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), (C3×D4).28C23, D4.28(C22×S3), C6.174(C22×D4), D4.S3.3C22, Q8.38(C22×S3), (C3×Q8).28C23, 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
Generators and relations for D12.35C23
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 >
Subgroups: 536 in 248 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, D4, Q8, Q8, Q8, Dic3, C12, C12, C12, D6, C2×C6, C2×C6, C2×C8, M4(2), D8, SD16, Q16, C2×Q8, C2×Q8, C4○D4, C4○D4, C4○D4, C3⋊C8, C3⋊C8, Dic6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×D4, C3×Q8, C3×Q8, C3×Q8, C8○D4, C2×Q16, C4○D8, C8.C22, 2- 1+4, 2- 1+4, C2×C3⋊C8, C4.Dic3, D4⋊S3, D4.S3, Q8⋊2S3, C3⋊Q16, C2×Dic6, C4○D12, D4⋊2S3, S3×Q8, C6×Q8, C6×Q8, C3×C4○D4, C3×C4○D4, C3×C4○D4, Q8○D8, Q8.11D6, C2×C3⋊Q16, D4.Dic3, Q8.13D6, Q8.14D6, Q8○D12, C3×2- 1+4, D12.35C23
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C3⋊D4, C22×S3, C22×D4, C2×C3⋊D4, S3×C23, Q8○D8, C22×C3⋊D4, D12.35C23
►On 96 points
Generators in S96
(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 21)(14 20)(15 19)(16 18)(22 24)(25 30)(26 29)(27 28)(31 36)(32 35)(33 34)(37 39)(40 48)(41 47)(42 46)(43 45)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(61 63)(64 72)(65 71)(66 70)(67 69)(73 78)(74 77)(75 76)(79 84)(80 83)(81 82)(85 87)(88 96)(89 95)(90 94)(91 93)
(1 55 7 49)(2 56 8 50)(3 57 9 51)(4 58 10 52)(5 59 11 53)(6 60 12 54)(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 79 31 73)(26 80 32 74)(27 81 33 75)(28 82 34 76)(29 83 35 77)(30 84 36 78)(37 85 43 91)(38 86 44 92)(39 87 45 93)(40 88 46 94)(41 89 47 95)(42 90 48 96)
(1 28 7 34)(2 29 8 35)(3 30 9 36)(4 31 10 25)(5 32 11 26)(6 33 12 27)(13 46 19 40)(14 47 20 41)(15 48 21 42)(16 37 22 43)(17 38 23 44)(18 39 24 45)(49 82 55 76)(50 83 56 77)(51 84 57 78)(52 73 58 79)(53 74 59 80)(54 75 60 81)(61 91 67 85)(62 92 68 86)(63 93 69 87)(64 94 70 88)(65 95 71 89)(66 96 72 90)
(1 19)(2 14)(3 21)(4 16)(5 23)(6 18)(7 13)(8 20)(9 15)(10 22)(11 17)(12 24)(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 70)(50 65)(51 72)(52 67)(53 62)(54 69)(55 64)(56 71)(57 66)(58 61)(59 68)(60 63)(73 85)(74 92)(75 87)(76 94)(77 89)(78 96)(79 91)(80 86)(81 93)(82 88)(83 95)(84 90)
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)(13,21)(14,20)(15,19)(16,18)(22,24)(25,30)(26,29)(27,28)(31,36)(32,35)(33,34)(37,39)(40,48)(41,47)(42,46)(43,45)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,63)(64,72)(65,71)(66,70)(67,69)(73,78)(74,77)(75,76)(79,84)(80,83)(81,82)(85,87)(88,96)(89,95)(90,94)(91,93), (1,55,7,49)(2,56,8,50)(3,57,9,51)(4,58,10,52)(5,59,11,53)(6,60,12,54)(13,70,19,64)(14,71,20,65)(15,72,21,66)(16,61,22,67)(17,62,23,68)(18,63,24,69)(25,79,31,73)(26,80,32,74)(27,81,33,75)(28,82,34,76)(29,83,35,77)(30,84,36,78)(37,85,43,91)(38,86,44,92)(39,87,45,93)(40,88,46,94)(41,89,47,95)(42,90,48,96), (1,28,7,34)(2,29,8,35)(3,30,9,36)(4,31,10,25)(5,32,11,26)(6,33,12,27)(13,46,19,40)(14,47,20,41)(15,48,21,42)(16,37,22,43)(17,38,23,44)(18,39,24,45)(49,82,55,76)(50,83,56,77)(51,84,57,78)(52,73,58,79)(53,74,59,80)(54,75,60,81)(61,91,67,85)(62,92,68,86)(63,93,69,87)(64,94,70,88)(65,95,71,89)(66,96,72,90), (1,19)(2,14)(3,21)(4,16)(5,23)(6,18)(7,13)(8,20)(9,15)(10,22)(11,17)(12,24)(25,43)(26,38)(27,45)(28,40)(29,47)(30,42)(31,37)(32,44)(33,39)(34,46)(35,41)(36,48)(49,70)(50,65)(51,72)(52,67)(53,62)(54,69)(55,64)(56,71)(57,66)(58,61)(59,68)(60,63)(73,85)(74,92)(75,87)(76,94)(77,89)(78,96)(79,91)(80,86)(81,93)(82,88)(83,95)(84,90)>;
