metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6⋊24D4, C42.111D6, C6.1042+ (1+4), (C4×D4)⋊15S3, Dic3⋊D4⋊8C2, (D4×C12)⋊17C2, C12⋊3(C4○D4), C4⋊1(C4○D12), C12⋊3D4⋊9C2, C4⋊C4.317D6, C3⋊1(Q8⋊6D4), C4.142(S3×D4), C12⋊7D4⋊19C2, C4⋊D12⋊12C2, (C4×Dic6)⋊32C2, (C2×D4).216D6, C12.348(C2×D4), (C2×C6).97C24, Dic3⋊5D4⋊15C2, C6.52(C22×D4), C2.16(D4○D12), D6⋊C4.54C22, C22⋊C4.112D6, Dic3.18(C2×D4), (C22×C4).226D6, (C2×C12).785C23, (C4×C12).154C22, (C6×D4).258C22, (C2×D12).138C22, (C22×S3).32C23, C4⋊Dic3.299C22, C22.122(S3×C23), (C22×C6).167C23, C23.107(C22×S3), (C4×Dic3).75C22, Dic3⋊C4.111C22, (C22×C12).109C22, (C2×Dic3).205C23, (C2×Dic6).316C22, C2.25(C2×S3×D4), C6.44(C2×C4○D4), (C2×C4○D12)⋊10C2, C2.48(C2×C4○D12), (S3×C2×C4).200C22, (C3×C4⋊C4).328C22, (C2×C4).580(C22×S3), (C2×C3⋊D4).14C22, (C3×C22⋊C4).124C22, SmallGroup(192,1112)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 888 in 312 conjugacy classes, 107 normal (29 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×4], C4 [×9], C22, C22 [×18], S3 [×4], C6 [×3], C6 [×2], C2×C4 [×3], C2×C4 [×2], C2×C4 [×16], D4 [×24], Q8 [×4], C23 [×2], C23 [×4], Dic3 [×4], Dic3 [×2], C12 [×4], C12 [×3], D6 [×12], C2×C6, C2×C6 [×6], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×3], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×14], C2×Q8, C4○D4 [×8], Dic6 [×4], C4×S3 [×8], D12 [×10], C2×Dic3 [×4], C3⋊D4 [×12], C2×C12 [×3], C2×C12 [×2], C2×C12 [×4], C3×D4 [×2], C22×S3 [×4], C22×C6 [×2], C4×D4, C4×D4 [×2], C4×Q8, C4⋊D4 [×6], C4⋊1D4 [×3], C2×C4○D4 [×2], C4×Dic3 [×2], Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×4], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6, S3×C2×C4 [×4], C2×D12 [×6], C4○D12 [×8], C2×C3⋊D4 [×8], C22×C12 [×2], C6×D4, Q8⋊6D4, C4×Dic6, C4⋊D12, Dic3⋊D4 [×4], Dic3⋊5D4 [×2], C12⋊7D4 [×2], C12⋊3D4 [×2], D4×C12, C2×C4○D12 [×2], Dic6⋊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, Q8⋊6D4, C2×C4○D12, C2×S3×D4, D4○D12, Dic6⋊24D4
Generators and relations
G = < a,b,c,d | a12=c4=d2=1, b2=a6, bab-1=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 16 7 22)(2 15 8 21)(3 14 9 20)(4 13 10 19)(5 24 11 18)(6 23 12 17)(25 47 31 41)(26 46 32 40)(27 45 33 39)(28 44 34 38)(29 43 35 37)(30 42 36 48)(49 67 55 61)(50 66 56 72)(51 65 57 71)(52 64 58 70)(53 63 59 69)(54 62 60 68)(73 96 79 90)(74 95 80 89)(75 94 81 88)(76 93 82 87)(77 92 83 86)(78 91 84 85)
(1 82 70 39)(2 83 71 40)(3 84 72 41)(4 73 61 42)(5 74 62 43)(6 75 63 44)(7 76 64 45)(8 77 65 46)(9 78 66 47)(10 79 67 48)(11 80 68 37)(12 81 69 38)(13 96 49 36)(14 85 50 25)(15 86 51 26)(16 87 52 27)(17 88 53 28)(18 89 54 29)(19 90 55 30)(20 91 56 31)(21 92 57 32)(22 93 58 33)(23 94 59 34)(24 95 60 35)
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 49)(11 50)(12 51)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 61)(20 62)(21 63)(22 64)(23 65)(24 66)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(73 90)(74 91)(75 92)(76 93)(77 94)(78 95)(79 96)(80 85)(81 86)(82 87)(83 88)(84 89)
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), (1,16,7,22)(2,15,8,21)(3,14,9,20)(4,13,10,19)(5,24,11,18)(6,23,12,17)(25,47,31,41)(26,46,32,40)(27,45,33,39)(28,44,34,38)(29,43,35,37)(30,42,36,48)(49,67,55,61)(50,66,56,72)(51,65,57,71)(52,64,58,70)(53,63,59,69)(54,62,60,68)(73,96,79,90)(74,95,80,89)(75,94,81,88)(76,93,82,87)(77,92,83,86)(78,91,84,85), (1,82,70,39)(2,83,71,40)(3,84,72,41)(4,73,61,42)(5,74,62,43)(6,75,63,44)(7,76,64,45)(8,77,65,46)(9,78,66,47)(10,79,67,48)(11,80,68,37)(12,81,69,38)(13,96,49,36)(14,85,50,25)(15,86,51,26)(16,87,52,27)(17,88,53,28)(18,89,54,29)(19,90,55,30)(20,91,56,31)(21,92,57,32)(22,93,58,33)(23,94,59,34)(24,95,60,35), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,49)(11,50)(12,51)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(73,90)(74,91)(75,92)(76,93)(77,94)(78,95)(79,96)(80,85)(81,86)(82,87)(83,88)(84,89)>;
