metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.18D6, C23.7Dic6, (C2×C12).50D4, (C22×C4).45D6, (C22×C6).13Q8, C6.11(C4⋊1D4), C2.5(C12⋊3D4), (C2×Dic3).55D4, C22.239(S3×D4), C6.57(C22⋊Q8), C3⋊2(C23.4Q8), C6.C42⋊29C2, (C23×C6).32C22, C22.46(C2×Dic6), C2.20(C23.9D6), C2.8(C12.48D4), C22.96(C4○D12), C23.378(C22×S3), (C22×C12).58C22, (C22×C6).324C23, C22.94(D4⋊2S3), C2.6(C23.23D6), C6.29(C22.D4), C2.22(Dic3.D4), (C22×Dic3).40C22, (C2×C4⋊Dic3)⋊9C2, (C2×C6).34(C2×Q8), (C2×Dic3⋊C4)⋊9C2, (C2×C6).318(C2×D4), (C2×C6).78(C4○D4), (C2×C4).29(C3⋊D4), (C6×C22⋊C4).14C2, (C2×C22⋊C4).12S3, C22.124(C2×C3⋊D4), (C2×C6.D4).11C2, SmallGroup(192,508)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.18D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=dc=cd, f2=c, ab=ba, ac=ca, eae-1=ad=da, faf-1=abd, bc=cb, bd=db, be=eb, bf=fb, ce=ec, cf=fc, de=ed, df=fd, fef-1=cde5 >
Subgroups: 472 in 186 conjugacy classes, 63 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C23, C23, C23, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C22×C6, C22×C6, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, Dic3⋊C4, C4⋊Dic3, C6.D4, C3×C22⋊C4, C22×Dic3, C22×C12, C23×C6, C23.4Q8, C6.C42, C2×Dic3⋊C4, C2×C4⋊Dic3, C2×C6.D4, C6×C22⋊C4, C24.18D6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, Dic6, C3⋊D4, C22×S3, C22⋊Q8, C22.D4, C4⋊1D4, C2×Dic6, C4○D12, S3×D4, D4⋊2S3, C2×C3⋊D4, C23.4Q8, Dic3.D4, C23.9D6, C12.48D4, C23.23D6, C12⋊3D4, C24.18D6
►On 96 points
Generators in S96
(2 24)(4 14)(6 16)(8 18)(10 20)(12 22)(25 48)(26 76)(27 38)(28 78)(29 40)(30 80)(31 42)(32 82)(33 44)(34 84)(35 46)(36 74)(37 89)(39 91)(41 93)(43 95)(45 85)(47 87)(50 63)(52 65)(54 67)(56 69)(58 71)(60 61)(73 86)(75 88)(77 90)(79 92)(81 94)(83 96)
(1 70)(2 71)(3 72)(4 61)(5 62)(6 63)(7 64)(8 65)(9 66)(10 67)(11 68)(12 69)(13 59)(14 60)(15 49)(16 50)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 57)(24 58)(25 48)(26 37)(27 38)(28 39)(29 40)(30 41)(31 42)(32 43)(33 44)(34 45)(35 46)(36 47)(73 86)(74 87)(75 88)(76 89)(77 90)(78 91)(79 92)(80 93)(81 94)(82 95)(83 96)(84 85)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 13)(10 14)(11 15)(12 16)(25 94)(26 95)(27 96)(28 85)(29 86)(30 87)(31 88)(32 89)(33 90)(34 91)(35 92)(36 93)(37 82)(38 83)(39 84)(40 73)(41 74)(42 75)(43 76)(44 77)(45 78)(46 79)(47 80)(48 81)(49 68)(50 69)(51 70)(52 71)(53 72)(54 61)(55 62)(56 63)(57 64)(58 65)(59 66)(60 67)
(1 23)(2 24)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(11 21)(12 22)(25 88)(26 89)(27 90)(28 91)(29 92)(30 93)(31 94)(32 95)(33 96)(34 85)(35 86)(36 87)(37 76)(38 77)(39 78)(40 79)(41 80)(42 81)(43 82)(44 83)(45 84)(46 73)(47 74)(48 75)(49 62)(50 63)(51 64)(52 65)(53 66)(54 67)(55 68)(56 69)(57 70)(58 71)(59 72)(60 61)
(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 76 17 43)(2 75 18 42)(3 74 19 41)(4 73 20 40)(5 84 21 39)(6 83 22 38)(7 82 23 37)(8 81 24 48)(9 80 13 47)(10 79 14 46)(11 78 15 45)(12 77 16 44)(25 65 94 58)(26 64 95 57)(27 63 96 56)(28 62 85 55)(29 61 86 54)(30 72 87 53)(31 71 88 52)(32 70 89 51)(33 69 90 50)(34 68 91 49)(35 67 92 60)(36 66 93 59)
