metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊6D12, C42.112D6, C6.612- (1+4), (C3×D4)⋊11D4, (C4×D4)⋊17S3, (C4×D12)⋊31C2, (D4×C12)⋊19C2, C3⋊3(D4⋊6D4), C4⋊C4.284D6, C12.55(C2×D4), C4.23(C2×D12), C12⋊14(C4○D4), C4⋊5(D4⋊2S3), C12⋊7D4⋊10C2, C4.D12⋊15C2, C12⋊2Q8⋊25C2, (C2×D4).249D6, (C2×C6).99C24, D6⋊C4.5C22, C6.17(C22×D4), C22.2(C2×D12), C22⋊C4.113D6, C2.18(Q8○D12), C2.19(C22×D12), (C22×C4).227D6, (C2×C12).160C23, (C4×C12).155C22, (C6×D4).260C22, C22.D12⋊6C2, C4⋊Dic3.39C22, (C2×D12).212C22, (C22×S3).34C23, C23.183(C22×S3), (C22×C6).169C23, (C22×C12).81C22, C22.124(S3×C23), (C2×Dic3).206C23, (C2×Dic6).144C22, (C22×Dic3).97C22, (C2×C6).2(C2×D4), C6.74(C2×C4○D4), (C2×D4⋊2S3)⋊4C2, (C2×C4⋊Dic3)⋊25C2, (S3×C2×C4).65C22, C2.22(C2×D4⋊2S3), (C3×C4⋊C4).329C22, (C2×C4).732(C22×S3), (C2×C3⋊D4).15C22, (C3×C22⋊C4).106C22, SmallGroup(192,1114)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 728 in 292 conjugacy classes, 115 normal (29 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×4], C4 [×9], C22, C22 [×4], C22 [×10], S3 [×2], C6 [×3], C6 [×4], C2×C4 [×3], C2×C4 [×2], C2×C4 [×22], D4 [×4], D4 [×10], Q8 [×4], C23 [×2], C23 [×2], Dic3 [×6], C12 [×4], C12 [×3], D6 [×6], C2×C6, C2×C6 [×4], C2×C6 [×4], C42, C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×9], C22×C4 [×2], C22×C4 [×6], C2×D4, C2×D4 [×5], C2×Q8 [×2], C4○D4 [×8], Dic6 [×4], C4×S3 [×4], D12 [×2], C2×Dic3 [×6], C2×Dic3 [×8], C3⋊D4 [×8], C2×C12 [×3], C2×C12 [×2], C2×C12 [×4], C3×D4 [×4], C22×S3 [×2], C22×C6 [×2], C2×C4⋊C4 [×2], C4×D4, C4×D4, C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×4], C4⋊Q8, C2×C4○D4 [×2], C4⋊Dic3, C4⋊Dic3 [×8], D6⋊C4 [×6], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6 [×2], S3×C2×C4 [×2], C2×D12, D4⋊2S3 [×8], C22×Dic3 [×4], C2×C3⋊D4 [×4], C22×C12 [×2], C6×D4, D4⋊6D4, C12⋊2Q8, C4×D12, C22.D12 [×4], C4.D12 [×2], C2×C4⋊Dic3 [×2], C12⋊7D4 [×2], D4×C12, C2×D4⋊2S3 [×2], D4⋊6D12
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×2], C24, D12 [×4], C22×S3 [×7], C22×D4, C2×C4○D4, 2- (1+4), C2×D12 [×6], D4⋊2S3 [×2], S3×C23, D4⋊6D4, C22×D12, C2×D4⋊2S3, Q8○D12, D4⋊6D12
Generators and relations
G = < a,b,c,d | a4=b2=c12=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >
►On 96 points
(1 19 72 33)(2 20 61 34)(3 21 62 35)(4 22 63 36)(5 23 64 25)(6 24 65 26)(7 13 66 27)(8 14 67 28)(9 15 68 29)(10 16 69 30)(11 17 70 31)(12 18 71 32)(37 51 73 85)(38 52 74 86)(39 53 75 87)(40 54 76 88)(41 55 77 89)(42 56 78 90)(43 57 79 91)(44 58 80 92)(45 59 81 93)(46 60 82 94)(47 49 83 95)(48 50 84 96)
(1 48)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 90)(14 91)(15 92)(16 93)(17 94)(18 95)(19 96)(20 85)(21 86)(22 87)(23 88)(24 89)(25 54)(26 55)(27 56)(28 57)(29 58)(30 59)(31 60)(32 49)(33 50)(34 51)(35 52)(36 53)(61 73)(62 74)(63 75)(64 76)(65 77)(66 78)(67 79)(68 80)(69 81)(70 82)(71 83)(72 84)
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)(25 28)(26 27)(29 36)(30 35)(31 34)(32 33)(37 82)(38 81)(39 80)(40 79)(41 78)(42 77)(43 76)(44 75)(45 74)(46 73)(47 84)(48 83)(49 96)(50 95)(51 94)(52 93)(53 92)(54 91)(55 90)(56 89)(57 88)(58 87)(59 86)(60 85)(61 70)(62 69)(63 68)(64 67)(65 66)(71 72)
G:=sub<Sym(96)| (1,19,72,33)(2,20,61,34)(3,21,62,35)(4,22,63,36)(5,23,64,25)(6,24,65,26)(7,13,66,27)(8,14,67,28)(9,15,68,29)(10,16,69,30)(11,17,70,31)(12,18,71,32)(37,51,73,85)(38,52,74,86)(39,53,75,87)(40,54,76,88)(41,55,77,89)(42,56,78,90)(43,57,79,91)(44,58,80,92)(45,59,81,93)(46,60,82,94)(47,49,83,95)(48,50,84,96), (1,48)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,90)(14,91)(15,92)(16,93)(17,94)(18,95)(19,96)(20,85)(21,86)(22,87)(23,88)(24,89)(25,54)(26,55)(27,56)(28,57)(29,58)(30,59)(31,60)(32,49)(33,50)(34,51)(35,52)(36,53)(61,73)(62,74)(63,75)(64,76)(65,77)(66,78)(67,79)(68,80)(69,81)(70,82)(71,83)(72,84), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(25,28)(26,27)(29,36)(30,35)(31,34)(32,33)(37,82)(38,81)(39,80)(40,79)(41,78)(42,77)(43,76)(44,75)(45,74)(46,73)(47,84)(48,83)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(57,88)(58,87)(59,86)(60,85)(61,70)(62,69)(63,68)(64,67)(65,66)(71,72)>;
