metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.13D12, Q8.18D12, C24.12C23, C12.63C24, D12.26C23, D24.14C22, M4(2).28D6, Dic6.26C23, Dic12.10C22, C8○D4⋊9S3, C3⋊1(Q8○D8), Q8○D12⋊4C2, C4○D24⋊13C2, C4○D4.56D6, (C3×D4).25D4, (C2×C8).102D6, C4.29(C2×D12), C12.75(C2×D4), (C3×Q8).25D4, C8.D6⋊12C2, C4.60(S3×C23), C8.54(C22×S3), C22.5(C2×D12), C6.30(C22×D4), (C2×Dic12)⋊15C2, C24⋊C2.2C22, (C2×C24).70C22, C2.32(C22×D12), (C2×C12).517C23, C4○D12.27C22, (C2×Dic6).194C22, (C3×M4(2)).30C22, (C3×C8○D4)⋊5C2, (C2×C6).10(C2×D4), (C2×C4).228(C22×S3), (C3×C4○D4).47C22, SmallGroup(192,1312)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 624 in 248 conjugacy classes, 107 normal (16 characteristic)
C1, C2, C2 [×5], C3, C4, C4 [×3], C4 [×6], C22 [×3], C22 [×2], S3 [×2], C6, C6 [×3], C8, C8 [×3], C2×C4 [×3], C2×C4 [×12], D4 [×3], D4 [×8], Q8, Q8 [×12], Dic3 [×6], C12, C12 [×3], D6 [×2], C2×C6 [×3], C2×C8 [×3], M4(2) [×3], D8, SD16 [×6], Q16 [×9], C2×Q8 [×8], C4○D4, C4○D4 [×12], C24, C24 [×3], Dic6 [×6], Dic6 [×6], C4×S3 [×6], D12 [×2], C2×Dic3 [×6], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, C8○D4, C2×Q16 [×3], C4○D8 [×3], C8.C22 [×6], 2- (1+4) [×2], C24⋊C2 [×6], D24, Dic12 [×9], C2×C24 [×3], C3×M4(2) [×3], C2×Dic6 [×6], C4○D12 [×6], D4⋊2S3 [×6], S3×Q8 [×2], C3×C4○D4, Q8○D8, C4○D24 [×3], C2×Dic12 [×3], C8.D6 [×6], C3×C8○D4, Q8○D12 [×2], D4.13D12
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, D12 [×4], C22×S3 [×7], C22×D4, C2×D12 [×6], S3×C23, Q8○D8, C22×D12, D4.13D12
Generators and relations
G = < a,b,c,d | a4=b2=d2=1, c12=a2, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=a2c11 >
►On 96 points
(1 28 13 40)(2 29 14 41)(3 30 15 42)(4 31 16 43)(5 32 17 44)(6 33 18 45)(7 34 19 46)(8 35 20 47)(9 36 21 48)(10 37 22 25)(11 38 23 26)(12 39 24 27)(49 88 61 76)(50 89 62 77)(51 90 63 78)(52 91 64 79)(53 92 65 80)(54 93 66 81)(55 94 67 82)(56 95 68 83)(57 96 69 84)(58 73 70 85)(59 74 71 86)(60 75 72 87)
(1 79)(2 80)(3 81)(4 82)(5 83)(6 84)(7 85)(8 86)(9 87)(10 88)(11 89)(12 90)(13 91)(14 92)(15 93)(16 94)(17 95)(18 96)(19 73)(20 74)(21 75)(22 76)(23 77)(24 78)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(37 49)(38 50)(39 51)(40 52)(41 53)(42 54)(43 55)(44 56)(45 57)(46 58)(47 59)(48 60)
(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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)(25 30)(26 29)(27 28)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 40)(49 66)(50 65)(51 64)(52 63)(53 62)(54 61)(55 60)(56 59)(57 58)(67 72)(68 71)(69 70)(73 96)(74 95)(75 94)(76 93)(77 92)(78 91)(79 90)(80 89)(81 88)(82 87)(83 86)(84 85)
G:=sub<Sym(96)| (1,28,13,40)(2,29,14,41)(3,30,15,42)(4,31,16,43)(5,32,17,44)(6,33,18,45)(7,34,19,46)(8,35,20,47)(9,36,21,48)(10,37,22,25)(11,38,23,26)(12,39,24,27)(49,88,61,76)(50,89,62,77)(51,90,63,78)(52,91,64,79)(53,92,65,80)(54,93,66,81)(55,94,67,82)(56,95,68,83)(57,96,69,84)(58,73,70,85)(59,74,71,86)(60,75,72,87), (1,79)(2,80)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,73)(20,74)(21,75)(22,76)(23,77)(24,78)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,49)(38,50)(39,51)(40,52)(41,53)(42,54)(43,55)(44,56)(45,57)(46,58)(47,59)(48,60), (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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(25,30)(26,29)(27,28)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(49,66)(50,65)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)(67,72)(68,71)(69,70)(73,96)(74,95)(75,94)(76,93)(77,92)(78,91)(79,90)(80,89)(81,88)(82,87)(83,86)(84,85)>;