G:=sub<Sym(96)| (2,24)(4,14)(6,16)(8,18)(10,20)(12,22)(25,48)(26,76)(27,38)(28,78)(29,40)(30,80)(31,42)(32,82)(33,44)(34,84)(35,46)(36,74)(37,89)(39,91)(41,93)(43,95)(45,85)(47,87)(50,63)(52,65)(54,67)(56,69)(58,71)(60,61)(73,86)(75,88)(77,90)(79,92)(81,94)(83,96), (1,70)(2,71)(3,72)(4,61)(5,62)(6,63)(7,64)(8,65)(9,66)(10,67)(11,68)(12,69)(13,59)(14,60)(15,49)(16,50)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,57)(24,58)(25,48)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43)(33,44)(34,45)(35,46)(36,47)(73,86)(74,87)(75,88)(76,89)(77,90)(78,91)(79,92)(80,93)(81,94)(82,95)(83,96)(84,85), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,13)(10,14)(11,15)(12,16)(25,94)(26,95)(27,96)(28,85)(29,86)(30,87)(31,88)(32,89)(33,90)(34,91)(35,92)(36,93)(37,82)(38,83)(39,84)(40,73)(41,74)(42,75)(43,76)(44,77)(45,78)(46,79)(47,80)(48,81)(49,68)(50,69)(51,70)(52,71)(53,72)(54,61)(55,62)(56,63)(57,64)(58,65)(59,66)(60,67), (1,23)(2,24)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(11,21)(12,22)(25,88)(26,89)(27,90)(28,91)(29,92)(30,93)(31,94)(32,95)(33,96)(34,85)(35,86)(36,87)(37,76)(38,77)(39,78)(40,79)(41,80)(42,81)(43,82)(44,83)(45,84)(46,73)(47,74)(48,75)(49,62)(50,63)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,70)(58,71)(59,72)(60,61), (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), (1,76,17,43)(2,75,18,42)(3,74,19,41)(4,73,20,40)(5,84,21,39)(6,83,22,38)(7,82,23,37)(8,81,24,48)(9,80,13,47)(10,79,14,46)(11,78,15,45)(12,77,16,44)(25,65,94,58)(26,64,95,57)(27,63,96,56)(28,62,85,55)(29,61,86,54)(30,72,87,53)(31,71,88,52)(32,70,89,51)(33,69,90,50)(34,68,91,49)(35,67,92,60)(36,66,93,59)>;
G:=Group( (2,24)(4,14)(6,16)(8,18)(10,20)(12,22)(25,48)(26,76)(27,38)(28,78)(29,40)(30,80)(31,42)(32,82)(33,44)(34,84)(35,46)(36,74)(37,89)(39,91)(41,93)(43,95)(45,85)(47,87)(50,63)(52,65)(54,67)(56,69)(58,71)(60,61)(73,86)(75,88)(77,90)(79,92)(81,94)(83,96), (1,70)(2,71)(3,72)(4,61)(5,62)(6,63)(7,64)(8,65)(9,66)(10,67)(11,68)(12,69)(13,59)(14,60)(15,49)(16,50)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,57)(24,58)(25,48)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43)(33,44)(34,45)(35,46)(36,47)(73,86)(74,87)(75,88)(76,89)(77,90)(78,91)(79,92)(80,93)(81,94)(82,95)(83,96)(84,85), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,13)(10,14)(11,15)(12,16)(25,94)(26,95)(27,96)(28,85)(29,86)(30,87)(31,88)(32,89)(33,90)(34,91)(35,92)(36,93)(37,82)(38,83)(39,84)(40,73)(41,74)(42,75)(43,76)(44,77)(45,78)(46,79)(47,80)(48,81)(49,68)(50,69)(51,70)(52,71)(53,72)(54,61)(55,62)(56,63)(57,64)(58,65)(59,66)(60,67), (1,23)(2,24)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(11,21)(12,22)(25,88)(26,89)(27,90)(28,91)(29,92)(30,93)(31,94)(32,95)(33,96)(34,85)(35,86)(36,87)(37,76)(38,77)(39,78)(40,79)(41,80)(42,81)(43,82)(44,83)(45,84)(46,73)(47,74)(48,75)(49,62)(50,63)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,70)(58,71)(59,72)(60,61), (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), (1,76,17,43)(2,75,18,42)(3,74,19,41)(4,73,20,40)(5,84,21,39)(6,83,22,38)(7,82,23,37)(8,81,24,48)(9,80,13,47)(10,79,14,46)(11,78,15,45)(12,77,16,44)(25,65,94,58)(26,64,95,57)(27,63,96,56)(28,62,85,55)(29,61,86,54)(30,72,87,53)(31,71,88,52)(32,70,89,51)(33,69,90,50)(34,68,91,49)(35,67,92,60)(36,66,93,59) );
