metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.5C42, C23.32D6, C22.11D12, C22.3Dic6, (C2×C12)⋊3C4, C6.9(C4⋊C4), (C2×C6).4Q8, (C2×C4)⋊2Dic3, (C2×C6).32D4, C2.2(D6⋊C4), (C2×Dic3)⋊2C4, C3⋊(C2.C42), (C22×C4).4S3, C2.5(C4×Dic3), (C22×C12).1C2, C22.12(C4×S3), C2.2(C4⋊Dic3), C6.11(C22⋊C4), C2.2(Dic3⋊C4), C2.2(C6.D4), C22.16(C3⋊D4), (C22×C6).31C22, (C22×Dic3).1C2, C22.10(C2×Dic3), (C2×C6).13(C2×C4), SmallGroup(96,38)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.C42
G = < a,b,c,d,e | a2=b2=c2=1, d6=b, e2=abc, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=cd5 >
Subgroups: 146 in 76 conjugacy classes, 45 normal (19 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, Dic3, C12, C2×C6, C2×C6, C22×C4, C22×C4, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2.C42, C22×Dic3, C22×C12, C6.C42
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C42, C22⋊C4, C4⋊C4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2.C42, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C6.C42
►Regular action on 96 points
(1 60)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 56)(10 57)(11 58)(12 59)(13 93)(14 94)(15 95)(16 96)(17 85)(18 86)(19 87)(20 88)(21 89)(22 90)(23 91)(24 92)(25 84)(26 73)(27 74)(28 75)(29 76)(30 77)(31 78)(32 79)(33 80)(34 81)(35 82)(36 83)(37 69)(38 70)(39 71)(40 72)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(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 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(25 70)(26 71)(27 72)(28 61)(29 62)(30 63)(31 64)(32 65)(33 66)(34 67)(35 68)(36 69)(37 83)(38 84)(39 73)(40 74)(41 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 81)(48 82)(49 88)(50 89)(51 90)(52 91)(53 92)(54 93)(55 94)(56 95)(57 96)(58 85)(59 86)(60 87)
(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 38 93 31)(2 77 94 69)(3 48 95 29)(4 75 96 67)(5 46 85 27)(6 73 86 65)(7 44 87 25)(8 83 88 63)(9 42 89 35)(10 81 90 61)(11 40 91 33)(12 79 92 71)(13 78 60 70)(14 37 49 30)(15 76 50 68)(16 47 51 28)(17 74 52 66)(18 45 53 26)(19 84 54 64)(20 43 55 36)(21 82 56 62)(22 41 57 34)(23 80 58 72)(24 39 59 32)
G:=sub<Sym(96)| (1,60)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(11,58)(12,59)(13,93)(14,94)(15,95)(16,96)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,92)(25,84)(26,73)(27,74)(28,75)(29,76)(30,77)(31,78)(32,79)(33,80)(34,81)(35,82)(36,83)(37,69)(38,70)(39,71)(40,72)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,70)(26,71)(27,72)(28,61)(29,62)(30,63)(31,64)(32,65)(33,66)(34,67)(35,68)(36,69)(37,83)(38,84)(39,73)(40,74)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,81)(48,82)(49,88)(50,89)(51,90)(52,91)(53,92)(54,93)(55,94)(56,95)(57,96)(58,85)(59,86)(60,87), (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), (1,38,93,31)(2,77,94,69)(3,48,95,29)(4,75,96,67)(5,46,85,27)(6,73,86,65)(7,44,87,25)(8,83,88,63)(9,42,89,35)(10,81,90,61)(11,40,91,33)(12,79,92,71)(13,78,60,70)(14,37,49,30)(15,76,50,68)(16,47,51,28)(17,74,52,66)(18,45,53,26)(19,84,54,64)(20,43,55,36)(21,82,56,62)(22,41,57,34)(23,80,58,72)(24,39,59,32)>;
G:=Group( (1,60)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(11,58)(12,59)(13,93)(14,94)(15,95)(16,96)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,92)(25,84)(26,73)(27,74)(28,75)(29,76)(30,77)(31,78)(32,79)(33,80)(34,81)(35,82)(36,83)(37,69)(38,70)(39,71)(40,72)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,70)(26,71)(27,72)(28,61)(29,62)(30,63)(31,64)(32,65)(33,66)(34,67)(35,68)(36,69)(37,83)(38,84)(39,73)(40,74)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,81)(48,82)(49,88)(50,89)(51,90)(52,91)(53,92)(54,93)(55,94)(56,95)(57,96)(58,85)(59,86)(60,87), (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), (1,38,93,31)(2,77,94,69)(3,48,95,29)(4,75,96,67)(5,46,85,27)(6,73,86,65)(7,44,87,25)(8,83,88,63)(9,42,89,35)(10,81,90,61)(11,40,91,33)(12,79,92,71)(13,78,60,70)(14,37,49,30)(15,76,50,68)(16,47,51,28)(17,74,52,66)(18,45,53,26)(19,84,54,64)(20,43,55,36)(21,82,56,62)(22,41,57,34)(23,80,58,72)(24,39,59,32) );
G=PermutationGroup([[(1,60),(2,49),(3,50),(4,51),(5,52),(6,53),(7,54),(8,55),(9,56),(10,57),(11,58),(12,59),(13,93),(14,94),(15,95),(16,96),(17,85),(18,86),(19,87),(20,88),(21,89),(22,90),(23,91),(24,92),(25,84),(26,73),(27,74),(28,75),(29,76),(30,77),(31,78),(32,79),(33,80),(34,81),(35,82),(36,83),(37,69),(38,70),(39,71),(40,72),(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48),(49,55),(50,56),(51,57),(52,58),(53,59),(54,60),(61,67),(62,68),(63,69),(64,70),(65,71),(66,72),(73,79),(74,80),(75,81),(76,82),(77,83),(78,84),(85,91),(86,92),(87,93),(88,94),(89,95),(90,96)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,13),(8,14),(9,15),(10,16),(11,17),(12,18),(25,70),(26,71),(27,72),(28,61),(29,62),(30,63),(31,64),(32,65),(33,66),(34,67),(35,68),(36,69),(37,83),(38,84),(39,73),(40,74),(41,75),(42,76),(43,77),(44,78),(45,79),(46,80),(47,81),(48,82),(49,88),(50,89),(51,90),(52,91),(53,92),(54,93),(55,94),(56,95),(57,96),(58,85),(59,86),(60,87)], [(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)], [(1,38,93,31),(2,77,94,69),(3,48,95,29),(4,75,96,67),(5,46,85,27),(6,73,86,65),(7,44,87,25),(8,83,88,63),(9,42,89,35),(10,81,90,61),(11,40,91,33),(12,79,92,71),(13,78,60,70),(14,37,49,30),(15,76,50,68),(16,47,51,28),(17,74,52,66),(18,45,53,26),(19,84,54,64),(20,43,55,36),(21,82,56,62),(22,41,57,34),(23,80,58,72),(24,39,59,32)]])
C6.C42 is a maximal subgroup of
C23.35D12 C22.2D24 (C6×D4)⋊C4 (C6×Q8)⋊C4 (C2×C12)⋊Q8 C6.(C4×Q8) Dic3.5C42 Dic3⋊C42 C3⋊(C42⋊8C4) C3⋊(C42⋊5C4) C6.(C4×D4) C2.(C4×D12) C2.(C4×Dic6) Dic3⋊C4⋊C4 (C2×C4)⋊Dic6 C6.(C4⋊Q8) (C2×Dic3).9D4 (C2×C4).17D12 (C2×C4).Dic6 (C22×C4).85D6 (C22×C4).30D6 S3×C2.C42 C22.58(S3×D4) (C2×C4)⋊9D12 D6⋊C42 D6⋊(C4⋊C4) D6⋊C4⋊C4 D6⋊C4⋊5C4 D6⋊C4⋊3C4 (C22×S3)⋊Q8 (C2×C4).21D12 C6.(C4⋊D4) C12⋊4(C4⋊C4) (C2×Dic6)⋊7C4 C4×Dic3⋊C4 C42⋊6Dic3 (C2×C42).6S3 C4×C4⋊Dic3 C42⋊11Dic3 C42⋊7Dic3 C4×D6⋊C4 (C2×C42)⋊3S3 Dic3×C22⋊C4 C24.55D6 C24.56D6 C24.14D6 C24.15D6 C24.57D6 C23⋊2Dic6 C24.17D6 C24.18D6 C24.58D6 C24.19D6 C24.20D6 C24.21D6 C24.23D6 C24.60D6 C24.25D6 C23⋊3D12 C24.27D6 C12⋊(C4⋊C4) C4.(D6⋊C4) Dic3×C4⋊C4 Dic3⋊(C4⋊C4) (C4×Dic3)⋊9C4 C6.67(C4×D4) (C2×Dic3)⋊Q8 C4⋊C4⋊5Dic3 (C2×C4).44D12 (C2×C12).54D4 (C2×Dic3).Q8 C4⋊C4⋊6Dic3 (C2×C12).288D4 (C2×C12).55D4 C4⋊(D6⋊C4) D6⋊C4⋊6C4 D6⋊C4⋊7C4 (C2×C4)⋊3D12 (C2×C12).289D4 (C2×C12).290D4 (C2×C12).56D4 C4×C6.D4 C24.73D6 C24.74D6 C24.75D6 C24.76D6 C24.29D6 C24.31D6 C24.32D6 (C6×Q8)⋊7C4 C22.52(S3×Q8) (C22×Q8)⋊9S3 C18.C42 C62.6Q8 C62.15Q8 C30.24C42 C30.29C42 D10.20D12 D10.10D12
C6.C42 is a maximal quotient of
C12.8C42 (C2×C12)⋊3C8 C24.12D6 C24.13D6 C12.C42 C12.(C4⋊C4) C42⋊3Dic3 C12.2C42 (C2×C12).Q8 (C2×C24)⋊5C4 C12.9C42 C12.10C42 M4(2)⋊Dic3 C12.3C42 (C2×C24)⋊C4 C12.20C42 C12.4C42 M4(2)⋊4Dic3 C12.21C42 C18.C42 C62.6Q8 C62.15Q8 C30.24C42 C30.29C42 D10.20D12 D10.10D12
36 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | ··· | 6G | 12A | ··· | 12H |
| order | 1 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | - | + | - | + | ||||
| image | C1 | C2 | C2 | C4 | C4 | S3 | D4 | Q8 | Dic3 | D6 | Dic6 | C4×S3 | D12 | C3⋊D4 |
| kernel | C6.C42 | C22×Dic3 | C22×C12 | C2×Dic3 | C2×C12 | C22×C4 | C2×C6 | C2×C6 | C2×C4 | C23 | C22 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 8 | 4 | 1 | 3 | 1 | 2 | 1 | 2 | 4 | 2 | 4 |
Matrix representation of C6.C42 ►in GL4(𝔽13) generated by
| 12 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 10 | 3 |
| 0 | 0 | 10 | 7 |
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 10 | 3 |
| 0 | 0 | 6 | 3 |
G:=sub<GL(4,GF(13))| [12,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,5,0,0,0,0,10,10,0,0,3,7],[8,0,0,0,0,8,0,0,0,0,10,6,0,0,3,3] >;
C6.C42 in GAP, Magma, Sage, TeX
C_6.C_4^2
% in TeX
G:=Group("C6.C4^2"); // GroupNames label
G:=SmallGroup(96,38);
// by ID
G=gap.SmallGroup(96,38);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,217,55,2309]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^6=b,e^2=a*b*c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^5>;
// generators/relations