metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.32D6, (C2×C12)⋊23D4, (C22×C6)⋊8D4, (C2×Dic3)⋊7D4, (C22×S3)⋊5D4, (C22×D4)⋊5S3, C6.68C22≀C2, C23⋊4(C3⋊D4), C3⋊3(C23⋊2D4), C6.37(C4⋊1D4), C22.284(S3×D4), (C22×C4).169D6, C2.35(C23⋊2D6), C2.26(D6⋊3D4), C6.131(C4⋊D4), C2.27(C12⋊3D4), C6.C42⋊46C2, C2.7(C24⋊4S3), (C23×C6).49C22, (S3×C23).25C22, C23.395(C22×S3), (C22×C6).368C23, C2.35(C23.14D6), (C22×C12).396C22, C22.107(D4⋊2S3), (C22×Dic3).69C22, (D4×C2×C6)⋊12C2, (C2×D6⋊C4)⋊39C2, (C2×C4)⋊5(C3⋊D4), (C2×C6).557(C2×D4), (C22×C3⋊D4)⋊2C2, (C2×C6).164(C4○D4), (C2×C6.D4)⋊12C2, C22.219(C2×C3⋊D4), SmallGroup(192,782)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.32D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=e6=1, f2=c, ab=ba, ac=ca, eae-1=ad=da, faf-1=abd, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce-1 >
Subgroups: 952 in 322 conjugacy classes, 69 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C22×C6, C22×C6, C2.C42, C2×C22⋊C4, C22×D4, C22×D4, D6⋊C4, C6.D4, C22×Dic3, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, S3×C23, C23×C6, C23⋊2D4, C6.C42, C2×D6⋊C4, C2×C6.D4, C22×C3⋊D4, D4×C2×C6, C24.32D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C22≀C2, C4⋊D4, C4⋊1D4, S3×D4, D4⋊2S3, C2×C3⋊D4, C23⋊2D4, C23⋊2D6, D6⋊3D4, C23.14D6, C12⋊3D4, C24⋊4S3, C24.32D6
►On 96 points
Generators in S96
(1 23)(2 86)(3 19)(4 88)(5 21)(6 90)(7 61)(8 42)(9 63)(10 38)(11 65)(12 40)(13 25)(14 94)(15 27)(16 96)(17 29)(18 92)(20 76)(22 78)(24 74)(26 82)(28 84)(30 80)(31 43)(32 70)(33 45)(34 72)(35 47)(36 68)(37 53)(39 49)(41 51)(44 58)(46 60)(48 56)(50 66)(52 62)(54 64)(55 67)(57 69)(59 71)(73 85)(75 87)(77 89)(79 91)(81 93)(83 95)
(1 17)(2 18)(3 13)(4 14)(5 15)(6 16)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(31 37)(32 38)(33 39)(34 40)(35 41)(36 42)(43 53)(44 54)(45 49)(46 50)(47 51)(48 52)(55 61)(56 62)(57 63)(58 64)(59 65)(60 66)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 29)(8 30)(9 25)(10 26)(11 27)(12 28)(13 63)(14 64)(15 65)(16 66)(17 61)(18 62)(19 69)(20 70)(21 71)(22 72)(23 67)(24 68)(31 75)(32 76)(33 77)(34 78)(35 73)(36 74)(37 81)(38 82)(39 83)(40 84)(41 79)(42 80)(43 87)(44 88)(45 89)(46 90)(47 85)(48 86)(49 95)(50 96)(51 91)(52 92)(53 93)(54 94)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 51)(8 52)(9 53)(10 54)(11 49)(12 50)(13 81)(14 82)(15 83)(16 84)(17 79)(18 80)(19 87)(20 88)(21 89)(22 90)(23 85)(24 86)(25 93)(26 94)(27 95)(28 96)(29 91)(30 92)(31 57)(32 58)(33 59)(34 60)(35 55)(36 56)(37 63)(38 64)(39 65)(40 66)(41 61)(42 62)(43 69)(44 70)(45 71)(46 72)(47 67)(48 68)
(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 6 55 60)(2 59 56 5)(3 4 57 58)(7 72 29 22)(8 21 30 71)(9 70 25 20)(10 19 26 69)(11 68 27 24)(12 23 28 67)(13 14 63 64)(15 18 65 62)(16 61 66 17)(31 32 75 76)(33 36 77 74)(34 73 78 35)(37 38 81 82)(39 42 83 80)(40 79 84 41)(43 54 87 94)(44 93 88 53)(45 52 89 92)(46 91 90 51)(47 50 85 96)(48 95 86 49)
