metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.5C22≀C2, (C2×Dic3)⋊3D4, (C2×C12).30D4, (C2×C4).19D12, C6.6(C4⋊D4), (C22×S3).7D4, (C22×C4).87D6, C2.9(Dic3⋊D4), C2.8(C12⋊D4), C2.8(D6⋊D4), C6.2(C4.4D4), (C22×D12).2C2, C22.156(S3×D4), C22.81(C2×D12), C2.C42⋊12S3, C2.6(C42⋊7S3), C2.9(D6.D4), C3⋊1(C23.10D4), (S3×C23).4C22, C22.89(C4○D12), (C22×C6).297C23, (C22×C12).16C22, C23.370(C22×S3), C22.45(Q8⋊3S3), C6.40(C22.D4), (C22×Dic3).19C22, (C2×D6⋊C4)⋊1C2, (C2×C6).205(C2×D4), (C2×Dic3⋊C4)⋊19C2, (C2×C6).59(C4○D4), (C3×C2.C42)⋊9C2, SmallGroup(192,231)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.C22≀C2
G = < a,b,c,d,e,f | a6=b2=d2=e2=1, c2=f2=a3, bab=faf-1=a-1, ac=ca, ad=da, ae=ea, cbc-1=a3b, fbf-1=bd=db, be=eb, cd=dc, ce=ec, fcf-1=a3ce, de=ed, df=fd, ef=fe >
Subgroups: 816 in 238 conjugacy classes, 61 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C22×D4, Dic3⋊C4, D6⋊C4, C2×D12, C22×Dic3, C22×C12, C22×C12, S3×C23, C23.10D4, C3×C2.C42, C2×Dic3⋊C4, C2×D6⋊C4, C22×D12, C6.C22≀C2
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C22×S3, C22≀C2, C4⋊D4, C22.D4, C4.4D4, C2×D12, C4○D12, S3×D4, Q8⋊3S3, C23.10D4, C42⋊7S3, D6⋊D4, Dic3⋊D4, D6.D4, C12⋊D4, C6.C22≀C2
►On 96 points
Generators in S96
(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 11)(2 10)(3 9)(4 8)(5 7)(6 12)(13 32)(14 31)(15 36)(16 35)(17 34)(18 33)(19 93)(20 92)(21 91)(22 96)(23 95)(24 94)(25 44)(26 43)(27 48)(28 47)(29 46)(30 45)(37 56)(38 55)(39 60)(40 59)(41 58)(42 57)(49 71)(50 70)(51 69)(52 68)(53 67)(54 72)(61 83)(62 82)(63 81)(64 80)(65 79)(66 84)(73 89)(74 88)(75 87)(76 86)(77 85)(78 90)
(1 74 4 77)(2 75 5 78)(3 76 6 73)(7 87 10 90)(8 88 11 85)(9 89 12 86)(13 21 16 24)(14 22 17 19)(15 23 18 20)(25 72 28 69)(26 67 29 70)(27 68 30 71)(31 93 34 96)(32 94 35 91)(33 95 36 92)(37 80 40 83)(38 81 41 84)(39 82 42 79)(43 50 46 53)(44 51 47 54)(45 52 48 49)(55 66 58 63)(56 61 59 64)(57 62 60 65)
(1 16)(2 17)(3 18)(4 13)(5 14)(6 15)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(19 75)(20 76)(21 77)(22 78)(23 73)(24 74)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 79)(50 80)(51 81)(52 82)(53 83)(54 84)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)
(1 35)(2 36)(3 31)(4 32)(5 33)(6 34)(7 18)(8 13)(9 14)(10 15)(11 16)(12 17)(19 86)(20 87)(21 88)(22 89)(23 90)(24 85)(25 48)(26 43)(27 44)(28 45)(29 46)(30 47)(37 56)(38 57)(39 58)(40 59)(41 60)(42 55)(49 72)(50 67)(51 68)(52 69)(53 70)(54 71)(61 80)(62 81)(63 82)(64 83)(65 84)(66 79)(73 96)(74 91)(75 92)(76 93)(77 94)(78 95)
(1 26 4 29)(2 25 5 28)(3 30 6 27)(7 41 10 38)(8 40 11 37)(9 39 12 42)(13 59 16 56)(14 58 17 55)(15 57 18 60)(19 82 22 79)(20 81 23 84)(21 80 24 83)(31 47 34 44)(32 46 35 43)(33 45 36 48)(49 75 52 78)(50 74 53 77)(51 73 54 76)(61 85 64 88)(62 90 65 87)(63 89 66 86)(67 91 70 94)(68 96 71 93)(69 95 72 92)
