metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.37D4, Dic6.36D4, C4⋊C4.66D6, C22⋊Q8⋊2S3, C4.102(S3×D4), (C2×Q8).52D6, C6.48C22≀C2, C6.D8⋊38C2, C12.153(C2×D4), (C2×C12).266D4, C3⋊4(D4.7D4), (C22×C6).93D4, C6.100(C4○D8), C6.SD16⋊37C2, (C22×C4).144D6, (C6×Q8).46C22, C12.55D4⋊14C2, C2.16(C23⋊2D6), (C2×C12).366C23, C6.90(C8.C22), C23.34(C3⋊D4), C2.19(Q8.13D6), (C2×D12).243C22, C2.11(Q8.11D6), (C22×C12).170C22, (C2×Dic6).270C22, (C2×C3⋊Q16)⋊8C2, (C3×C22⋊Q8)⋊2C2, (C2×Q8⋊2S3)⋊9C2, (C2×C6).497(C2×D4), (C2×C4○D12).10C2, (C2×C3⋊C8).115C22, (C2×C4).173(C3⋊D4), (C3×C4⋊C4).113C22, (C2×C4).466(C22×S3), C22.172(C2×C3⋊D4), SmallGroup(192,606)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.37D4
G = < a,b,c,d | a12=b2=c4=1, d2=a6, bab=a-1, cac-1=a7, ad=da, cbc-1=a3b, bd=db, dcd-1=c-1 >
Subgroups: 448 in 152 conjugacy classes, 43 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, C3⋊C8, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×Q8, C22×S3, C22×C6, C22⋊C8, D4⋊C4, Q8⋊C4, C22⋊Q8, C2×SD16, C2×Q16, C2×C4○D4, C2×C3⋊C8, Q8⋊2S3, C3⋊Q16, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C22×C12, C6×Q8, D4.7D4, C6.D8, C6.SD16, C12.55D4, C2×Q8⋊2S3, C2×C3⋊Q16, C3×C22⋊Q8, C2×C4○D12, D12.37D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C22≀C2, C4○D8, C8.C22, S3×D4, C2×C3⋊D4, D4.7D4, C23⋊2D6, Q8.11D6, Q8.13D6, D12.37D4
►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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 20)(14 19)(15 18)(16 17)(21 24)(22 23)(25 26)(27 36)(28 35)(29 34)(30 33)(31 32)(37 44)(38 43)(39 42)(40 41)(45 48)(46 47)(49 55)(50 54)(51 53)(56 60)(57 59)(61 65)(62 64)(66 72)(67 71)(68 70)(73 77)(74 76)(78 84)(79 83)(80 82)(85 93)(86 92)(87 91)(88 90)(94 96)
(1 85 41 71)(2 92 42 66)(3 87 43 61)(4 94 44 68)(5 89 45 63)(6 96 46 70)(7 91 47 65)(8 86 48 72)(9 93 37 67)(10 88 38 62)(11 95 39 69)(12 90 40 64)(13 79 34 56)(14 74 35 51)(15 81 36 58)(16 76 25 53)(17 83 26 60)(18 78 27 55)(19 73 28 50)(20 80 29 57)(21 75 30 52)(22 82 31 59)(23 77 32 54)(24 84 33 49)
(1 32 7 26)(2 33 8 27)(3 34 9 28)(4 35 10 29)(5 36 11 30)(6 25 12 31)(13 37 19 43)(14 38 20 44)(15 39 21 45)(16 40 22 46)(17 41 23 47)(18 42 24 48)(49 72 55 66)(50 61 56 67)(51 62 57 68)(52 63 58 69)(53 64 59 70)(54 65 60 71)(73 87 79 93)(74 88 80 94)(75 89 81 95)(76 90 82 96)(77 91 83 85)(78 92 84 86)
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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,20)(14,19)(15,18)(16,17)(21,24)(22,23)(25,26)(27,36)(28,35)(29,34)(30,33)(31,32)(37,44)(38,43)(39,42)(40,41)(45,48)(46,47)(49,55)(50,54)(51,53)(56,60)(57,59)(61,65)(62,64)(66,72)(67,71)(68,70)(73,77)(74,76)(78,84)(79,83)(80,82)(85,93)(86,92)(87,91)(88,90)(94,96), (1,85,41,71)(2,92,42,66)(3,87,43,61)(4,94,44,68)(5,89,45,63)(6,96,46,70)(7,91,47,65)(8,86,48,72)(9,93,37,67)(10,88,38,62)(11,95,39,69)(12,90,40,64)(13,79,34,56)(14,74,35,51)(15,81,36,58)(16,76,25,53)(17,83,26,60)(18,78,27,55)(19,73,28,50)(20,80,29,57)(21,75,30,52)(22,82,31,59)(23,77,32,54)(24,84,33,49), (1,32,7,26)(2,33,8,27)(3,34,9,28)(4,35,10,29)(5,36,11,30)(6,25,12,31)(13,37,19,43)(14,38,20,44)(15,39,21,45)(16,40,22,46)(17,41,23,47)(18,42,24,48)(49,72,55,66)(50,61,56,67)(51,62,57,68)(52,63,58,69)(53,64,59,70)(54,65,60,71)(73,87,79,93)(74,88,80,94)(75,89,81,95)(76,90,82,96)(77,91,83,85)(78,92,84,86)>;
