metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.174D6, C6.822+ (1+4), C6.372- (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), (C2×Q8).110D6, D6⋊Q8.4C2, Dic3.7(C2×Q8), C6.49(C22×Q8), Dic3.Q8⋊41C2, (C2×C6).273C24, C42⋊2S3.8C2, D6⋊C4.52C22, D6⋊3Q8.12C2, C12.6Q8⋊24C2, C12.3Q8⋊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×Dic6).191C22, (C2×Dic3).144C23, (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
Subgroups: 464 in 206 conjugacy classes, 103 normal (43 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×14], C22, C22 [×4], S3 [×2], C6 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×13], Q8 [×4], C23, Dic3 [×2], Dic3 [×6], C12 [×2], C12 [×6], D6 [×2], D6 [×2], C2×C6, C42, C42 [×3], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×16], C22×C4 [×3], C2×Q8 [×2], C2×Q8 [×2], Dic6 [×2], C4×S3 [×4], C4×S3 [×2], C2×Dic3 [×3], C2×Dic3 [×4], C2×C12 [×3], C2×C12 [×4], C3×Q8 [×2], C22×S3, C2×C4⋊C4, C42⋊C2 [×2], C22⋊Q8 [×4], C42.C2 [×4], C4⋊Q8, C4⋊Q8 [×3], C4×Dic3, C4×Dic3 [×2], Dic3⋊C4 [×2], Dic3⋊C4 [×10], C4⋊Dic3 [×2], C4⋊Dic3 [×2], D6⋊C4 [×2], D6⋊C4 [×2], C4×C12, C3×C4⋊C4 [×2], C3×C4⋊C4 [×2], C2×Dic6 [×2], S3×C2×C4, S3×C2×C4 [×2], C6×Q8 [×2], C23.41C23, C12.6Q8, C42⋊2S3, C12⋊Q8, Dic3.Q8 [×2], C12.3Q8, S3×C4⋊C4, C4⋊C4⋊7S3, D6⋊Q8 [×2], Dic3⋊Q8 [×2], D6⋊3Q8 [×2], C3×C4⋊Q8, C42.174D6
Quotients:
C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D6 [×7], C2×Q8 [×6], C24, C22×S3 [×7], C22×Q8, 2+ (1+4), 2- (1+4), S3×Q8 [×2], S3×C23, C23.41C23, D4⋊6D6, C2×S3×Q8, Q8.15D6, C42.174D6
Generators and relations
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 >
►On 96 points
(1 13 28 76)(2 77 29 14)(3 15 30 78)(4 79 31 16)(5 17 32 80)(6 81 33 18)(7 19 34 82)(8 83 35 20)(9 21 36 84)(10 73 25 22)(11 23 26 74)(12 75 27 24)(37 90 68 57)(38 58 69 91)(39 92 70 59)(40 60 71 93)(41 94 72 49)(42 50 61 95)(43 96 62 51)(44 52 63 85)(45 86 64 53)(46 54 65 87)(47 88 66 55)(48 56 67 89)
(1 87 34 60)(2 49 35 88)(3 89 36 50)(4 51 25 90)(5 91 26 52)(6 53 27 92)(7 93 28 54)(8 55 29 94)(9 95 30 56)(10 57 31 96)(11 85 32 58)(12 59 33 86)(13 46 82 71)(14 72 83 47)(15 48 84 61)(16 62 73 37)(17 38 74 63)(18 64 75 39)(19 40 76 65)(20 66 77 41)(21 42 78 67)(22 68 79 43)(23 44 80 69)(24 70 81 45)
(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 18 19 24)(14 23 20 17)(15 16 21 22)(25 30 31 36)(26 35 32 29)(27 28 33 34)(37 61 43 67)(38 66 44 72)(39 71 45 65)(40 64 46 70)(41 69 47 63)(42 62 48 68)(49 91 55 85)(50 96 56 90)(51 89 57 95)(52 94 58 88)(53 87 59 93)(54 92 60 86)(73 78 79 84)(74 83 80 77)(75 76 81 82)
G:=sub<Sym(96)| (1,13,28,76)(2,77,29,14)(3,15,30,78)(4,79,31,16)(5,17,32,80)(6,81,33,18)(7,19,34,82)(8,83,35,20)(9,21,36,84)(10,73,25,22)(11,23,26,74)(12,75,27,24)(37,90,68,57)(38,58,69,91)(39,92,70,59)(40,60,71,93)(41,94,72,49)(42,50,61,95)(43,96,62,51)(44,52,63,85)(45,86,64,53)(46,54,65,87)(47,88,66,55)(48,56,67,89), (1,87,34,60)(2,49,35,88)(3,89,36,50)(4,51,25,90)(5,91,26,52)(6,53,27,92)(7,93,28,54)(8,55,29,94)(9,95,30,56)(10,57,31,96)(11,85,32,58)(12,59,33,86)(13,46,82,71)(14,72,83,47)(15,48,84,61)(16,62,73,37)(17,38,74,63)(18,64,75,39)(19,40,76,65)(20,66,77,41)(21,42,78,67)(22,68,79,43)(23,44,80,69)(24,70,81,45), (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)(13,18,19,24)(14,23,20,17)(15,16,21,22)(25,30,31,36)(26,35,32,29)(27,28,33,34)(37,61,43,67)(38,66,44,72)(39,71,45,65)(40,64,46,70)(41,69,47,63)(42,62,48,68)(49,91,55,85)(50,96,56,90)(51,89,57,95)(52,94,58,88)(53,87,59,93)(54,92,60,86)(73,78,79,84)(74,83,80,77)(75,76,81,82)>;
G:=Group( (1,13,28,76)(2,77,29,14)(3,15,30,78)(4,79,31,16)(5,17,32,80)(6,81,33,18)(7,19,34,82)(8,83,35,20)(9,21,36,84)(10,73,25,22)(11,23,26,74)(12,75,27,24)(37,90,68,57)(38,58,69,91)(39,92,70,59)(40,60,71,93)(41,94,72,49)(42,50,61,95)(43,96,62,51)(44,52,63,85)(45,86,64,53)(46,54,65,87)(47,88,66,55)(48,56,67,89), (1,87,34,60)(2,49,35,88)(3,89,36,50)(4,51,25,90)(5,91,26,52)(6,53,27,92)(7,93,28,54)(8,55,29,94)(9,95,30,56)(10,57,31,96)(11,85,32,58)(12,59,33,86)(13,46,82,71)(14,72,83,47)(15,48,84,61)(16,62,73,37)(17,38,74,63)(18,64,75,39)(19,40,76,65)(20,66,77,41)(21,42,78,67)(22,68,79,43)(23,44,80,69)(24,70,81,45), (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)(13,18,19,24)(14,23,20,17)(15,16,21,22)(25,30,31,36)(26,35,32,29)(27,28,33,34)(37,61,43,67)(38,66,44,72)(39,71,45,65)(40,64,46,70)(41,69,47,63)(42,62,48,68)(49,91,55,85)(50,96,56,90)(51,89,57,95)(52,94,58,88)(53,87,59,93)(54,92,60,86)(73,78,79,84)(74,83,80,77)(75,76,81,82) );
G=PermutationGroup([(1,13,28,76),(2,77,29,14),(3,15,30,78),(4,79,31,16),(5,17,32,80),(6,81,33,18),(7,19,34,82),(8,83,35,20),(9,21,36,84),(10,73,25,22),(11,23,26,74),(12,75,27,24),(37,90,68,57),(38,58,69,91),(39,92,70,59),(40,60,71,93),(41,94,72,49),(42,50,61,95),(43,96,62,51),(44,52,63,85),(45,86,64,53),(46,54,65,87),(47,88,66,55),(48,56,67,89)], [(1,87,34,60),(2,49,35,88),(3,89,36,50),(4,51,25,90),(5,91,26,52),(6,53,27,92),(7,93,28,54),(8,55,29,94),(9,95,30,56),(10,57,31,96),(11,85,32,58),(12,59,33,86),(13,46,82,71),(14,72,83,47),(15,48,84,61),(16,62,73,37),(17,38,74,63),(18,64,75,39),(19,40,76,65),(20,66,77,41),(21,42,78,67),(22,68,79,43),(23,44,80,69),(24,70,81,45)], [(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),(13,18,19,24),(14,23,20,17),(15,16,21,22),(25,30,31,36),(26,35,32,29),(27,28,33,34),(37,61,43,67),(38,66,44,72),(39,71,45,65),(40,64,46,70),(41,69,47,63),(42,62,48,68),(49,91,55,85),(50,96,56,90),(51,89,57,95),(52,94,58,88),(53,87,59,93),(54,92,60,86),(73,78,79,84),(74,83,80,77),(75,76,81,82)])
Matrix representation ►G ⊆ 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] >;
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 | C12.3Q8 | 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 |
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