G=PermutationGroup([[(2,24),(4,14),(6,16),(8,18),(10,20),(12,22),(25,48),(26,76),(27,38),(28,78),(29,40),(30,80),(31,42),(32,82),(33,44),(34,84),(35,46),(36,74),(37,89),(39,91),(41,93),(43,95),(45,85),(47,87),(50,63),(52,65),(54,67),(56,69),(58,71),(60,61),(73,86),(75,88),(77,90),(79,92),(81,94),(83,96)], [(1,70),(2,71),(3,72),(4,61),(5,62),(6,63),(7,64),(8,65),(9,66),(10,67),(11,68),(12,69),(13,59),(14,60),(15,49),(16,50),(17,51),(18,52),(19,53),(20,54),(21,55),(22,56),(23,57),(24,58),(25,48),(26,37),(27,38),(28,39),(29,40),(30,41),(31,42),(32,43),(33,44),(34,45),(35,46),(36,47),(73,86),(74,87),(75,88),(76,89),(77,90),(78,91),(79,92),(80,93),(81,94),(82,95),(83,96),(84,85)], [(1,17),(2,18),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,13),(10,14),(11,15),(12,16),(25,94),(26,95),(27,96),(28,85),(29,86),(30,87),(31,88),(32,89),(33,90),(34,91),(35,92),(36,93),(37,82),(38,83),(39,84),(40,73),(41,74),(42,75),(43,76),(44,77),(45,78),(46,79),(47,80),(48,81),(49,68),(50,69),(51,70),(52,71),(53,72),(54,61),(55,62),(56,63),(57,64),(58,65),(59,66),(60,67)], [(1,23),(2,24),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(11,21),(12,22),(25,88),(26,89),(27,90),(28,91),(29,92),(30,93),(31,94),(32,95),(33,96),(34,85),(35,86),(36,87),(37,76),(38,77),(39,78),(40,79),(41,80),(42,81),(43,82),(44,83),(45,84),(46,73),(47,74),(48,75),(49,62),(50,63),(51,64),(52,65),(53,66),(54,67),(55,68),(56,69),(57,70),(58,71),(59,72),(60,61)], [(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)], [(1,76,17,43),(2,75,18,42),(3,74,19,41),(4,73,20,40),(5,84,21,39),(6,83,22,38),(7,82,23,37),(8,81,24,48),(9,80,13,47),(10,79,14,46),(11,78,15,45),(12,77,16,44),(25,65,94,58),(26,64,95,57),(27,63,96,56),(28,62,85,55),(29,61,86,54),(30,72,87,53),(31,71,88,52),(32,70,89,51),(33,69,90,50),(34,68,91,49),(35,67,92,60),(36,66,93,59)]])
42 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | ··· | 6G | 6H | 6I | 6J | 6K | 12A | ··· | 12H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 2 | 4 | 4 | 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 | Q8 | D6 | D6 | C4○D4 | C3⋊D4 | Dic6 | C4○D12 | S3×D4 | D4⋊2S3 |
| kernel | C24.18D6 | C6.C42 | C2×Dic3⋊C4 | C2×C4⋊Dic3 | C2×C6.D4 | C6×C22⋊C4 | C2×C22⋊C4 | C2×Dic3 | C2×C12 | C22×C6 | C22×C4 | C24 | C2×C6 | C2×C4 | C23 | C22 | C22 | C22 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 2 | 2 | 2 | 1 | 6 | 4 | 4 | 4 | 2 | 2 |
Matrix representation of C24.18D6 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 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 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 10 | 6 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 1 |
| 0 | 0 | 0 | 0 | 8 | 2 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,11,10,0,0,0,0,0,6],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,0,0,5,0,0,0,0,0,0,0,11,8,0,0,0,0,1,2] >;
C24.18D6 in GAP, Magma, Sage, TeX
C_2^4._{18}D_6 % in TeX
G:=Group("C2^4.18D6"); // GroupNames label
G:=SmallGroup(192,508);
// by ID
G=gap.SmallGroup(192,508);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,701,344,254,387,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=d*c=c*d,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*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*d*e^5>;
// generators/relations