metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊9Q8, C42.175D6, C6.832+ (1+4), C4⋊Q8⋊13S3, C4.19(S3×Q8), C4⋊C4.220D6, C3⋊8(D4⋊3Q8), D6.12(C2×Q8), C12.56(C2×Q8), C4.D12⋊44C2, D6⋊3Q8⋊37C2, (C4×Dic6)⋊53C2, (C4×D12).27C2, (C2×Q8).111D6, C6.50(C22×Q8), (C2×C6).274C24, C12.3Q8⋊44C2, Dic3⋊5D4.14C2, C12.137(C4○D4), C2.87(D4⋊6D6), (C2×C12).107C23, (C4×C12).215C22, D6⋊C4.153C22, C4.40(Q8⋊3S3), (C6×Q8).141C22, (C2×D12).273C22, Dic3⋊C4.62C22, C4⋊Dic3.253C22, C22.295(S3×C23), (C22×S3).235C23, (C2×Dic6).303C22, (C2×Dic3).145C23, (C4×Dic3).163C22, (S3×C4⋊C4)⋊45C2, C2.33(C2×S3×Q8), (C3×C4⋊Q8)⋊16C2, C6.122(C2×C4○D4), (S3×C2×C4).147C22, C2.30(C2×Q8⋊3S3), (C3×C4⋊C4).217C22, (C2×C4).220(C22×S3), SmallGroup(192,1289)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 544 in 228 conjugacy classes, 107 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C4 [×11], C22, C22 [×8], S3 [×4], C6 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×14], D4 [×4], Q8 [×4], C23 [×2], Dic3 [×6], C12 [×4], C12 [×5], D6 [×4], D6 [×4], C2×C6, C42, C42 [×2], C22⋊C4 [×6], C4⋊C4 [×4], C4⋊C4 [×12], C22×C4 [×6], C2×D4, C2×Q8 [×2], C2×Q8, Dic6 [×2], C4×S3 [×8], D12 [×4], C2×Dic3 [×6], C2×C12 [×3], C2×C12 [×4], C3×Q8 [×2], C22×S3 [×2], C2×C4⋊C4 [×2], C4×D4 [×3], C4×Q8, C22⋊Q8 [×6], C42.C2 [×2], C4⋊Q8, C4×Dic3 [×2], Dic3⋊C4 [×6], C4⋊Dic3 [×2], C4⋊Dic3 [×4], D6⋊C4 [×6], C4×C12, C3×C4⋊C4 [×4], C2×Dic6, S3×C2×C4 [×6], C2×D12, C6×Q8 [×2], D4⋊3Q8, C4×Dic6, C4×D12, C12.3Q8 [×2], S3×C4⋊C4 [×2], Dic3⋊5D4 [×2], C4.D12 [×2], D6⋊3Q8 [×4], C3×C4⋊Q8, D12⋊9Q8
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], Q8⋊3S3 [×2], S3×C23, D4⋊3Q8, D4⋊6D6, C2×S3×Q8, C2×Q8⋊3S3, D12⋊9Q8
Generators and relations
G = < a,b,c,d | a12=b2=c4=1, d2=c2, bab=a-1, cac-1=a7, ad=da, cbc-1=dbd-1=a6b, 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 3)(4 12)(5 11)(6 10)(7 9)(13 21)(14 20)(15 19)(16 18)(22 24)(25 33)(26 32)(27 31)(28 30)(34 36)(37 43)(38 42)(39 41)(44 48)(45 47)(49 55)(50 54)(51 53)(56 60)(57 59)(61 67)(62 66)(63 65)(68 72)(69 71)(73 81)(74 80)(75 79)(76 78)(82 84)(85 93)(86 92)(87 91)(88 90)(94 96)
(1 91 57 25)(2 86 58 32)(3 93 59 27)(4 88 60 34)(5 95 49 29)(6 90 50 36)(7 85 51 31)(8 92 52 26)(9 87 53 33)(10 94 54 28)(11 89 55 35)(12 96 56 30)(13 82 66 45)(14 77 67 40)(15 84 68 47)(16 79 69 42)(17 74 70 37)(18 81 71 44)(19 76 72 39)(20 83 61 46)(21 78 62 41)(22 73 63 48)(23 80 64 43)(24 75 65 38)
(1 73 57 48)(2 74 58 37)(3 75 59 38)(4 76 60 39)(5 77 49 40)(6 78 50 41)(7 79 51 42)(8 80 52 43)(9 81 53 44)(10 82 54 45)(11 83 55 46)(12 84 56 47)(13 28 66 94)(14 29 67 95)(15 30 68 96)(16 31 69 85)(17 32 70 86)(18 33 71 87)(19 34 72 88)(20 35 61 89)(21 36 62 90)(22 25 63 91)(23 26 64 92)(24 27 65 93)
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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,21)(14,20)(15,19)(16,18)(22,24)(25,33)(26,32)(27,31)(28,30)(34,36)(37,43)(38,42)(39,41)(44,48)(45,47)(49,55)(50,54)(51,53)(56,60)(57,59)(61,67)(62,66)(63,65)(68,72)(69,71)(73,81)(74,80)(75,79)(76,78)(82,84)(85,93)(86,92)(87,91)(88,90)(94,96), (1,91,57,25)(2,86,58,32)(3,93,59,27)(4,88,60,34)(5,95,49,29)(6,90,50,36)(7,85,51,31)(8,92,52,26)(9,87,53,33)(10,94,54,28)(11,89,55,35)(12,96,56,30)(13,82,66,45)(14,77,67,40)(15,84,68,47)(16,79,69,42)(17,74,70,37)(18,81,71,44)(19,76,72,39)(20,83,61,46)(21,78,62,41)(22,73,63,48)(23,80,64,43)(24,75,65,38), (1,73,57,48)(2,74,58,37)(3,75,59,38)(4,76,60,39)(5,77,49,40)(6,78,50,41)(7,79,51,42)(8,80,52,43)(9,81,53,44)(10,82,54,45)(11,83,55,46)(12,84,56,47)(13,28,66,94)(14,29,67,95)(15,30,68,96)(16,31,69,85)(17,32,70,86)(18,33,71,87)(19,34,72,88)(20,35,61,89)(21,36,62,90)(22,25,63,91)(23,26,64,92)(24,27,65,93)>;