G:=Group( (1,19,72,33)(2,20,61,34)(3,21,62,35)(4,22,63,36)(5,23,64,25)(6,24,65,26)(7,13,66,27)(8,14,67,28)(9,15,68,29)(10,16,69,30)(11,17,70,31)(12,18,71,32)(37,51,73,85)(38,52,74,86)(39,53,75,87)(40,54,76,88)(41,55,77,89)(42,56,78,90)(43,57,79,91)(44,58,80,92)(45,59,81,93)(46,60,82,94)(47,49,83,95)(48,50,84,96), (1,48)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,90)(14,91)(15,92)(16,93)(17,94)(18,95)(19,96)(20,85)(21,86)(22,87)(23,88)(24,89)(25,54)(26,55)(27,56)(28,57)(29,58)(30,59)(31,60)(32,49)(33,50)(34,51)(35,52)(36,53)(61,73)(62,74)(63,75)(64,76)(65,77)(66,78)(67,79)(68,80)(69,81)(70,82)(71,83)(72,84), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(25,28)(26,27)(29,36)(30,35)(31,34)(32,33)(37,82)(38,81)(39,80)(40,79)(41,78)(42,77)(43,76)(44,75)(45,74)(46,73)(47,84)(48,83)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(57,88)(58,87)(59,86)(60,85)(61,70)(62,69)(63,68)(64,67)(65,66)(71,72) );
G=PermutationGroup([(1,19,72,33),(2,20,61,34),(3,21,62,35),(4,22,63,36),(5,23,64,25),(6,24,65,26),(7,13,66,27),(8,14,67,28),(9,15,68,29),(10,16,69,30),(11,17,70,31),(12,18,71,32),(37,51,73,85),(38,52,74,86),(39,53,75,87),(40,54,76,88),(41,55,77,89),(42,56,78,90),(43,57,79,91),(44,58,80,92),(45,59,81,93),(46,60,82,94),(47,49,83,95),(48,50,84,96)], [(1,48),(2,37),(3,38),(4,39),(5,40),(6,41),(7,42),(8,43),(9,44),(10,45),(11,46),(12,47),(13,90),(14,91),(15,92),(16,93),(17,94),(18,95),(19,96),(20,85),(21,86),(22,87),(23,88),(24,89),(25,54),(26,55),(27,56),(28,57),(29,58),(30,59),(31,60),(32,49),(33,50),(34,51),(35,52),(36,53),(61,73),(62,74),(63,75),(64,76),(65,77),(66,78),(67,79),(68,80),(69,81),(70,82),(71,83),(72,84)], [(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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,24),(14,23),(15,22),(16,21),(17,20),(18,19),(25,28),(26,27),(29,36),(30,35),(31,34),(32,33),(37,82),(38,81),(39,80),(40,79),(41,78),(42,77),(43,76),(44,75),(45,74),(46,73),(47,84),(48,83),(49,96),(50,95),(51,94),(52,93),(53,92),(54,91),(55,90),(56,89),(57,88),(58,87),(59,86),(60,85),(61,70),(62,69),(63,68),(64,67),(65,66),(71,72)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 5 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 8 | 0 | 0 |
| 5 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 10 |
| 0 | 0 | 3 | 6 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 3 | 10 |
| 0 | 0 | 7 | 10 |
G:=sub<GL(4,GF(13))| [5,0,0,0,0,8,0,0,0,0,12,0,0,0,0,12],[0,5,0,0,8,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,3,3,0,0,10,6],[1,0,0,0,0,12,0,0,0,0,3,7,0,0,10,10] >;
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | D6 | D6 | C4○D4 | D12 | 2- (1+4) | D4⋊2S3 | Q8○D12 |
| kernel | D4⋊6D12 | C12⋊2Q8 | C4×D12 | C22.D12 | C4.D12 | C2×C4⋊Dic3 | C12⋊7D4 | D4×C12 | C2×D4⋊2S3 | C4×D4 | C3×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C12 | D4 | C6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 1 | 2 | 1 | 4 | 1 | 2 | 1 | 2 | 1 | 4 | 8 | 1 | 2 | 2 |
In GAP, Magma, Sage, TeX
D_4\rtimes_6D_{12} % in TeX
G:=Group("D4:6D12"); // GroupNames label
G:=SmallGroup(192,1114);
// by ID
G=gap.SmallGroup(192,1114);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,387,675,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^12=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations