metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.40C24, D12.35C23, 2- (1+4)⋊6S3, Dic6.35C23, C3⋊5(Q8○D8), Q8○D12⋊10C2, C4○D4.33D6, (C3×D4).39D4, C3⋊C8.19C23, (C3×Q8).39D4, C12.272(C2×D4), C4.40(S3×C23), D4⋊S3.2C22, (C2×Q8).117D6, Q8.14D6⋊12C2, Q8.13D6⋊11C2, D4.21(C3⋊D4), D4.Dic3⋊13C2, Q8.28(C3⋊D4), D4.28(C22×S3), (C3×D4).28C23, C6.174(C22×D4), D4.S3.3C22, (C3×Q8).28C23, Q8.38(C22×S3), Q8.11D6⋊12C2, C3⋊Q16.4C22, (C2×C12).121C23, C4○D12.34C22, (C3×2- (1+4))⋊3C2, (C6×Q8).154C22, Q8⋊2S3.3C22, C4.Dic3.32C22, (C2×Dic6).205C22, (C2×C6).88(C2×D4), C4.78(C2×C3⋊D4), (C2×C3⋊Q16)⋊32C2, C22.9(C2×C3⋊D4), (C2×C3⋊C8).185C22, C2.47(C22×C3⋊D4), (C2×C4).105(C22×S3), (C3×C4○D4).30C22, SmallGroup(192,1397)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 536 in 248 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2 [×5], C3, C4, C4 [×3], C4 [×6], C22 [×3], C22 [×2], S3, C6, C6 [×4], C8 [×4], C2×C4 [×3], C2×C4 [×12], D4, D4 [×3], D4 [×7], Q8, Q8 [×3], Q8 [×9], Dic3 [×3], C12, C12 [×3], C12 [×3], D6, C2×C6 [×3], C2×C6, C2×C8 [×3], M4(2) [×3], D8, SD16 [×6], Q16 [×9], C2×Q8 [×3], C2×Q8 [×5], C4○D4, C4○D4 [×3], C4○D4 [×9], C3⋊C8, C3⋊C8 [×3], Dic6 [×3], Dic6 [×3], C4×S3 [×3], D12, C2×Dic3 [×3], C3⋊D4 [×3], C2×C12 [×3], C2×C12 [×6], C3×D4, C3×D4 [×3], C3×D4 [×3], C3×Q8, C3×Q8 [×3], C3×Q8 [×3], C8○D4, C2×Q16 [×3], C4○D8 [×3], C8.C22 [×6], 2- (1+4), 2- (1+4), C2×C3⋊C8 [×3], C4.Dic3 [×3], D4⋊S3, D4.S3 [×3], Q8⋊2S3 [×3], C3⋊Q16 [×9], C2×Dic6 [×3], C4○D12 [×3], D4⋊2S3 [×3], S3×Q8, C6×Q8 [×3], C6×Q8, C3×C4○D4, C3×C4○D4 [×3], C3×C4○D4 [×3], Q8○D8, Q8.11D6 [×3], C2×C3⋊Q16 [×3], D4.Dic3, Q8.13D6 [×3], Q8.14D6 [×3], Q8○D12, C3×2- (1+4), D12.35C23
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C3⋊D4 [×4], C22×S3 [×7], C22×D4, C2×C3⋊D4 [×6], S3×C23, Q8○D8, C22×C3⋊D4, D12.35C23
Generators and relations
G = < a,b,c,d,e | a12=b2=e2=1, c2=d2=a6, bab=a-1, ac=ca, ad=da, eae=a7, bc=cb, bd=db, ebe=a3b, dcd-1=a6c, ce=ec, de=ed >
►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 15)(16 24)(17 23)(18 22)(19 21)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(37 45)(38 44)(39 43)(40 42)(46 48)(49 54)(50 53)(51 52)(55 60)(56 59)(57 58)(61 69)(62 68)(63 67)(64 66)(70 72)(73 78)(74 77)(75 76)(79 84)(80 83)(81 82)(85 87)(88 96)(89 95)(90 94)(91 93)
(1 52 7 58)(2 53 8 59)(3 54 9 60)(4 55 10 49)(5 56 11 50)(6 57 12 51)(13 70 19 64)(14 71 20 65)(15 72 21 66)(16 61 22 67)(17 62 23 68)(18 63 24 69)(25 76 31 82)(26 77 32 83)(27 78 33 84)(28 79 34 73)(29 80 35 74)(30 81 36 75)(37 94 43 88)(38 95 44 89)(39 96 45 90)(40 85 46 91)(41 86 47 92)(42 87 48 93)
(1 25 7 31)(2 26 8 32)(3 27 9 33)(4 28 10 34)(5 29 11 35)(6 30 12 36)(13 40 19 46)(14 41 20 47)(15 42 21 48)(16 43 22 37)(17 44 23 38)(18 45 24 39)(49 79 55 73)(50 80 56 74)(51 81 57 75)(52 82 58 76)(53 83 59 77)(54 84 60 78)(61 94 67 88)(62 95 68 89)(63 96 69 90)(64 85 70 91)(65 86 71 92)(66 87 72 93)
(1 16)(2 23)(3 18)(4 13)(5 20)(6 15)(7 22)(8 17)(9 24)(10 19)(11 14)(12 21)(25 43)(26 38)(27 45)(28 40)(29 47)(30 42)(31 37)(32 44)(33 39)(34 46)(35 41)(36 48)(49 64)(50 71)(51 66)(52 61)(53 68)(54 63)(55 70)(56 65)(57 72)(58 67)(59 62)(60 69)(73 91)(74 86)(75 93)(76 88)(77 95)(78 90)(79 85)(80 92)(81 87)(82 94)(83 89)(84 96)