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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,20)(14,19)(15,18)(16,17)(21,24)(22,23)(25,26)(27,36)(28,35)(29,34)(30,33)(31,32)(37,44)(38,43)(39,42)(40,41)(45,48)(46,47)(49,55)(50,54)(51,53)(56,60)(57,59)(61,65)(62,64)(66,72)(67,71)(68,70)(73,77)(74,76)(78,84)(79,83)(80,82)(85,93)(86,92)(87,91)(88,90)(94,96), (1,85,41,71)(2,92,42,66)(3,87,43,61)(4,94,44,68)(5,89,45,63)(6,96,46,70)(7,91,47,65)(8,86,48,72)(9,93,37,67)(10,88,38,62)(11,95,39,69)(12,90,40,64)(13,79,34,56)(14,74,35,51)(15,81,36,58)(16,76,25,53)(17,83,26,60)(18,78,27,55)(19,73,28,50)(20,80,29,57)(21,75,30,52)(22,82,31,59)(23,77,32,54)(24,84,33,49), (1,32,7,26)(2,33,8,27)(3,34,9,28)(4,35,10,29)(5,36,11,30)(6,25,12,31)(13,37,19,43)(14,38,20,44)(15,39,21,45)(16,40,22,46)(17,41,23,47)(18,42,24,48)(49,72,55,66)(50,61,56,67)(51,62,57,68)(52,63,58,69)(53,64,59,70)(54,65,60,71)(73,87,79,93)(74,88,80,94)(75,89,81,95)(76,90,82,96)(77,91,83,85)(78,92,84,86) );
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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,20),(14,19),(15,18),(16,17),(21,24),(22,23),(25,26),(27,36),(28,35),(29,34),(30,33),(31,32),(37,44),(38,43),(39,42),(40,41),(45,48),(46,47),(49,55),(50,54),(51,53),(56,60),(57,59),(61,65),(62,64),(66,72),(67,71),(68,70),(73,77),(74,76),(78,84),(79,83),(80,82),(85,93),(86,92),(87,91),(88,90),(94,96)], [(1,85,41,71),(2,92,42,66),(3,87,43,61),(4,94,44,68),(5,89,45,63),(6,96,46,70),(7,91,47,65),(8,86,48,72),(9,93,37,67),(10,88,38,62),(11,95,39,69),(12,90,40,64),(13,79,34,56),(14,74,35,51),(15,81,36,58),(16,76,25,53),(17,83,26,60),(18,78,27,55),(19,73,28,50),(20,80,29,57),(21,75,30,52),(22,82,31,59),(23,77,32,54),(24,84,33,49)], [(1,32,7,26),(2,33,8,27),(3,34,9,28),(4,35,10,29),(5,36,11,30),(6,25,12,31),(13,37,19,43),(14,38,20,44),(15,39,21,45),(16,40,22,46),(17,41,23,47),(18,42,24,48),(49,72,55,66),(50,61,56,67),(51,62,57,68),(52,63,58,69),(53,64,59,70),(54,65,60,71),(73,87,79,93),(74,88,80,94),(75,89,81,95),(76,90,82,96),(77,91,83,85),(78,92,84,86)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D4 | D6 | D6 | D6 | C3⋊D4 | C3⋊D4 | C4○D8 | C8.C22 | S3×D4 | Q8.11D6 | Q8.13D6 |
| kernel | D12.37D4 | C6.D8 | C6.SD16 | C12.55D4 | C2×Q8⋊2S3 | C2×C3⋊Q16 | C3×C22⋊Q8 | C2×C4○D12 | C22⋊Q8 | Dic6 | D12 | C2×C12 | C22×C6 | C4⋊C4 | C22×C4 | C2×Q8 | C2×C4 | C23 | C6 | C6 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 2 | 2 | 2 |
Matrix representation of D12.37D4 ►in GL6(𝔽73)
| 1 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 71 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 71 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 56 | 1 |
| 43 | 13 | 0 | 0 | 0 | 0 |
| 60 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 41 | 32 | 0 | 0 |
| 0 | 0 | 57 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 71 |
| 0 | 0 | 0 | 0 | 72 | 56 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 46 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 56 | 1 |
G:=sub<GL(6,GF(73))| [1,72,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,71,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,71,72,0,0,0,0,0,0,72,56,0,0,0,0,0,1],[43,60,0,0,0,0,13,30,0,0,0,0,0,0,41,57,0,0,0,0,32,32,0,0,0,0,0,0,17,72,0,0,0,0,71,56],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,72,56,0,0,0,0,0,1] >;
D12.37D4 in GAP, Magma, Sage, TeX
D_{12}._{37}D_4 % in TeX
G:=Group("D12.37D4"); // GroupNames label
G:=SmallGroup(192,606);
// by ID
G=gap.SmallGroup(192,606);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,219,184,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=1,d^2=a^6,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations