metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.39D12, C8⋊Dic3⋊6C2, (C2×C12).40D4, (C2×C8).107D6, (C2×C4).29D12, C22⋊C8.6S3, C6.6(C2×SD16), C2.Dic12⋊9C2, (C2×C6).13SD16, (C22×C6).48D4, (C22×C4).91D6, C6.7(C8.C22), C12.279(C4○D4), (C2×C24).118C22, (C2×C12).738C23, C22.8(C24⋊C2), C12.48D4.2C2, C2.10(C8.D6), C22.101(C2×D12), C3⋊1(C23.47D4), C4.103(D4⋊2S3), C4⋊Dic3.268C22, (C22×C12).48C22, (C2×Dic6).10C22, C6.14(C22.D4), C2.10(C23.21D6), C2.9(C2×C24⋊C2), (C2×C6).121(C2×D4), (C3×C22⋊C8).8C2, (C2×C4⋊Dic3).11C2, (C2×C4).683(C22×S3), SmallGroup(192,280)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.39D12
G = < a,b,c,d,e | a2=b2=c2=1, d12=c, e2=cb=bc, dad-1=eae-1=ab=ba, ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=d11 >
Subgroups: 288 in 104 conjugacy classes, 43 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C24, Dic6, C2×Dic3, C2×C12, C2×C12, C22×C6, C22⋊C8, Q8⋊C4, C4.Q8, C2×C4⋊C4, C22⋊Q8, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C4⋊Dic3, C6.D4, C2×C24, C2×Dic6, C22×Dic3, C22×C12, C23.47D4, C2.Dic12, C8⋊Dic3, C3×C22⋊C8, C12.48D4, C2×C4⋊Dic3, C23.39D12
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C4○D4, D12, C22×S3, C22.D4, C2×SD16, C8.C22, C24⋊C2, C2×D12, D4⋊2S3, C23.47D4, C23.21D6, C2×C24⋊C2, C8.D6, C23.39D12
►On 96 points
Generators in S96
(2 35)(4 37)(6 39)(8 41)(10 43)(12 45)(14 47)(16 25)(18 27)(20 29)(22 31)(24 33)(50 73)(52 75)(54 77)(56 79)(58 81)(60 83)(62 85)(64 87)(66 89)(68 91)(70 93)(72 95)
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 41)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 25)(17 26)(18 27)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(49 96)(50 73)(51 74)(52 75)(53 76)(54 77)(55 78)(56 79)(57 80)(58 81)(59 82)(60 83)(61 84)(62 85)(63 86)(64 87)(65 88)(66 89)(67 90)(68 91)(69 92)(70 93)(71 94)(72 95)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(49 61)(50 62)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(73 85)(74 86)(75 87)(76 88)(77 89)(78 90)(79 91)(80 92)(81 93)(82 94)(83 95)(84 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 70 46 81)(2 57 47 92)(3 68 48 79)(4 55 25 90)(5 66 26 77)(6 53 27 88)(7 64 28 75)(8 51 29 86)(9 62 30 73)(10 49 31 84)(11 60 32 95)(12 71 33 82)(13 58 34 93)(14 69 35 80)(15 56 36 91)(16 67 37 78)(17 54 38 89)(18 65 39 76)(19 52 40 87)(20 63 41 74)(21 50 42 85)(22 61 43 96)(23 72 44 83)(24 59 45 94)
G:=sub<Sym(96)| (2,35)(4,37)(6,39)(8,41)(10,43)(12,45)(14,47)(16,25)(18,27)(20,29)(22,31)(24,33)(50,73)(52,75)(54,77)(56,79)(58,81)(60,83)(62,85)(64,87)(66,89)(68,91)(70,93)(72,95), (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,25)(17,26)(18,27)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(49,96)(50,73)(51,74)(52,75)(53,76)(54,77)(55,78)(56,79)(57,80)(58,81)(59,82)(60,83)(61,84)(62,85)(63,86)(64,87)(65,88)(66,89)(67,90)(68,91)(69,92)(70,93)(71,94)(72,95), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96), (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), (1,70,46,81)(2,57,47,92)(3,68,48,79)(4,55,25,90)(5,66,26,77)(6,53,27,88)(7,64,28,75)(8,51,29,86)(9,62,30,73)(10,49,31,84)(11,60,32,95)(12,71,33,82)(13,58,34,93)(14,69,35,80)(15,56,36,91)(16,67,37,78)(17,54,38,89)(18,65,39,76)(19,52,40,87)(20,63,41,74)(21,50,42,85)(22,61,43,96)(23,72,44,83)(24,59,45,94)>;
