metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊7Q8, C42.149D6, C6.1312+ (1+4), C12⋊Q8⋊36C2, C4.16(S3×Q8), C4⋊C4.205D6, C3⋊7(D4⋊3Q8), D6.11(C2×Q8), C42.C2⋊5S3, C12.51(C2×Q8), C4.D12⋊35C2, D6⋊Q8⋊34C2, (C4×Dic6)⋊47C2, (C4×D12).24C2, C2.56(D4○D12), C6.43(C22×Q8), (C2×C12).88C23, (C2×C6).234C24, D6⋊C4.40C22, C12.3Q8⋊34C2, Dic3⋊5D4.11C2, (C4×C12).194C22, Dic3.29(C4○D4), (C2×D12).267C22, C4⋊Dic3.379C22, C22.255(S3×C23), Dic3⋊C4.144C22, (C22×S3).221C23, (C2×Dic3).122C23, (C4×Dic3).141C22, (C2×Dic6).251C22, (S3×C4⋊C4)⋊35C2, C2.26(C2×S3×Q8), C2.85(S3×C4○D4), C6.196(C2×C4○D4), (C3×C42.C2)⋊7C2, (S3×C2×C4).217C22, (C2×C4).78(C22×S3), (C3×C4⋊C4).189C22, SmallGroup(192,1249)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 560 in 228 conjugacy classes, 105 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×13], C22, C22 [×8], S3 [×4], C6 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×14], D4 [×4], Q8 [×4], C23 [×2], Dic3 [×2], Dic3 [×5], C12 [×2], C12 [×6], D6 [×4], D6 [×4], C2×C6, C42, C42 [×2], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×10], C22×C4 [×6], C2×D4, C2×Q8 [×3], Dic6 [×4], C4×S3 [×8], D12 [×4], C2×Dic3 [×4], C2×Dic3 [×2], C2×C12 [×3], C2×C12 [×4], C22×S3 [×2], C2×C4⋊C4 [×2], C4×D4 [×3], C4×Q8, C22⋊Q8 [×6], C42.C2, C42.C2, C4⋊Q8, C4×Dic3 [×2], Dic3⋊C4 [×2], Dic3⋊C4 [×4], C4⋊Dic3 [×2], C4⋊Dic3 [×2], D6⋊C4 [×6], C4×C12, C3×C4⋊C4 [×2], C3×C4⋊C4 [×4], C2×Dic6, C2×Dic6 [×2], S3×C2×C4 [×6], C2×D12, D4⋊3Q8, C4×Dic6, C4×D12, C12⋊Q8, C12.3Q8, S3×C4⋊C4 [×2], Dic3⋊5D4 [×2], D6⋊Q8 [×4], C4.D12 [×2], C3×C42.C2, D12⋊7Q8
Quotients:
C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D6 [×7], C2×Q8 [×6], C4○D4 [×2], C24, C22×S3 [×7], C22×Q8, C2×C4○D4, 2+ (1+4), S3×Q8 [×2], S3×C23, D4⋊3Q8, C2×S3×Q8, S3×C4○D4, D4○D12, D12⋊7Q8
Generators and relations
G = < a,b,c,d | a12=b2=c4=1, d2=c2, bab=a-1, ac=ca, dad-1=a5, cbc-1=a6b, dbd-1=a10b, dcd-1=c-1 >
►On 96 points
(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 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 15)(10 14)(11 13)(12 24)(25 42)(26 41)(27 40)(28 39)(29 38)(30 37)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)(49 85)(50 96)(51 95)(52 94)(53 93)(54 92)(55 91)(56 90)(57 89)(58 88)(59 87)(60 86)(61 76)(62 75)(63 74)(64 73)(65 84)(66 83)(67 82)(68 81)(69 80)(70 79)(71 78)(72 77)
(1 44 15 33)(2 45 16 34)(3 46 17 35)(4 47 18 36)(5 48 19 25)(6 37 20 26)(7 38 21 27)(8 39 22 28)(9 40 23 29)(10 41 24 30)(11 42 13 31)(12 43 14 32)(49 61 89 74)(50 62 90 75)(51 63 91 76)(52 64 92 77)(53 65 93 78)(54 66 94 79)(55 67 95 80)(56 68 96 81)(57 69 85 82)(58 70 86 83)(59 71 87 84)(60 72 88 73)
(1 80 15 67)(2 73 16 72)(3 78 17 65)(4 83 18 70)(5 76 19 63)(6 81 20 68)(7 74 21 61)(8 79 22 66)(9 84 23 71)(10 77 24 64)(11 82 13 69)(12 75 14 62)(25 51 48 91)(26 56 37 96)(27 49 38 89)(28 54 39 94)(29 59 40 87)(30 52 41 92)(31 57 42 85)(32 50 43 90)(33 55 44 95)(34 60 45 88)(35 53 46 93)(36 58 47 86)
G:=sub<Sym(96)| (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)(12,24)(25,42)(26,41)(27,40)(28,39)(29,38)(30,37)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(49,85)(50,96)(51,95)(52,94)(53,93)(54,92)(55,91)(56,90)(57,89)(58,88)(59,87)(60,86)(61,76)(62,75)(63,74)(64,73)(65,84)(66,83)(67,82)(68,81)(69,80)(70,79)(71,78)(72,77), (1,44,15,33)(2,45,16,34)(3,46,17,35)(4,47,18,36)(5,48,19,25)(6,37,20,26)(7,38,21,27)(8,39,22,28)(9,40,23,29)(10,41,24,30)(11,42,13,31)(12,43,14,32)(49,61,89,74)(50,62,90,75)(51,63,91,76)(52,64,92,77)(53,65,93,78)(54,66,94,79)(55,67,95,80)(56,68,96,81)(57,69,85,82)(58,70,86,83)(59,71,87,84)(60,72,88,73), (1,80,15,67)(2,73,16,72)(3,78,17,65)(4,83,18,70)(5,76,19,63)(6,81,20,68)(7,74,21,61)(8,79,22,66)(9,84,23,71)(10,77,24,64)(11,82,13,69)(12,75,14,62)(25,51,48,91)(26,56,37,96)(27,49,38,89)(28,54,39,94)(29,59,40,87)(30,52,41,92)(31,57,42,85)(32,50,43,90)(33,55,44,95)(34,60,45,88)(35,53,46,93)(36,58,47,86)>;
G:=Group( (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)(12,24)(25,42)(26,41)(27,40)(28,39)(29,38)(30,37)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(49,85)(50,96)(51,95)(52,94)(53,93)(54,92)(55,91)(56,90)(57,89)(58,88)(59,87)(60,86)(61,76)(62,75)(63,74)(64,73)(65,84)(66,83)(67,82)(68,81)(69,80)(70,79)(71,78)(72,77), (1,44,15,33)(2,45,16,34)(3,46,17,35)(4,47,18,36)(5,48,19,25)(6,37,20,26)(7,38,21,27)(8,39,22,28)(9,40,23,29)(10,41,24,30)(11,42,13,31)(12,43,14,32)(49,61,89,74)(50,62,90,75)(51,63,91,76)(52,64,92,77)(53,65,93,78)(54,66,94,79)(55,67,95,80)(56,68,96,81)(57,69,85,82)(58,70,86,83)(59,71,87,84)(60,72,88,73), (1,80,15,67)(2,73,16,72)(3,78,17,65)(4,83,18,70)(5,76,19,63)(6,81,20,68)(7,74,21,61)(8,79,22,66)(9,84,23,71)(10,77,24,64)(11,82,13,69)(12,75,14,62)(25,51,48,91)(26,56,37,96)(27,49,38,89)(28,54,39,94)(29,59,40,87)(30,52,41,92)(31,57,42,85)(32,50,43,90)(33,55,44,95)(34,60,45,88)(35,53,46,93)(36,58,47,86) );
G=PermutationGroup([(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)], [(1,23),(2,22),(3,21),(4,20),(5,19),(6,18),(7,17),(8,16),(9,15),(10,14),(11,13),(12,24),(25,42),(26,41),(27,40),(28,39),(29,38),(30,37),(31,48),(32,47),(33,46),(34,45),(35,44),(36,43),(49,85),(50,96),(51,95),(52,94),(53,93),(54,92),(55,91),(56,90),(57,89),(58,88),(59,87),(60,86),(61,76),(62,75),(63,74),(64,73),(65,84),(66,83),(67,82),(68,81),(69,80),(70,79),(71,78),(72,77)], [(1,44,15,33),(2,45,16,34),(3,46,17,35),(4,47,18,36),(5,48,19,25),(6,37,20,26),(7,38,21,27),(8,39,22,28),(9,40,23,29),(10,41,24,30),(11,42,13,31),(12,43,14,32),(49,61,89,74),(50,62,90,75),(51,63,91,76),(52,64,92,77),(53,65,93,78),(54,66,94,79),(55,67,95,80),(56,68,96,81),(57,69,85,82),(58,70,86,83),(59,71,87,84),(60,72,88,73)], [(1,80,15,67),(2,73,16,72),(3,78,17,65),(4,83,18,70),(5,76,19,63),(6,81,20,68),(7,74,21,61),(8,79,22,66),(9,84,23,71),(10,77,24,64),(11,82,13,69),(12,75,14,62),(25,51,48,91),(26,56,37,96),(27,49,38,89),(28,54,39,94),(29,59,40,87),(30,52,41,92),(31,57,42,85),(32,50,43,90),(33,55,44,95),(34,60,45,88),(35,53,46,93),(36,58,47,86)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 12 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 10 | 0 | 0 |
| 0 | 0 | 10 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(6,GF(13))| [12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,8,0,0,0,0,8,0],[12,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,4,10,0,0,0,0,10,9,0,0,0,0,0,0,0,12,0,0,0,0,12,0] >;
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 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 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | C4○D4 | 2+ (1+4) | S3×Q8 | S3×C4○D4 | D4○D12 |
| kernel | D12⋊7Q8 | C4×Dic6 | C4×D12 | C12⋊Q8 | C12.3Q8 | S3×C4⋊C4 | Dic3⋊5D4 | D6⋊Q8 | C4.D12 | C3×C42.C2 | C42.C2 | D12 | C42 | C4⋊C4 | Dic3 | C6 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 1 | 1 | 4 | 1 | 6 | 4 | 1 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
D_{12}\rtimes_7Q_8 % in TeX
G:=Group("D12:7Q8"); // GroupNames label
G:=SmallGroup(192,1249);
// by ID
G=gap.SmallGroup(192,1249);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,219,184,1571,297,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^5,c*b*c^-1=a^6*b,d*b*d^-1=a^10*b,d*c*d^-1=c^-1>;
// generators/relations