G:=Group( (1,28,13,40)(2,29,14,41)(3,30,15,42)(4,31,16,43)(5,32,17,44)(6,33,18,45)(7,34,19,46)(8,35,20,47)(9,36,21,48)(10,37,22,25)(11,38,23,26)(12,39,24,27)(49,88,61,76)(50,89,62,77)(51,90,63,78)(52,91,64,79)(53,92,65,80)(54,93,66,81)(55,94,67,82)(56,95,68,83)(57,96,69,84)(58,73,70,85)(59,74,71,86)(60,75,72,87), (1,79)(2,80)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,73)(20,74)(21,75)(22,76)(23,77)(24,78)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,49)(38,50)(39,51)(40,52)(41,53)(42,54)(43,55)(44,56)(45,57)(46,58)(47,59)(48,60), (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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(25,30)(26,29)(27,28)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(49,66)(50,65)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)(67,72)(68,71)(69,70)(73,96)(74,95)(75,94)(76,93)(77,92)(78,91)(79,90)(80,89)(81,88)(82,87)(83,86)(84,85) );
G=PermutationGroup([(1,28,13,40),(2,29,14,41),(3,30,15,42),(4,31,16,43),(5,32,17,44),(6,33,18,45),(7,34,19,46),(8,35,20,47),(9,36,21,48),(10,37,22,25),(11,38,23,26),(12,39,24,27),(49,88,61,76),(50,89,62,77),(51,90,63,78),(52,91,64,79),(53,92,65,80),(54,93,66,81),(55,94,67,82),(56,95,68,83),(57,96,69,84),(58,73,70,85),(59,74,71,86),(60,75,72,87)], [(1,79),(2,80),(3,81),(4,82),(5,83),(6,84),(7,85),(8,86),(9,87),(10,88),(11,89),(12,90),(13,91),(14,92),(15,93),(16,94),(17,95),(18,96),(19,73),(20,74),(21,75),(22,76),(23,77),(24,78),(25,61),(26,62),(27,63),(28,64),(29,65),(30,66),(31,67),(32,68),(33,69),(34,70),(35,71),(36,72),(37,49),(38,50),(39,51),(40,52),(41,53),(42,54),(43,55),(44,56),(45,57),(46,58),(47,59),(48,60)], [(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)], [(1,24),(2,23),(3,22),(4,21),(5,20),(6,19),(7,18),(8,17),(9,16),(10,15),(11,14),(12,13),(25,30),(26,29),(27,28),(31,48),(32,47),(33,46),(34,45),(35,44),(36,43),(37,42),(38,41),(39,40),(49,66),(50,65),(51,64),(52,63),(53,62),(54,61),(55,60),(56,59),(57,58),(67,72),(68,71),(69,70),(73,96),(74,95),(75,94),(76,93),(77,92),(78,91),(79,90),(80,89),(81,88),(82,87),(83,86),(84,85)])
Matrix representation ►G ⊆ GL4(𝔽73) generated by
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 42 | 11 | 42 | 11 |
| 62 | 31 | 62 | 31 |
| 42 | 11 | 31 | 62 |
| 62 | 31 | 11 | 42 |
| 50 | 55 | 0 | 0 |
| 18 | 68 | 0 | 0 |
| 0 | 0 | 50 | 55 |
| 0 | 0 | 18 | 68 |
| 23 | 18 | 0 | 0 |
| 68 | 50 | 0 | 0 |
| 0 | 0 | 23 | 18 |
| 0 | 0 | 68 | 50 |
G:=sub<GL(4,GF(73))| [0,0,72,0,0,0,0,72,1,0,0,0,0,1,0,0],[42,62,42,62,11,31,11,31,42,62,31,11,11,31,62,42],[50,18,0,0,55,68,0,0,0,0,50,18,0,0,55,68],[23,68,0,0,18,50,0,0,0,0,23,68,0,0,18,50] >;
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 12A | 12B | 12C | 12D | 12E | 24A | 24B | 24C | 24D | 24E | ··· | 24J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 2 | 2 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 12 | ··· | 12 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | D12 | D12 | Q8○D8 | D4.13D12 |
| kernel | D4.13D12 | C4○D24 | C2×Dic12 | C8.D6 | C3×C8○D4 | Q8○D12 | C8○D4 | C3×D4 | C3×Q8 | C2×C8 | M4(2) | C4○D4 | D4 | Q8 | C3 | C1 |
| # reps | 1 | 3 | 3 | 6 | 1 | 2 | 1 | 3 | 1 | 3 | 3 | 1 | 6 | 2 | 2 | 4 |
In GAP, Magma, Sage, TeX
D_4._{13}D_{12} % in TeX
G:=Group("D4.13D12"); // GroupNames label
G:=SmallGroup(192,1312);
// by ID
G=gap.SmallGroup(192,1312);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,184,675,192,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=d^2=1,c^12=a^2,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=a^2*c^11>;
// generators/relations