metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊24D4, C42.109D6, C6.592- (1+4), (C4×D4)⋊13S3, D6⋊3D4⋊8C2, (C4×D12)⋊29C2, (D4×C12)⋊15C2, C4⋊3(C4○D12), C12⋊2(C4○D4), C3⋊2(D4⋊6D4), C4⋊C4.316D6, D6.15(C2×D4), C4.140(S3×D4), C12⋊2Q8⋊24C2, C23.9D6⋊6C2, (C2×D4).214D6, C12.346(C2×D4), (C2×C6).95C24, C6.50(C22×D4), D6⋊C4.98C22, C2.16(Q8○D12), C22⋊C4.110D6, (C22×C4).224D6, C12.48D4⋊20C2, (C2×C12).783C23, (C4×C12).152C22, (C6×D4).257C22, (C2×D12).287C22, Dic3⋊C4.65C22, C4⋊Dic3.199C22, C22.120(S3×C23), C23.105(C22×S3), (C22×C6).165C23, (C2×Dic3).41C23, (C22×S3).173C23, (C22×C12).107C22, (C2×Dic6).239C22, C6.D4.12C22, (S3×C4⋊C4)⋊15C2, C2.23(C2×S3×D4), (C2×C4○D12)⋊8C2, C6.42(C2×C4○D4), C2.46(C2×C4○D12), (S3×C2×C4).64C22, (C3×C4⋊C4).326C22, (C2×C4).579(C22×S3), (C2×C3⋊D4).113C22, (C3×C22⋊C4).122C22, SmallGroup(192,1110)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 728 in 292 conjugacy classes, 107 normal (29 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×4], C4 [×9], C22, C22 [×14], S3 [×4], C6 [×3], C6 [×2], C2×C4 [×3], C2×C4 [×2], C2×C4 [×22], D4 [×14], Q8 [×4], C23 [×2], C23 [×2], Dic3 [×6], C12 [×4], C12 [×3], D6 [×4], D6 [×4], C2×C6, C2×C6 [×6], 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 [×12], D12 [×4], C2×Dic3 [×6], C3⋊D4 [×8], C2×C12 [×3], C2×C12 [×2], C2×C12 [×4], C3×D4 [×2], 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], Dic3⋊C4 [×4], C4⋊Dic3, C4⋊Dic3 [×4], D6⋊C4 [×2], C6.D4 [×4], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6 [×2], S3×C2×C4 [×6], C2×D12, C4○D12 [×8], C2×C3⋊D4 [×4], C22×C12 [×2], C6×D4, D4⋊6D4, C12⋊2Q8, C4×D12, C23.9D6 [×4], S3×C4⋊C4 [×2], C12.48D4 [×2], D6⋊3D4 [×2], D4×C12, C2×C4○D12 [×2], D12⋊24D4
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×2], C24, C22×S3 [×7], C22×D4, C2×C4○D4, 2- (1+4), C4○D12 [×2], S3×D4 [×2], S3×C23, D4⋊6D4, C2×C4○D12, C2×S3×D4, Q8○D12, D12⋊24D4
Generators and relations
G = < a,b,c,d | a12=b2=c4=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a6b, dcd=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 17)(14 16)(18 24)(19 23)(20 22)(25 33)(26 32)(27 31)(28 30)(34 36)(37 39)(40 48)(41 47)(42 46)(43 45)(49 55)(50 54)(51 53)(56 60)(57 59)(61 69)(62 68)(63 67)(64 66)(70 72)(73 79)(74 78)(75 77)(80 84)(81 83)(85 91)(86 90)(87 89)(92 96)(93 95)
(1 81 20 57)(2 82 21 58)(3 83 22 59)(4 84 23 60)(5 73 24 49)(6 74 13 50)(7 75 14 51)(8 76 15 52)(9 77 16 53)(10 78 17 54)(11 79 18 55)(12 80 19 56)(25 67 96 40)(26 68 85 41)(27 69 86 42)(28 70 87 43)(29 71 88 44)(30 72 89 45)(31 61 90 46)(32 62 91 47)(33 63 92 48)(34 64 93 37)(35 65 94 38)(36 66 95 39)
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 89)(14 90)(15 91)(16 92)(17 93)(18 94)(19 95)(20 96)(21 85)(22 86)(23 87)(24 88)(37 78)(38 79)(39 80)(40 81)(41 82)(42 83)(43 84)(44 73)(45 74)(46 75)(47 76)(48 77)(49 71)(50 72)(51 61)(52 62)(53 63)(54 64)(55 65)(56 66)(57 67)(58 68)(59 69)(60 70)
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)(13,17)(14,16)(18,24)(19,23)(20,22)(25,33)(26,32)(27,31)(28,30)(34,36)(37,39)(40,48)(41,47)(42,46)(43,45)(49,55)(50,54)(51,53)(56,60)(57,59)(61,69)(62,68)(63,67)(64,66)(70,72)(73,79)(74,78)(75,77)(80,84)(81,83)(85,91)(86,90)(87,89)(92,96)(93,95), (1,81,20,57)(2,82,21,58)(3,83,22,59)(4,84,23,60)(5,73,24,49)(6,74,13,50)(7,75,14,51)(8,76,15,52)(9,77,16,53)(10,78,17,54)(11,79,18,55)(12,80,19,56)(25,67,96,40)(26,68,85,41)(27,69,86,42)(28,70,87,43)(29,71,88,44)(30,72,89,45)(31,61,90,46)(32,62,91,47)(33,63,92,48)(34,64,93,37)(35,65,94,38)(36,66,95,39), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,89)(14,90)(15,91)(16,92)(17,93)(18,94)(19,95)(20,96)(21,85)(22,86)(23,87)(24,88)(37,78)(38,79)(39,80)(40,81)(41,82)(42,83)(43,84)(44,73)(45,74)(46,75)(47,76)(48,77)(49,71)(50,72)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66)(57,67)(58,68)(59,69)(60,70)>;