G:=sub<Sym(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), (1,11)(2,10)(3,9)(4,8)(5,7)(6,12)(13,32)(14,31)(15,36)(16,35)(17,34)(18,33)(19,93)(20,92)(21,91)(22,96)(23,95)(24,94)(25,44)(26,43)(27,48)(28,47)(29,46)(30,45)(37,56)(38,55)(39,60)(40,59)(41,58)(42,57)(49,71)(50,70)(51,69)(52,68)(53,67)(54,72)(61,83)(62,82)(63,81)(64,80)(65,79)(66,84)(73,89)(74,88)(75,87)(76,86)(77,85)(78,90), (1,74,4,77)(2,75,5,78)(3,76,6,73)(7,87,10,90)(8,88,11,85)(9,89,12,86)(13,21,16,24)(14,22,17,19)(15,23,18,20)(25,72,28,69)(26,67,29,70)(27,68,30,71)(31,93,34,96)(32,94,35,91)(33,95,36,92)(37,80,40,83)(38,81,41,84)(39,82,42,79)(43,50,46,53)(44,51,47,54)(45,52,48,49)(55,66,58,63)(56,61,59,64)(57,62,60,65), (1,16)(2,17)(3,18)(4,13)(5,14)(6,15)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(19,75)(20,76)(21,77)(22,78)(23,73)(24,74)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96), (1,35)(2,36)(3,31)(4,32)(5,33)(6,34)(7,18)(8,13)(9,14)(10,15)(11,16)(12,17)(19,86)(20,87)(21,88)(22,89)(23,90)(24,85)(25,48)(26,43)(27,44)(28,45)(29,46)(30,47)(37,56)(38,57)(39,58)(40,59)(41,60)(42,55)(49,72)(50,67)(51,68)(52,69)(53,70)(54,71)(61,80)(62,81)(63,82)(64,83)(65,84)(66,79)(73,96)(74,91)(75,92)(76,93)(77,94)(78,95), (1,26,4,29)(2,25,5,28)(3,30,6,27)(7,41,10,38)(8,40,11,37)(9,39,12,42)(13,59,16,56)(14,58,17,55)(15,57,18,60)(19,82,22,79)(20,81,23,84)(21,80,24,83)(31,47,34,44)(32,46,35,43)(33,45,36,48)(49,75,52,78)(50,74,53,77)(51,73,54,76)(61,85,64,88)(62,90,65,87)(63,89,66,86)(67,91,70,94)(68,96,71,93)(69,95,72,92)>;
G:=Group( (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), (1,11)(2,10)(3,9)(4,8)(5,7)(6,12)(13,32)(14,31)(15,36)(16,35)(17,34)(18,33)(19,93)(20,92)(21,91)(22,96)(23,95)(24,94)(25,44)(26,43)(27,48)(28,47)(29,46)(30,45)(37,56)(38,55)(39,60)(40,59)(41,58)(42,57)(49,71)(50,70)(51,69)(52,68)(53,67)(54,72)(61,83)(62,82)(63,81)(64,80)(65,79)(66,84)(73,89)(74,88)(75,87)(76,86)(77,85)(78,90), (1,74,4,77)(2,75,5,78)(3,76,6,73)(7,87,10,90)(8,88,11,85)(9,89,12,86)(13,21,16,24)(14,22,17,19)(15,23,18,20)(25,72,28,69)(26,67,29,70)(27,68,30,71)(31,93,34,96)(32,94,35,91)(33,95,36,92)(37,80,40,83)(38,81,41,84)(39,82,42,79)(43,50,46,53)(44,51,47,54)(45,52,48,49)(55,66,58,63)(56,61,59,64)(57,62,60,65), (1,16)(2,17)(3,18)(4,13)(5,14)(6,15)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(19,75)(20,76)(21,77)(22,78)(23,73)(24,74)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96), (1,35)(2,36)(3,31)(4,32)(5,33)(6,34)(7,18)(8,13)(9,14)(10,15)(11,16)(12,17)(19,86)(20,87)(21,88)(22,89)(23,90)(24,85)(25,48)(26,43)(27,44)(28,45)(29,46)(30,47)(37,56)(38,57)(39,58)(40,59)(41,60)(42,55)(49,72)(50,67)(51,68)(52,69)(53,70)(54,71)(61,80)(62,81)(63,82)(64,83)(65,84)(66,79)(73,96)(74,91)(75,92)(76,93)(77,94)(78,95), (1,26,4,29)(2,25,5,28)(3,30,6,27)(7,41,10,38)(8,40,11,37)(9,39,12,42)(13,59,16,56)(14,58,17,55)(15,57,18,60)(19,82,22,79)(20,81,23,84)(21,80,24,83)(31,47,34,44)(32,46,35,43)(33,45,36,48)(49,75,52,78)(50,74,53,77)(51,73,54,76)(61,85,64,88)(62,90,65,87)(63,89,66,86)(67,91,70,94)(68,96,71,93)(69,95,72,92) );