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)(13,15)(16,24)(17,23)(18,22)(19,21)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(37,45)(38,44)(39,43)(40,42)(46,48)(49,54)(50,53)(51,52)(55,60)(56,59)(57,58)(61,69)(62,68)(63,67)(64,66)(70,72)(73,78)(74,77)(75,76)(79,84)(80,83)(81,82)(85,87)(88,96)(89,95)(90,94)(91,93), (1,52,7,58)(2,53,8,59)(3,54,9,60)(4,55,10,49)(5,56,11,50)(6,57,12,51)(13,70,19,64)(14,71,20,65)(15,72,21,66)(16,61,22,67)(17,62,23,68)(18,63,24,69)(25,76,31,82)(26,77,32,83)(27,78,33,84)(28,79,34,73)(29,80,35,74)(30,81,36,75)(37,94,43,88)(38,95,44,89)(39,96,45,90)(40,85,46,91)(41,86,47,92)(42,87,48,93), (1,25,7,31)(2,26,8,32)(3,27,9,33)(4,28,10,34)(5,29,11,35)(6,30,12,36)(13,40,19,46)(14,41,20,47)(15,42,21,48)(16,43,22,37)(17,44,23,38)(18,45,24,39)(49,79,55,73)(50,80,56,74)(51,81,57,75)(52,82,58,76)(53,83,59,77)(54,84,60,78)(61,94,67,88)(62,95,68,89)(63,96,69,90)(64,85,70,91)(65,86,71,92)(66,87,72,93), (1,16)(2,23)(3,18)(4,13)(5,20)(6,15)(7,22)(8,17)(9,24)(10,19)(11,14)(12,21)(25,43)(26,38)(27,45)(28,40)(29,47)(30,42)(31,37)(32,44)(33,39)(34,46)(35,41)(36,48)(49,64)(50,71)(51,66)(52,61)(53,68)(54,63)(55,70)(56,65)(57,72)(58,67)(59,62)(60,69)(73,91)(74,86)(75,93)(76,88)(77,95)(78,90)(79,85)(80,92)(81,87)(82,94)(83,89)(84,96)>;
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)(13,15)(16,24)(17,23)(18,22)(19,21)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(37,45)(38,44)(39,43)(40,42)(46,48)(49,54)(50,53)(51,52)(55,60)(56,59)(57,58)(61,69)(62,68)(63,67)(64,66)(70,72)(73,78)(74,77)(75,76)(79,84)(80,83)(81,82)(85,87)(88,96)(89,95)(90,94)(91,93), (1,52,7,58)(2,53,8,59)(3,54,9,60)(4,55,10,49)(5,56,11,50)(6,57,12,51)(13,70,19,64)(14,71,20,65)(15,72,21,66)(16,61,22,67)(17,62,23,68)(18,63,24,69)(25,76,31,82)(26,77,32,83)(27,78,33,84)(28,79,34,73)(29,80,35,74)(30,81,36,75)(37,94,43,88)(38,95,44,89)(39,96,45,90)(40,85,46,91)(41,86,47,92)(42,87,48,93), (1,25,7,31)(2,26,8,32)(3,27,9,33)(4,28,10,34)(5,29,11,35)(6,30,12,36)(13,40,19,46)(14,41,20,47)(15,42,21,48)(16,43,22,37)(17,44,23,38)(18,45,24,39)(49,79,55,73)(50,80,56,74)(51,81,57,75)(52,82,58,76)(53,83,59,77)(54,84,60,78)(61,94,67,88)(62,95,68,89)(63,96,69,90)(64,85,70,91)(65,86,71,92)(66,87,72,93), (1,16)(2,23)(3,18)(4,13)(5,20)(6,15)(7,22)(8,17)(9,24)(10,19)(11,14)(12,21)(25,43)(26,38)(27,45)(28,40)(29,47)(30,42)(31,37)(32,44)(33,39)(34,46)(35,41)(36,48)(49,64)(50,71)(51,66)(52,61)(53,68)(54,63)(55,70)(56,65)(57,72)(58,67)(59,62)(60,69)(73,91)(74,86)(75,93)(76,88)(77,95)(78,90)(79,85)(80,92)(81,87)(82,94)(83,89)(84,96) );
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),(13,15),(16,24),(17,23),(18,22),(19,21),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(37,45),(38,44),(39,43),(40,42),(46,48),(49,54),(50,53),(51,52),(55,60),(56,59),(57,58),(61,69),(62,68),(63,67),(64,66),(70,72),(73,78),(74,77),(75,76),(79,84),(80,83),(81,82),(85,87),(88,96),(89,95),(90,94),(91,93)], [(1,52,7,58),(2,53,8,59),(3,54,9,60),(4,55,10,49),(5,56,11,50),(6,57,12,51),(13,70,19,64),(14,71,20,65),