G:=Group( (2,35)(4,37)(6,39)(8,41)(10,43)(12,45)(14,47)(16,25)(18,27)(20,29)(22,31)(24,33)(50,73)(52,75)(54,77)(56,79)(58,81)(60,83)(62,85)(64,87)(66,89)(68,91)(70,93)(72,95), (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,25)(17,26)(18,27)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(49,96)(50,73)(51,74)(52,75)(53,76)(54,77)(55,78)(56,79)(57,80)(58,81)(59,82)(60,83)(61,84)(62,85)(63,86)(64,87)(65,88)(66,89)(67,90)(68,91)(69,92)(70,93)(71,94)(72,95), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96), (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), (1,70,46,81)(2,57,47,92)(3,68,48,79)(4,55,25,90)(5,66,26,77)(6,53,27,88)(7,64,28,75)(8,51,29,86)(9,62,30,73)(10,49,31,84)(11,60,32,95)(12,71,33,82)(13,58,34,93)(14,69,35,80)(15,56,36,91)(16,67,37,78)(17,54,38,89)(18,65,39,76)(19,52,40,87)(20,63,41,74)(21,50,42,85)(22,61,43,96)(23,72,44,83)(24,59,45,94) );
G=PermutationGroup([[(2,35),(4,37),(6,39),(8,41),(10,43),(12,45),(14,47),(16,25),(18,27),(20,29),(22,31),(24,33),(50,73),(52,75),(54,77),(56,79),(58,81),(60,83),(62,85),(64,87),(66,89),(68,91),(70,93),(72,95)], [(1,34),(2,35),(3,36),(4,37),(5,38),(6,39),(7,40),(8,41),(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,25),(17,26),(18,27),(19,28),(20,29),(21,30),(22,31),(23,32),(24,33),(49,96),(50,73),(51,74),(52,75),(53,76),(54,77),(55,78),(56,79),(57,80),(58,81),(59,82),(60,83),(61,84),(62,85),(63,86),(64,87),(65,88),(66,89),(67,90),(68,91),(69,92),(70,93),(71,94),(72,95)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48),(49,61),(50,62),(51,63),(52,64),(53,65),(54,66),(55,67),(56,68),(57,69),(58,70),(59,71),(60,72),(73,85),(74,86),(75,87),(76,88),(77,89),(78,90),(79,91),(80,92),(81,93),(82,94),(83,95),(84,96)], [(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)], [(1,70,46,81),(2,57,47,92),(3,68,48,79),(4,55,25,90),(5,66,26,77),(6,53,27,88),(7,64,28,75),(8,51,29,86),(9,62,30,73),(10,49,31,84),(11,60,32,95),(12,71,33,82),(13,58,34,93),(14,69,35,80),(15,56,36,91),(16,67,37,78),(17,54,38,89),(18,65,39,76),(19,52,40,87),(20,63,41,74),(21,50,42,85),(22,61,43,96),(23,72,44,83),(24,59,45,94)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 12 | 12 | 12 | 12 | 24 | 24 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C4○D4 | SD16 | D12 | D12 | C24⋊C2 | C8.C22 | D4⋊2S3 | C8.D6 |
| kernel | C23.39D12 | C2.Dic12 | C8⋊Dic3 | C3×C22⋊C8 | C12.48D4 | C2×C4⋊Dic3 | C22⋊C8 | C2×C12 | C22×C6 | C2×C8 | C22×C4 | C12 | C2×C6 | C2×C4 | C23 | C22 | C6 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 4 | 4 | 2 | 2 | 8 | 1 | 2 | 2 |
Matrix representation of C23.39D12 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 67 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 56 | 57 | 0 | 0 |
| 0 | 0 | 9 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 11 |
| 0 | 0 | 0 | 0 | 71 | 13 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,67,0,0,0,0,12,0,0,0,0,0,0,0,0,72,0,0,0,0,1,1],[0,27,0,0,0,0,27,0,0,0,0,0,0,0,56,9,0,0,0,0,57,17,0,0,0,0,0,0,60,71,0,0,0,0,11,13] >;
C23.39D12 in GAP, Magma, Sage, TeX
C_2^3._{39}D_{12} % in TeX
G:=Group("C2^3.39D12"); // GroupNames label
G:=SmallGroup(192,280);
// by ID
G=gap.SmallGroup(192,280);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,254,219,58,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^12=c,e^2=c*b=b*c,d*a*d^-1=e*a*e^-1=a*b=b*a,a*c=c*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^11>;
// generators/relations