60 20 54)(15 53 21 59)(16 58 22 52)(17 51 23 57)(18 56 24 50)(25 72 31 66)(26 65 32 71)(27 70 33 64)(28 63 34 69)(29 68 35 62)(30 61 36 67)(37 38 43 44)(39 48 45 42)(40 41 46 47)(73 78 79 84)(74 83 80 77)(75 76 81 82)(85 94 91 88)(86 87 92 93)(89 90 95 96)
G:=sub<Sym(96)| (1,87,41,73)(2,74,42,88)(3,89,43,75)(4,76,44,90)(5,91,45,77)(6,78,46,92)(7,93,47,79)(8,80,48,94)(9,95,37,81)(10,82,38,96)(11,85,39,83)(12,84,40,86)(13,71,50,33)(14,34,51,72)(15,61,52,35)(16,36,53,62)(17,63,54,25)(18,26,55,64)(19,65,56,27)(20,28,57,66)(21,67,58,29)(22,30,59,68)(23,69,60,31)(24,32,49,70), (1,33,47,65)(2,66,48,34)(3,35,37,67)(4,68,38,36)(5,25,39,69)(6,70,40,26)(7,27,41,71)(8,72,42,28)(9,29,43,61)(10,62,44,30)(11,31,45,63)(12,64,46,32)(13,79,56,87)(14,88,57,80)(15,81,58,89)(16,90,59,82)(17,83,60,91)(18,92,49,84)(19,73,50,93)(20,94,51,74)(21,75,52,95)(22,96,53,76)(23,77,54,85)(24,86,55,78), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,55,19,49)(14,60,20,54)(15,53,21,59)(16,58,22,52)(17,51,23,57)(18,56,24,50)(25,72,31,66)(26,65,32,71)(27,70,33,64)(28,63,34,69)(29,68,35,62)(30,61,36,67)(37,38,43,44)(39,48,45,42)(40,41,46,47)(73,78,79,84)(74,83,80,77)(75,76,81,82)(85,94,91,88)(86,87,92,93)(89,90,95,96)>;
G:=Group( (1,87,41,73)(2,74,42,88)(3,89,43,75)(4,76,44,90)(5,91,45,77)(6,78,46,92)(7,93,47,79)(8,80,48,94)(9,95,37,81)(10,82,38,96)(11,85,39,83)(12,84,40,86)(13,71,50,33)(14,34,51,72)(15,61,52,35)(16,36,53,62)(17,63,54,25)(18,26,55,64)(19,65,56,27)(20,28,57,66)(21,67,58,29)(22,30,59,68)(23,69,60,31)(24,32,49,70), (1,33,47,65)(2,66,48,34)(3,35,37,67)(4,68,38,36)(5,25,39,69)(6,70,40,26)(7,27,41,71)(8,72,42,28)(9,29,43,61)(10,62,44,30)(11,31,45,63)(12,64,46,32)(13,79,56,87)(14,88,57,80)(15,81,58,89)(16,90,59,82)(17,83,60,91)(18,92,49,84)(19,73,50,93)(20,94,51,74)(21,75,52,95)(22,96,53,76)(23,77,54,85)(24,86,55,78), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,55,19,49)(14,60,20,54)(15,53,21,59)(16,58,22,52)(17,51,23,57)(18,56,24,50)(25,72,31,66)(26,65,32,71)(27,70,33,64)(28,63,34,69)(29,68,35,62)(30,61,36,67)(37,38,43,44)(39,48,45,42)(40,41,46,47)(73,78,79,84)(74,83,80,77)(75,76,81,82)(85,94,91,88)(86,87,92,93)(89,90,95,96) );
G=PermutationGroup([[(1,87,41,73),(2,74,42,88),(3,89,43,75),(4,76,44,90),(5,91,45,77),(6,78,46,92),(7,93,47,79),(8,80,48,94),(9,95,37,81),(10,82,38,96),(11,85,39,83),(12,84,40,86),(13,71,50,33),(14,34,51,72),(15,61,52,35),(16,36,53,62),(17,63,54,25),(18,26,55,64),(19,65,56,27),(20,28,57,66),(21,67,58,29),(22,30,59,68),(23,69,60,31),(24,32,49,70)], [(1,33,47,65),(2,66,48,34),(3,35,37,67),(4,68,38,36),(5,25,39,69),(6,70,40,26),(7,27,41,71),(8,72,42,28),(9,29,43,61),(10,62,44,30),(11,31,45,63),(12,64,46,32),(13,79,56,87),(14,88,57,80),(15,81,58,89),(16,90,59,82),(17,83,60,91),(18,92,49,84),(19,73,50,93),(20,94,51,74),(21,75,52,95),(22,96,53,76),(23,77,54,85),(24,86,55,78)], [(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)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,55,19,49),(14,60,20,54),(15,53,21,59),(16,58,22,52),(17,51,23,57),(18,56,24,50),(25,72,31,66),(26,65,32,71),(27,70,33,64),(28,63,34,69),(29,68,35,62),(30,61,36,67),(37,38,43,44),(39,48,45,42),(40,41,46,47),(73,78,79,84),(74,83,80,77),(75,76,81,82),(85,94,91,88),(86,87,92,93),(89,90,95,96)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 4K | ··· | 4P | 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 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | D6 | 2+ 1+4 | 2- 1+4 | S3×Q8 | D4⋊6D6 | Q8.15D6 |
| kernel | C42.174D6 | C12.6Q8 | C42⋊2S3 | C12⋊Q8 | Dic3.Q8 | C4.Dic6 | S3×C4⋊C4 | C4⋊C4⋊7S3 | D6⋊Q8 | Dic3⋊Q8 | D6⋊3Q8 | C3×C4⋊Q8 | C4⋊Q8 | C4×S3 | C42 | C4⋊C4 | C2×Q8 | C6 | C6 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 4 | 1 | 4 | 2 | 1 | 1 | 2 | 2 | 2 |
Matrix representation of C42.174D6 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 8 | 1 | 1 |
| 0 | 0 | 5 | 2 | 11 | 1 |
| 0 | 0 | 12 | 9 | 11 | 5 |
| 0 | 0 | 3 | 12 | 8 | 3 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 9 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 9 |
| 0 | 0 | 0 | 0 | 4 | 11 |
| 4 | 3 | 0 | 0 | 0 | 0 |
| 3 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 2 | 5 | 3 |
| 0 | 0 | 11 | 1 | 5 | 5 |
| 0 | 0 | 0 | 6 | 12 | 11 |
| 0 | 0 | 7 | 0 | 2 | 10 |
| 9 | 10 | 0 | 0 | 0 | 0 |
| 10 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 2 | 5 | 3 |
| 0 | 0 | 12 | 10 | 3 | 5 |
| 0 | 0 | 4 | 8 | 3 | 11 |
| 0 | 0 | 8 | 4 | 1 | 10 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,5,12,3,0,0,8,2,9,12,0,0,1,11,11,8,0,0,1,1,5,3],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,11,4,0,0,0,0,9,2,0,0,0,0,0,0,2,4,0,0,0,0,9,11],[4,3,0,0,0,0,3,9,0,0,0,0,0,0,3,11,0,7,0,0,2,1,6,0,0,0,5,5,12,2,0,0,3,5,11,10],[9,10,0,0,0,0,10,4,0,0,0,0,0,0,3,12,4,8,0,0,2,10,8,4,0,0,5,3,3,1,0,0,3,5,11,10] >;
C42.174D6 in GAP, Magma, Sage, TeX
C_4^2._{174}D_6 % in TeX
G:=Group("C4^2.174D6"); // GroupNames label
G:=SmallGroup(192,1288);
// by ID
G=gap.SmallGroup(192,1288);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,268,675,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^2*b^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