(15,72,21,66),(16,61,22,67),(17,62,23,68),(18,63,24,69),(25,76,31,82),(26,77,32,83),(27,78,33,84),(28,79,34,73),(29,80,35,74),(30,81,36,75),(37,94,43,88),(38,95,44,89),(39,96,45,90),(40,85,46,91),(41,86,47,92),(42,87,48,93)], [(1,25,7,31),(2,26,8,32),(3,27,9,33),(4,28,10,34),(5,29,11,35),(6,30,12,36),(13,40,19,46),(14,41,20,47),(15,42,21,48),(16,43,22,37),(17,44,23,38),(18,45,24,39),(49,79,55,73),(50,80,56,74),(51,81,57,75),(52,82,58,76),(53,83,59,77),(54,84,60,78),(61,94,67,88),(62,95,68,89),(63,96,69,90),(64,85,70,91),(65,86,71,92),(66,87,72,93)], [(1,16),(2,23),(3,18),(4,13),(5,20),(6,15),(7,22),(8,17),(9,24),(10,19),(11,14),(12,21),(25,43),(26,38),(27,45),(28,40),(29,47),(30,42),(31,37),(32,44),(33,39),(34,46),(35,41),(36,48),(49,64),(50,71),(51,66),(52,61),(53,68),(54,63),(55,70),(56,65),(57,72),(58,67),(59,62),(60,69),(73,91),(74,86),(75,93),(76,88),(77,95),(78,90),(79,85),(80,92),(81,87),(82,94),(83,89),(84,96)])
Matrix representation ►G ⊆ GL6(𝔽73)
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 22 | 51 | 0 | 1 |
| 0 | 0 | 51 | 51 | 72 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 59 | 51 | 0 | 1 |
| 0 | 0 | 22 | 14 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 66 | 64 | 36 | 0 |
| 0 | 0 | 71 | 4 | 0 | 36 |
| 0 | 0 | 63 | 54 | 7 | 9 |
| 0 | 0 | 12 | 70 | 2 | 69 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 61 | 38 | 67 | 0 |
| 0 | 0 | 49 | 47 | 0 | 67 |
| 0 | 0 | 6 | 27 | 12 | 35 |
| 0 | 0 | 6 | 46 | 24 | 26 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 57 | 57 | 0 | 0 |
| 0 | 0 | 57 | 16 | 0 | 0 |
| 0 | 0 | 5 | 68 | 57 | 57 |
| 0 | 0 | 68 | 68 | 57 | 16 |
G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,72,22,51,0,0,1,0,51,51,0,0,0,0,0,72,0,0,0,0,1,0],[1,0,0,0,0,0,72,72,0,0,0,0,0,0,0,1,59,22,0,0,1,0,51,14,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,66,71,63,12,0,0,64,4,54,70,0,0,36,0,7,2,0,0,0,36,9,69],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,61,49,6,6,0,0,38,47,27,46,0,0,67,0,12,24,0,0,0,67,35,26],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,57,57,5,68,0,0,57,16,68,68,0,0,0,0,57,57,0,0,0,0,57,16] >;
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | ··· | 6F | 8A | 8B | 8C | 8D | 8E | 12A | ··· | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 12 | 12 | 12 | 2 | 4 | ··· | 4 | 6 | 6 | 12 | 12 | 12 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C3⋊D4 | C3⋊D4 | Q8○D8 | D12.35C23 |
| kernel | D12.35C23 | Q8.11D6 | C2×C3⋊Q16 | D4.Dic3 | Q8.13D6 | Q8.14D6 | Q8○D12 | C3×2- (1+4) | 2- (1+4) | C3×D4 | C3×Q8 | C2×Q8 | C4○D4 | D4 | Q8 | C3 | C1 |
| # reps | 1 | 3 | 3 | 1 | 3 | 3 | 1 | 1 | 1 | 3 | 1 | 3 | 4 | 6 | 2 | 2 | 1 |
In GAP, Magma, Sage, TeX
D_{12}._{35}C_2^3 % in TeX
G:=Group("D12.35C2^3"); // GroupNames label
G:=SmallGroup(192,1397);
// by ID
G=gap.SmallGroup(192,1397);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,184,675,1684,235,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^12=b^2=e^2=1,c^2=d^2=a^6,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e=a^7,b*c=c*b,b*d=d*b,e*b*e=a^3*b,d*c*d^-1=a^6*c,c*e=e*c,d*e=e*d>;
// generators/relations