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)(13,21)(14,20)(15,19)(16,18)(22,24)(25,30)(26,29)(27,28)(31,36)(32,35)(33,34)(37,39)(40,48)(41,47)(42,46)(43,45)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,63)(64,72)(65,71)(66,70)(67,69)(73,78)(74,77)(75,76)(79,84)(80,83)(81,82)(85,87)(88,96)(89,95)(90,94)(91,93), (1,55,7,49)(2,56,8,50)(3,57,9,51)(4,58,10,52)(5,59,11,53)(6,60,12,54)(13,70,19,64)(14,71,20,65)(15,72,21,66)(16,61,22,67)(17,62,23,68)(18,63,24,69)(25,79,31,73)(26,80,32,74)(27,81,33,75)(28,82,34,76)(29,83,35,77)(30,84,36,78)(37,85,43,91)(38,86,44,92)(39,87,45,93)(40,88,46,94)(41,89,47,95)(42,90,48,96), (1,28,7,34)(2,29,8,35)(3,30,9,36)(4,31,10,25)(5,32,11,26)(6,33,12,27)(13,46,19,40)(14,47,20,41)(15,48,21,42)(16,37,22,43)(17,38,23,44)(18,39,24,45)(49,82,55,76)(50,83,56,77)(51,84,57,78)(52,73,58,79)(53,74,59,80)(54,75,60,81)(61,91,67,85)(62,92,68,86)(63,93,69,87)(64,94,70,88)(65,95,71,89)(66,96,72,90), (1,19)(2,14)(3,21)(4,16)(5,23)(6,18)(7,13)(8,20)(9,15)(10,22)(11,17)(12,24)(25,43)(26,38)(27,45)(28,40)(29,47)(30,42)(31,37)(32,44)(33,39)(34,46)(35,41)(36,48)(49,70)(50,65)(51,72)(52,67)(53,62)(54,69)(55,64)(56,71)(57,66)(58,61)(59,68)(60,63)(73,85)(74,92)(75,87)(76,94)(77,89)(78,96)(79,91)(80,86)(81,93)(82,88)(83,95)(84,90) );
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),(13,21),(14,20),(15,19),(16,18),(22,24),(25,30),(26,29),(27,28),(31,36),(32,35),(33,34),(37,39),(40,48),(41,47),(42,46),(43,45),(49,60),(50,59),(51,58),(52,57),(53,56),(54,55),(61,63),(64,72),(65,71),(66,70),(67,69),(73,78),(74,77),(75,76),(79,84),(80,83),(81,82),(85,87),(88,96),(89,95),(90,94),(91,93)], [(1,55,7,49),(2,56,8,50),(3,57,9,51),(4,58,10,52),(5,59,11,53),(6,60,12,54),(13,70,19,64),(14,71,20,65),(15,72,21,66),(16,61,22,67),(17,62,23,68),(18,63,24,69),(25,79,31,73),(26,80,32,74),(27,81,33,75),(28,82,34,76),(29,83,35,77),(30,84,36,78),(37,85,43,91),(38,86,44,92),(39,87,45,93),(40,88,46,94),(41,89,47,95),(42,90,48,96)], [(1,28,7,34),(2,29,8,35),(3,30,9,36),(4,31,10,25),(5,32,11,26),(6,33,12,27),(13,46,19,40),(14,47,20,41),(15,48,21,42),(16,37,22,43),(17,38,23,44),(18,39,24,45),(49,82,55,76),(50,83,56,77),(51,84,57,78),(52,73,58,79),(53,74,59,80),(54,75,60,81),(61,91,67,85),(62,92,68,86),(63,93,69,87),(64,94,70,88),(65,95,71,89),(66,96,72,90)], [(1,19),(2,14),(3,21),(4,16),(5,23),(6,18),(7,13),(8,20),(9,15),(10,22),(11,17),(12,24),(25,43),(26,38),(27,45),(28,40),(29,47),(30,42),(31,37),(32,44),(33,39),(34,46),(35,41),(36,48),(49,70),(50,65),(51,72),(52,67),(53,62),(54,69),(55,64),(56,71),(57,66),(58,61),(59,68),(60,63),(73,85),(74,92),(75,87),(76,94),(77,89),(78,96),(79,91),(80,86),(81,93),(82,88),(83,95),(84,90)]])
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 |
Matrix representation of D12.35C23 ►in 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] >;
D12.35C23 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