G:=sub<Sym(96)| (1,23)(2,86)(3,19)(4,88)(5,21)(6,90)(7,61)(8,42)(9,63)(10,38)(11,65)(12,40)(13,25)(14,94)(15,27)(16,96)(17,29)(18,92)(20,76)(22,78)(24,74)(26,82)(28,84)(30,80)(31,43)(32,70)(33,45)(34,72)(35,47)(36,68)(37,53)(39,49)(41,51)(44,58)(46,60)(48,56)(50,66)(52,62)(54,64)(55,67)(57,69)(59,71)(73,85)(75,87)(77,89)(79,91)(81,93)(83,95), (1,17)(2,18)(3,13)(4,14)(5,15)(6,16)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(31,37)(32,38)(33,39)(34,40)(35,41)(36,42)(43,53)(44,54)(45,49)(46,50)(47,51)(48,52)(55,61)(56,62)(57,63)(58,64)(59,65)(60,66)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96), (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,29)(8,30)(9,25)(10,26)(11,27)(12,28)(13,63)(14,64)(15,65)(16,66)(17,61)(18,62)(19,69)(20,70)(21,71)(22,72)(23,67)(24,68)(31,75)(32,76)(33,77)(34,78)(35,73)(36,74)(37,81)(38,82)(39,83)(40,84)(41,79)(42,80)(43,87)(44,88)(45,89)(46,90)(47,85)(48,86)(49,95)(50,96)(51,91)(52,92)(53,93)(54,94), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,51)(8,52)(9,53)(10,54)(11,49)(12,50)(13,81)(14,82)(15,83)(16,84)(17,79)(18,80)(19,87)(20,88)(21,89)(22,90)(23,85)(24,86)(25,93)(26,94)(27,95)(28,96)(29,91)(30,92)(31,57)(32,58)(33,59)(34,60)(35,55)(36,56)(37,63)(38,64)(39,65)(40,66)(41,61)(42,62)(43,69)(44,70)(45,71)(46,72)(47,67)(48,68), (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), (1,6,55,60)(2,59,56,5)(3,4,57,58)(7,72,29,22)(8,21,30,71)(9,70,25,20)(10,19,26,69)(11,68,27,24)(12,23,28,67)(13,14,63,64)(15,18,65,62)(16,61,66,17)(31,32,75,76)(33,36,77,74)(34,73,78,35)(37,38,81,82)(39,42,83,80)(40,79,84,41)(43,54,87,94)(44,93,88,53)(45,52,89,92)(46,91,90,51)(47,50,85,96)(48,95,86,49)>;
G:=Group( (1,23)(2,86)(3,19)(4,88)(5,21)(6,90)(7,61)(8,42)(9,63)(10,38)(11,65)(12,40)(13,25)(14,94)(15,27)(16,96)(17,29)(18,92)(20,76)(22,78)(24,74)(26,82)(28,84)(30,80)(31,43)(32,70)(33,45)(34,72)(35,47)(36,68)(37,53)(39,49)(41,51)(44,58)(46,60)(48,56)(50,66)(52,62)(54,64)(55,67)(57,69)(59,71)(73,85)(75,87)(77,89)(79,91)(81,93)(83,95), (1,17)(2,18)(3,13)(4,14)(5,15)(6,16)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(31,37)(32,38)(33,39)(34,40)(35,41)(36,42)(43,53)(44,54)(45,49)(46,50)(47,51)(48,52)(55,61)(56,62)(57,63)(58,64)(59,65)(60,66)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96), (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,29)(8,30)(9,25)(10,26)(11,27)(12,28)(13,63)(14,64)(15,65)(16,66)(17,61)(18,62)(19,69)(20,70)(21,71)(22,72)(23,67)(24,68)(31,75)(32,76)(33,77)(34,78)(35,73)(36,74)(37,81)(38,82)(39,83)(40,84)(41,79)(42,80)(43,87)(44,88)(45,89)(46,90)(47,85)(48,86)(49,95)(50,96)(51,91)(52,92)(53,93)(54,94), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,51)(8,52)(9,53)(10,54)(11,49)(12,50)(13,81)(14,82)(15,83)(16,84)(17,79)(18,80)(19,87)(20,88)(21,89)(22,90)(23,85)(24,86)(25,93)(26,94)(27,95)(28,96)(29,91)(30,92)(31,57)(32,58)(33,59)(34,60)(35,55)(36,56)(37,63)(38,64)(39,65)(40,66)(41,61)(42,62)(43,69)(44,70)(45,71)(46,72)(47,67)(48,68), (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), (1,6,55,60)(2,59,56,5)(3,4,57,58)(7,72,29,22)(8,21,30,71)(9,70,25,20)(10,19,26,69)(11,68,27,24)(12,23,28,67)(13,14,63,64)(15,18,65,62)(16,61,66,17)(31,32,75,76)(33,36,77,74)(34,73,78,35)(37,38,81,82)(39,42,83,80)(40,79,84,41)(43,54,87,94)(44,93,88,53)(45,52,89,92)(46,91,90,51)(47,50,85,96)(48,95,86,49) );
