metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.40D12, C22.3Dic12, (C2×C8).1D6, C24⋊1C4⋊2C2, (C2×C6).4Q16, C6.5(C2×Q16), (C2×C4).30D12, (C2×C12).41D4, C22⋊C8.3S3, C6.7(C8⋊C22), (C2×C24).1C22, C2.Dic12⋊4C2, C2.7(C2×Dic12), (C22×C4).92D6, (C22×C6).49D4, C2.10(C8⋊D6), C12.280(C4○D4), (C2×C12).739C23, C12.48D4.3C2, C22.102(C2×D12), C3⋊1(C23.48D4), C4⋊Dic3.11C22, C4.104(D4⋊2S3), (C22×C12).49C22, (C2×Dic6).11C22, C6.15(C22.D4), C2.11(C23.21D6), (C2×C6).122(C2×D4), (C3×C22⋊C8).5C2, (C2×C4⋊Dic3).12C2, (C2×C4).684(C22×S3), SmallGroup(192,281)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.40D12
G = < a,b,c,d,e | a2=b2=c2=1, d12=e2=c, ab=ba, ac=ca, dad-1=eae-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd11 >
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, C2.D8, C2×C4⋊C4, C22⋊Q8, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C4⋊Dic3, C6.D4, C2×C24, C2×Dic6, C22×Dic3, C22×C12, C23.48D4, C2.Dic12, C24⋊1C4, C3×C22⋊C8, C12.48D4, C2×C4⋊Dic3, C23.40D12
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, C4○D4, D12, C22×S3, C22.D4, C2×Q16, C8⋊C22, Dic12, C2×D12, D4⋊2S3, C23.48D4, C23.21D6, C2×Dic12, C8⋊D6, C23.40D12
►On 96 points
Generators in S96
(2 82)(4 84)(6 86)(8 88)(10 90)(12 92)(14 94)(16 96)(18 74)(20 76)(22 78)(24 80)(26 62)(28 64)(30 66)(32 68)(34 70)(36 72)(38 50)(40 52)(42 54)(44 56)(46 58)(48 60)
(1 93)(2 94)(3 95)(4 96)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 79)(12 80)(13 81)(14 82)(15 83)(16 84)(17 85)(18 86)(19 87)(20 88)(21 89)(22 90)(23 91)(24 92)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)(41 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 71)(48 72)
(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 48 13 36)(2 59 14 71)(3 46 15 34)(4 57 16 69)(5 44 17 32)(6 55 18 67)(7 42 19 30)(8 53 20 65)(9 40 21 28)(10 51 22 63)(11 38 23 26)(12 49 24 61)(25 92 37 80)(27 90 39 78)(29 88 41 76)(31 86 43 74)(33 84 45 96)(35 82 47 94)(50 79 62 91)(52 77 64 89)(54 75 66 87)(56 73 68 85)(58 95 70 83)(60 93 72 81)
G:=sub<Sym(96)| (2,82)(4,84)(6,86)(8,88)(10,90)(12,92)(14,94)(16,96)(18,74)(20,76)(22,78)(24,80)(26,62)(28,64)(30,66)(32,68)(34,70)(36,72)(38,50)(40,52)(42,54)(44,56)(46,58)(48,60), (1,93)(2,94)(3,95)(4,96)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,79)(12,80)(13,81)(14,82)(15,83)(16,84)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,92)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72), (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), (1,48,13,36)(2,59,14,71)(3,46,15,34)(4,57,16,69)(5,44,17,32)(6,55,18,67)(7,42,19,30)(8,53,20,65)(9,40,21,28)(10,51,22,63)(11,38,23,26)(12,49,24,61)(25,92,37,80)(27,90,39,78)(29,88,41,76)(31,86,43,74)(33,84,45,96)(35,82,47,94)(50,79,62,91)(52,77,64,89)(54,75,66,87)(56,73,68,85)(58,95,70,83)(60,93,72,81)>;
G:=Group( (2,82)(4,84)(6,86)(8,88)(10,90)(12,92)(14,94)(16,96)(18,74)(20,76)(22,78)(24,80)(26,62)(28,64)(30,66)(32,68)(34,70)(36,72)(38,50)(40,52)(42,54)(44,56)(46,58)(48,60), (1,93)(2,94)(3,95)(4,96)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,79)(12,80)(13,81)(14,82)(15,83)(16,84)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,92)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72), (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), (1,48,13,36)(2,59,14,71)(3,46,15,34)(4,57,16,69)(5,44,17,32)(6,55,18,67)(7,42,19,30)(8,53,20,65)(9,40,21,28)(10,51,22,63)(11,38,23,26)(12,49,24,61)(25,92,37,80)(27,90,39,78)(29,88,41,76)(31,86,43,74)(33,84,45,96)(35,82,47,94)(50,79,62,91)(52,77,64,89)(54,75,66,87)(56,73,68,85)(58,95,70,83)(60,93,72,81) );
G=PermutationGroup([[(2,82),(4,84),(6,86),(8,88),(10,90),(12,92),(14,94),(16,96),(18,74),(20,76),(22,78),(24,80),(26,62),(28,64),(30,66),(32,68),(34,70),(36,72),(38,50),(40,52),(42,54),(44,56),(46,58),(48,60)], [(1,93),(2,94),(3,95),(4,96),(5,73),(6,74),(7,75),(8,76),(9,77),(10,78),(11,79),(12,80),(13,81),(14,82),(15,83),(16,84),(17,85),(18,86),(19,87),(20,88),(21,89),(22,90),(23,91),(24,92),(25,49),(26,50),(27,51),(28,52),(29,53),(30,54),(31,55),(32,56),(33,57),(34,58),(35,59),(36,60),(37,61),(38,62),(39,63),(40,64),(41,65),(42,66),(43,67),(44,68),(45,69),(46,70),(47,71),(48,72)], [(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)], [(1,48,13,36),(2,59,14,71),(3,46,15,34),(4,57,16,69),(5,44,17,32),(6,55,18,67),(7,42,19,30),(8,53,20,65),(9,40,21,28),(10,51,22,63),(11,38,23,26),(12,49,24,61),(25,92,37,80),(27,90,39,78),(29,88,41,76),(31,86,43,74),(33,84,45,96),(35,82,47,94),(50,79,62,91),(52,77,64,89),(54,75,66,87),(56,73,68,85),(58,95,70,83),(60,93,72,81)]])
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 | Q16 | D12 | D12 | Dic12 | C8⋊C22 | D4⋊2S3 | C8⋊D6 |
| kernel | C23.40D12 | C2.Dic12 | C24⋊1C4 | 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 C23.40D12 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 68 | 55 |
| 0 | 0 | 18 | 50 |
| 0 | 46 | 0 | 0 |
| 27 | 0 | 0 | 0 |
| 0 | 0 | 37 | 11 |
| 0 | 0 | 48 | 36 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[0,72,0,0,1,0,0,0,0,0,68,18,0,0,55,50],[0,27,0,0,46,0,0,0,0,0,37,48,0,0,11,36] >;
C23.40D12 in GAP, Magma, Sage, TeX
C_2^3._{40}D_{12} % in TeX
G:=Group("C2^3.40D12"); // GroupNames label
G:=SmallGroup(192,281);
// by ID
G=gap.SmallGroup(192,281);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,336,254,219,310,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^12=e^2=c,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e^-1=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^11>;
// generators/relations