G=PermutationGroup([[(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)], [(1,11),(2,10),(3,9),(4,8),(5,7),(6,12),(13,32),(14,31),(15,36),(16,35),(17,34),(18,33),(19,93),(20,92),(21,91),(22,96),(23,95),(24,94),(25,44),(26,43),(27,48),(28,47),(29,46),(30,45),(37,56),(38,55),(39,60),(40,59),(41,58),(42,57),(49,71),(50,70),(51,69),(52,68),(53,67),(54,72),(61,83),(62,82),(63,81),(64,80),(65,79),(66,84),(73,89),(74,88),(75,87),(76,86),(77,85),(78,90)], [(1,74,4,77),(2,75,5,78),(3,76,6,73),(7,87,10,90),(8,88,11,85),(9,89,12,86),(13,21,16,24),(14,22,17,19),(15,23,18,20),(25,72,28,69),(26,67,29,70),(27,68,30,71),(31,93,34,96),(32,94,35,91),(33,95,36,92),(37,80,40,83),(38,81,41,84),(39,82,42,79),(43,50,46,53),(44,51,47,54),(45,52,48,49),(55,66,58,63),(56,61,59,64),(57,62,60,65)], [(1,16),(2,17),(3,18),(4,13),(5,14),(6,15),(7,31),(8,32),(9,33),(10,34),(11,35),(12,36),(19,75),(20,76),(21,77),(22,78),(23,73),(24,74),(25,55),(26,56),(27,57),(28,58),(29,59),(30,60),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48),(49,79),(50,80),(51,81),(52,82),(53,83),(54,84),(61,67),(62,68),(63,69),(64,70),(65,71),(66,72),(85,91),(86,92),(87,93),(88,94),(89,95),(90,96)], [(1,35),(2,36),(3,31),(4,32),(5,33),(6,34),(7,18),(8,13),(9,14),(10,15),(11,16),(12,17),(19,86),(20,87),(21,88),(22,89),(23,90),(24,85),(25,48),(26,43),(27,44),(28,45),(29,46),(30,47),(37,56),(38,57),(39,58),(40,59),(41,60),(42,55),(49,72),(50,67),(51,68),(52,69),(53,70),(54,71),(61,80),(62,81),(63,82),(64,83),(65,84),(66,79),(73,96),(74,91),(75,92),(76,93),(77,94),(78,95)], [(1,26,4,29),(2,25,5,28),(3,30,6,27),(7,41,10,38),(8,40,11,37),(9,39,12,42),(13,59,16,56),(14,58,17,55),(15,57,18,60),(19,82,22,79),(20,81,23,84),(21,80,24,83),(31,47,34,44),(32,46,35,43),(33,45,36,48),(49,75,52,78),(50,74,53,77),(51,73,54,76),(61,85,64,88),(62,90,65,87),(63,89,66,86),(67,91,70,94),(68,96,71,93),(69,95,72,92)]])
42 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 6A | ··· | 6G | 12A | ··· | 12L |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 12 | 12 | 12 | 12 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | C4○D4 | D12 | C4○D12 | S3×D4 | Q8⋊3S3 |
| kernel | C6.C22≀C2 | C3×C2.C42 | C2×Dic3⋊C4 | C2×D6⋊C4 | C22×D12 | C2.C42 | C2×Dic3 | C2×C12 | C22×S3 | C22×C4 | C2×C6 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 4 | 3 | 6 | 4 | 8 | 3 | 1 |
Matrix representation of C6.C22≀C2 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 7 | 0 | 0 |
| 0 | 0 | 6 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 7 |
| 0 | 0 | 0 | 0 | 6 | 3 |
| 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 | 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 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 2 | 0 | 0 |
| 0 | 0 | 11 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 5 | 5 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,1,12,0,0,0,0,1,0],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,3,6,0,0,0,0,7,10,0,0,0,0,0,0,10,6,0,0,0,0,7,3],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,9,11,0,0,0,0,2,4,0,0,0,0,0,0,8,5,0,0,0,0,0,5] >;
C6.C22≀C2 in GAP, Magma, Sage, TeX
C_6.C_2^2\wr C_2
% in TeX
G:=Group("C6.C2^2wrC2"); // GroupNames label
G:=SmallGroup(192,231);
// by ID
G=gap.SmallGroup(192,231);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,64,254,387,268,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^6=b^2=d^2=e^2=1,c^2=f^2=a^3,b*a*b=f*a*f^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=a^3*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,f*c*f^-1=a^3*c*e,d*e=e*d,d*f=f*d,e*f=f*e>;
// generators/relations