G=PermutationGroup([[(1,23),(2,86),(3,19),(4,88),(5,21),(6,90),(7,61),(8,42),(9,63),(10,38),(11,65),(12,40),(13,25),(14,94),(15,27),(16,96),(17,29),(18,92),(20,76),(22,78),(24,74),(26,82),(28,84),(30,80),(31,43),(32,70),(33,45),(34,72),(35,47),(36,68),(37,53),(39,49),(41,51),(44,58),(46,60),(48,56),(50,66),(52,62),(54,64),(55,67),(57,69),(59,71),(73,85),(75,87),(77,89),(79,91),(81,93),(83,95)], [(1,17),(2,18),(3,13),(4,14),(5,15),(6,16),(7,67),(8,68),(9,69),(10,70),(11,71),(12,72),(19,25),(20,26),(21,27),(22,28),(23,29),(24,30),(31,37),(32,38),(33,39),(34,40),(35,41),(36,42),(43,53),(44,54),(45,49),(46,50),(47,51),(48,52),(55,61),(56,62),(57,63),(58,64),(59,65),(60,66),(73,79),(74,80),(75,81),(76,82),(77,83),(78,84),(85,91),(86,92),(87,93),(88,94),(89,95),(90,96)], [(1,55),(2,56),(3,57),(4,58),(5,59),(6,60),(7,29),(8,30),(9,25),(10,26),(11,27),(12,28),(13,63),(14,64),(15,65),(16,66),(17,61),(18,62),(19,69),(20,70),(21,71),(22,72),(23,67),(24,68),(31,75),(32,76),(33,77),(34,78),(35,73),(36,74),(37,81),(38,82),(39,83),(40,84),(41,79),(42,80),(43,87),(44,88),(45,89),(46,90),(47,85),(48,86),(49,95),(50,96),(51,91),(52,92),(53,93),(54,94)], [(1,73),(2,74),(3,75),(4,76),(5,77),(6,78),(7,51),(8,52),(9,53),(10,54),(11,49),(12,50),(13,81),(14,82),(15,83),(16,84),(17,79),(18,80),(19,87),(20,88),(21,89),(22,90),(23,85),(24,86),(25,93),(26,94),(27,95),(28,96),(29,91),(30,92),(31,57),(32,58),(33,59),(34,60),(35,55),(36,56),(37,63),(38,64),(39,65),(40,66),(41,61),(42,62),(43,69),(44,70),(45,71),(46,72),(47,67),(48,68)], [(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)], [(1,6,55,60),(2,59,56,5),(3,4,57,58),(7,72,29,22),(8,21,30,71),(9,70,25,20),(10,19,26,69),(11,68,27,24),(12,23,28,67),(13,14,63,64),(15,18,65,62),(16,61,66,17),(31,32,75,76),(33,36,77,74),(34,73,78,35),(37,38,81,82),(39,42,83,80),(40,79,84,41),(43,54,87,94),(44,93,88,53),(45,52,89,92),(46,91,90,51),(47,50,85,96),(48,95,86,49)]])
42 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 3 | 4A | 4B | 4C | ··· | 4H | 6A | ··· | 6G | 6H | ··· | 6O | 12A | 12B | 12C | 12D |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 12 | 12 | 2 | 4 | 4 | 12 | ··· | 12 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D4 | D6 | D6 | C4○D4 | C3⋊D4 | C3⋊D4 | S3×D4 | D4⋊2S3 |
| kernel | C24.32D6 | C6.C42 | C2×D6⋊C4 | C2×C6.D4 | C22×C3⋊D4 | D4×C2×C6 | C22×D4 | C2×Dic3 | C2×C12 | C22×S3 | C22×C6 | C22×C4 | C24 | C2×C6 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 2 | 2 | 4 | 1 | 2 | 2 | 4 | 8 | 3 | 1 |
Matrix representation of C24.32D6 ►in GL6(𝔽13)
| 2 | 4 | 0 | 0 | 0 | 0 |
| 9 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 11 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 3 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 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 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 11 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 11 |
| 0 | 0 | 0 | 0 | 12 | 8 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 2 |
| 0 | 0 | 0 | 0 | 0 | 5 |
G:=sub<GL(6,GF(13))| [2,9,0,0,0,0,4,11,0,0,0,0,0,0,12,0,0,0,0,0,11,1,0,0,0,0,0,0,12,0,0,0,0,0,3,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,11,1,0,0,0,0,0,0,5,12,0,0,0,0,11,8],[12,0,0,0,0,0,12,1,0,0,0,0,0,0,1,12,0,0,0,0,2,12,0,0,0,0,0,0,8,0,0,0,0,0,2,5] >;
C24.32D6 in GAP, Magma, Sage, TeX
C_2^4._{32}D_6 % in TeX
G:=Group("C2^4.32D6"); // GroupNames label
G:=SmallGroup(192,782);
// by ID
G=gap.SmallGroup(192,782);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,254,387,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^6=1,f^2=c,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*b*d,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*e^-1>;
// generators/relations