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)(13,17)(14,16)(18,24)(19,23)(20,22)(25,33)(26,32)(27,31)(28,30)(34,36)(37,39)(40,48)(41,47)(42,46)(43,45)(49,55)(50,54)(51,53)(56,60)(57,59)(61,69)(62,68)(63,67)(64,66)(70,72)(73,79)(74,78)(75,77)(80,84)(81,83)(85,91)(86,90)(87,89)(92,96)(93,95), (1,81,20,57)(2,82,21,58)(3,83,22,59)(4,84,23,60)(5,73,24,49)(6,74,13,50)(7,75,14,51)(8,76,15,52)(9,77,16,53)(10,78,17,54)(11,79,18,55)(12,80,19,56)(25,67,96,40)(26,68,85,41)(27,69,86,42)(28,70,87,43)(29,71,88,44)(30,72,89,45)(31,61,90,46)(32,62,91,47)(33,63,92,48)(34,64,93,37)(35,65,94,38)(36,66,95,39), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,89)(14,90)(15,91)(16,92)(17,93)(18,94)(19,95)(20,96)(21,85)(22,86)(23,87)(24,88)(37,78)(38,79)(39,80)(40,81)(41,82)(42,83)(43,84)(44,73)(45,74)(46,75)(47,76)(48,77)(49,71)(50,72)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66)(57,67)(58,68)(59,69)(60,70) );
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),(13,17),(14,16),(18,24),(19,23),(20,22),(25,33),(26,32),(27,31),(28,30),(34,36),(37,39),(40,48),(41,47),(42,46),(43,45),(49,55),(50,54),(51,53),(56,60),(57,59),(61,69),(62,68),(63,67),(64,66),(70,72),(73,79),(74,78),(75,77),(80,84),(81,83),(85,91),(86,90),(87,89),(92,96),(93,95)], [(1,81,20,57),(2,82,21,58),(3,83,22,59),(4,84,23,60),(5,73,24,49),(6,74,13,50),(7,75,14,51),(8,76,15,52),(9,77,16,53),(10,78,17,54),(11,79,18,55),(12,80,19,56),(25,67,96,40),(26,68,85,41),(27,69,86,42),(28,70,87,43),(29,71,88,44),(30,72,89,45),(31,61,90,46),(32,62,91,47),(33,63,92,48),(34,64,93,37),(35,65,94,38),(36,66,95,39)], [(1,25),(2,26),(3,27),(4,28),(5,29),(6,30),(7,31),(8,32),(9,33),(10,34),(11,35),(12,36),(13,89),(14,90),(15,91),(16,92),(17,93),(18,94),(19,95),(20,96),(21,85),(22,86),(23,87),(24,88),(37,78),(38,79),(39,80),(40,81),(41,82),(42,83),(43,84),(44,73),(45,74),(46,75),(47,76),(48,77),(49,71),(50,72),(51,61),(52,62),(53,63),(54,64),(55,65),(56,66),(57,67),(58,68),(59,69),(60,70)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 3 | 10 | 0 | 0 |
| 3 | 6 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 12 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 9 | 11 |
| 0 | 0 | 2 | 4 |
| 2 | 4 | 0 | 0 |
| 9 | 11 | 0 | 0 |
| 0 | 0 | 2 | 4 |
| 0 | 0 | 9 | 11 |
G:=sub<GL(4,GF(13))| [3,3,0,0,10,6,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,12,1,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,9,2,0,0,11,4],[2,9,0,0,4,11,0,0,0,0,2,9,0,0,4,11] >;
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | ··· | 4H | 4I | 4J | ··· | 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 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | 2 | ··· | 2 | 4 | 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 | C4○D12 | 2- (1+4) | S3×D4 | Q8○D12 |
| kernel | D12⋊24D4 | C12⋊2Q8 | C4×D12 | C23.9D6 | S3×C4⋊C4 | C12.48D4 | D6⋊3D4 | D4×C12 | C2×C4○D12 | C4×D4 | D12 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C12 | C4 | 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_{12}\rtimes_{24}D_4 % in TeX
G:=Group("D12:24D4"); // GroupNames label
G:=SmallGroup(192,1110);
// by ID
G=gap.SmallGroup(192,1110);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,100,675,570,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^6*b,d*c*d=c^-1>;
// generators/relations