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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,21)(14,20)(15,19)(16,18)(22,24)(25,33)(26,32)(27,31)(28,30)(34,36)(37,43)(38,42)(39,41)(44,48)(45,47)(49,55)(50,54)(51,53)(56,60)(57,59)(61,67)(62,66)(63,65)(68,72)(69,71)(73,81)(74,80)(75,79)(76,78)(82,84)(85,93)(86,92)(87,91)(88,90)(94,96), (1,91,57,25)(2,86,58,32)(3,93,59,27)(4,88,60,34)(5,95,49,29)(6,90,50,36)(7,85,51,31)(8,92,52,26)(9,87,53,33)(10,94,54,28)(11,89,55,35)(12,96,56,30)(13,82,66,45)(14,77,67,40)(15,84,68,47)(16,79,69,42)(17,74,70,37)(18,81,71,44)(19,76,72,39)(20,83,61,46)(21,78,62,41)(22,73,63,48)(23,80,64,43)(24,75,65,38), (1,73,57,48)(2,74,58,37)(3,75,59,38)(4,76,60,39)(5,77,49,40)(6,78,50,41)(7,79,51,42)(8,80,52,43)(9,81,53,44)(10,82,54,45)(11,83,55,46)(12,84,56,47)(13,28,66,94)(14,29,67,95)(15,30,68,96)(16,31,69,85)(17,32,70,86)(18,33,71,87)(19,34,72,88)(20,35,61,89)(21,36,62,90)(22,25,63,91)(23,26,64,92)(24,27,65,93) );
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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,21),(14,20),(15,19),(16,18),(22,24),(25,33),(26,32),(27,31),(28,30),(34,36),(37,43),(38,42),(39,41),(44,48),(45,47),(49,55),(50,54),(51,53),(56,60),(57,59),(61,67),(62,66),(63,65),(68,72),(69,71),(73,81),(74,80),(75,79),(76,78),(82,84),(85,93),(86,92),(87,91),(88,90),(94,96)], [(1,91,57,25),(2,86,58,32),(3,93,59,27),(4,88,60,34),(5,95,49,29),(6,90,50,36),(7,85,51,31),(8,92,52,26),(9,87,53,33),(10,94,54,28),(11,89,55,35),(12,96,56,30),(13,82,66,45),(14,77,67,40),(15,84,68,47),(16,79,69,42),(17,74,70,37),(18,81,71,44),(19,76,72,39),(20,83,61,46),(21,78,62,41),(22,73,63,48),(23,80,64,43),(24,75,65,38)], [(1,73,57,48),(2,74,58,37),(3,75,59,38),(4,76,60,39),(5,77,49,40),(6,78,50,41),(7,79,51,42),(8,80,52,43),(9,81,53,44),(10,82,54,45),(11,83,55,46),(12,84,56,47),(13,28,66,94),(14,29,67,95),(15,30,68,96),(16,31,69,85),(17,32,70,86),(18,33,71,87),(19,34,72,88),(20,35,61,89),(21,36,62,90),(22,25,63,91),(23,26,64,92),(24,27,65,93)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 8 | 10 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 11 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 8 | 10 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 3 |
| 0 | 0 | 0 | 0 | 3 | 9 |
G:=sub<GL(6,GF(13))| [8,0,0,0,0,0,10,5,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,12,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,11,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[8,0,0,0,0,0,10,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,3,0,0,0,0,3,9] >;
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 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | D6 | C4○D4 | 2+ (1+4) | S3×Q8 | Q8⋊3S3 | D4⋊6D6 |
| kernel | D12⋊9Q8 | C4×Dic6 | C4×D12 | C12.3Q8 | S3×C4⋊C4 | Dic3⋊5D4 | C4.D12 | D6⋊3Q8 | C3×C4⋊Q8 | C4⋊Q8 | D12 | C42 | C4⋊C4 | C2×Q8 | C12 | C6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 1 | 4 | 1 | 4 | 2 | 4 | 1 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
D_{12}\rtimes_9Q_8 % in TeX
G:=Group("D12:9Q8"); // GroupNames label
G:=SmallGroup(192,1289);
// by ID
G=gap.SmallGroup(192,1289);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,232,100,570,185,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=d*b*d^-1=a^6*b,d*c*d^-1=c^-1>;
// generators/relations