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), (1,16,7,22)(2,15,8,21)(3,14,9,20)(4,13,10,19)(5,24,11,18)(6,23,12,17)(25,47,31,41)(26,46,32,40)(27,45,33,39)(28,44,34,38)(29,43,35,37)(30,42,36,48)(49,67,55,61)(50,66,56,72)(51,65,57,71)(52,64,58,70)(53,63,59,69)(54,62,60,68)(73,96,79,90)(74,95,80,89)(75,94,81,88)(76,93,82,87)(77,92,83,86)(78,91,84,85), (1,82,70,39)(2,83,71,40)(3,84,72,41)(4,73,61,42)(5,74,62,43)(6,75,63,44)(7,76,64,45)(8,77,65,46)(9,78,66,47)(10,79,67,48)(11,80,68,37)(12,81,69,38)(13,96,49,36)(14,85,50,25)(15,86,51,26)(16,87,52,27)(17,88,53,28)(18,89,54,29)(19,90,55,30)(20,91,56,31)(21,92,57,32)(22,93,58,33)(23,94,59,34)(24,95,60,35), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,49)(11,50)(12,51)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(73,90)(74,91)(75,92)(76,93)(77,94)(78,95)(79,96)(80,85)(81,86)(82,87)(83,88)(84,89) );
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)], [(1,16,7,22),(2,15,8,21),(3,14,9,20),(4,13,10,19),(5,24,11,18),(6,23,12,17),(25,47,31,41),(26,46,32,40),(27,45,33,39),(28,44,34,38),(29,43,35,37),(30,42,36,48),(49,67,55,61),(50,66,56,72),(51,65,57,71),(52,64,58,70),(53,63,59,69),(54,62,60,68),(73,96,79,90),(74,95,80,89),(75,94,81,88),(76,93,82,87),(77,92,83,86),(78,91,84,85)], [(1,82,70,39),(2,83,71,40),(3,84,72,41),(4,73,61,42),(5,74,62,43),(6,75,63,44),(7,76,64,45),(8,77,65,46),(9,78,66,47),(10,79,67,48),(11,80,68,37),(12,81,69,38),(13,96,49,36),(14,85,50,25),(15,86,51,26),(16,87,52,27),(17,88,53,28),(18,89,54,29),(19,90,55,30),(20,91,56,31),(21,92,57,32),(22,93,58,33),(23,94,59,34),(24,95,60,35)], [(1,52),(2,53),(3,54),(4,55),(5,56),(6,57),(7,58),(8,59),(9,60),(10,49),(11,50),(12,51),(13,67),(14,68),(15,69),(16,70),(17,71),(18,72),(19,61),(20,62),(21,63),(22,64),(23,65),(24,66),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48),(73,90),(74,91),(75,92),(76,93),(77,94),(78,95),(79,96),(80,85),(81,86),(82,87),(83,88),(84,89)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 3 | 3 |
| 0 | 0 | 10 | 6 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 4 | 11 |
| 0 | 0 | 2 | 9 |
| 0 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 11 | 4 |
| 0 | 0 | 9 | 2 |
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,3,10,0,0,3,6],[12,0,0,0,0,12,0,0,0,0,4,2,0,0,11,9],[0,12,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,1,0,0,0,0,11,9,0,0,4,2] >;
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | ··· | 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 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | ··· | 2 | 4 | 6 | 6 | 6 | 6 | 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 | D4○D12 |
| kernel | Dic6⋊24D4 | C4×Dic6 | C4⋊D12 | Dic3⋊D4 | Dic3⋊5D4 | C12⋊7D4 | C12⋊3D4 | D4×C12 | C2×C4○D12 | C4×D4 | Dic6 | 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
Dic_6\rtimes_{24}D_4 % in TeX
G:=Group("Dic6:24D4"); // GroupNames label
G:=SmallGroup(192,1112);
// by ID
G=gap.SmallGroup(192,1112);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,232,387,100,675,570,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^4=d^2=1,b^2=a^6,b*a*b^-1=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