metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.214D6, C3⋊C8.17D4, C4.12(S3×D4), (C2×D4).49D6, C4.4D4⋊3S3, C12.26(C2×D4), (C2×Q8).63D6, (C2×C12).272D4, C3⋊2(C8.12D4), C42⋊7S3⋊14C2, C6.106(C4○D8), C6.18(C4⋊1D4), C2.9(C12⋊3D4), (C6×D4).65C22, (C6×Q8).57C22, (C4×C12).108C22, (C2×C12).377C23, C2.25(Q8.13D6), (C2×D12).102C22, (C2×Dic6).107C22, (C4×C3⋊C8)⋊12C2, (C2×D4⋊S3).7C2, (C2×D4.S3)⋊12C2, (C2×C3⋊Q16)⋊13C2, (C3×C4.4D4)⋊3C2, (C2×C6).508(C2×D4), (C2×Q8⋊2S3)⋊14C2, (C2×C3⋊C8).253C22, (C2×C4).110(C3⋊D4), (C2×C4).477(C22×S3), C22.183(C2×C3⋊D4), SmallGroup(192,618)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.214D6
G = < a,b,c,d | a4=b4=c6=1, d2=cbc-1=b-1, ab=ba, cac-1=a-1b2, ad=da, bd=db, dcd-1=b-1c-1 >
Subgroups: 400 in 130 conjugacy classes, 43 normal (31 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C2×C8, D8, SD16, Q16, C2×D4, C2×D4, C2×Q8, C2×Q8, C3⋊C8, Dic6, D12, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×C6, C4×C8, C4.4D4, C4.4D4, C2×D8, C2×SD16, C2×Q16, C2×C3⋊C8, D6⋊C4, D4⋊S3, D4.S3, Q8⋊2S3, C3⋊Q16, C4×C12, C3×C22⋊C4, C2×Dic6, C2×D12, C6×D4, C6×Q8, C8.12D4, C4×C3⋊C8, C42⋊7S3, C2×D4⋊S3, C2×D4.S3, C2×Q8⋊2S3, C2×C3⋊Q16, C3×C4.4D4, C42.214D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C4⋊1D4, C4○D8, S3×D4, C2×C3⋊D4, C8.12D4, C12⋊3D4, Q8.13D6, C42.214D6
►On 96 points
(1 28 85 90)(2 29 86 91)(3 30 87 92)(4 31 88 93)(5 32 81 94)(6 25 82 95)(7 26 83 96)(8 27 84 89)(9 66 36 73)(10 67 37 74)(11 68 38 75)(12 69 39 76)(13 70 40 77)(14 71 33 78)(15 72 34 79)(16 65 35 80)(17 47 52 62)(18 48 53 63)(19 41 54 64)(20 42 55 57)(21 43 56 58)(22 44 49 59)(23 45 50 60)(24 46 51 61)
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)(33 39 37 35)(34 40 38 36)(41 47 45 43)(42 48 46 44)(49 55 53 51)(50 56 54 52)(57 63 61 59)(58 64 62 60)(65 71 69 67)(66 72 70 68)(73 79 77 75)(74 80 78 76)(81 87 85 83)(82 88 86 84)(89 95 93 91)(90 96 94 92)
(1 24 67 68 23 2)(3 22 69 66 17 8)(4 7 18 65 70 21)(5 20 71 72 19 6)(9 43 89 26 59 35)(10 34 60 25 90 42)(11 41 91 32 61 33)(12 40 62 31 92 48)(13 47 93 30 63 39)(14 38 64 29 94 46)(15 45 95 28 57 37)(16 36 58 27 96 44)(49 76 73 52 84 87)(50 86 85 51 74 75)(53 80 77 56 88 83)(54 82 81 55 78 79)
(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)
G:=sub<Sym(96)| (1,28,85,90)(2,29,86,91)(3,30,87,92)(4,31,88,93)(5,32,81,94)(6,25,82,95)(7,26,83,96)(8,27,84,89)(9,66,36,73)(10,67,37,74)(11,68,38,75)(12,69,39,76)(13,70,40,77)(14,71,33,78)(15,72,34,79)(16,65,35,80)(17,47,52,62)(18,48,53,63)(19,41,54,64)(20,42,55,57)(21,43,56,58)(22,44,49,59)(23,45,50,60)(24,46,51,61), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28)(33,39,37,35)(34,40,38,36)(41,47,45,43)(42,48,46,44)(49,55,53,51)(50,56,54,52)(57,63,61,59)(58,64,62,60)(65,71,69,67)(66,72,70,68)(73,79,77,75)(74,80,78,76)(81,87,85,83)(82,88,86,84)(89,95,93,91)(90,96,94,92), (1,24,67,68,23,2)(3,22,69,66,17,8)(4,7,18,65,70,21)(5,20,71,72,19,6)(9,43,89,26,59,35)(10,34,60,25,90,42)(11,41,91,32,61,33)(12,40,62,31,92,48)(13,47,93,30,63,39)(14,38,64,29,94,46)(15,45,95,28,57,37)(16,36,58,27,96,44)(49,76,73,52,84,87)(50,86,85,51,74,75)(53,80,77,56,88,83)(54,82,81,55,78,79), (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)>;
G:=Group( (1,28,85,90)(2,29,86,91)(3,30,87,92)(4,31,88,93)(5,32,81,94)(6,25,82,95)(7,26,83,96)(8,27,84,89)(9,66,36,73)(10,67,37,74)(11,68,38,75)(12,69,39,76)(13,70,40,77)(14,71,33,78)(15,72,34,79)(16,65,35,80)(17,47,52,62)(18,48,53,63)(19,41,54,64)(20,42,55,57)(21,43,56,58)(22,44,49,59)(23,45,50,60)(24,46,51,61), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28)(33,39,37,35)(34,40,38,36)(41,47,45,43)(42,48,46,44)(49,55,53,51)(50,56,54,52)(57,63,61,59)(58,64,62,60)(65,71,69,67)(66,72,70,68)(73,79,77,75)(74,80,78,76)(81,87,85,83)(82,88,86,84)(89,95,93,91)(90,96,94,92), (1,24,67,68,23,2)(3,22,69,66,17,8)(4,7,18,65,70,21)(5,20,71,72,19,6)(9,43,89,26,59,35)(10,34,60,25,90,42)(11,41,91,32,61,33)(12,40,62,31,92,48)(13,47,93,30,63,39)(14,38,64,29,94,46)(15,45,95,28,57,37)(16,36,58,27,96,44)(49,76,73,52,84,87)(50,86,85,51,74,75)(53,80,77,56,88,83)(54,82,81,55,78,79), (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) );
G=PermutationGroup([[(1,28,85,90),(2,29,86,91),(3,30,87,92),(4,31,88,93),(5,32,81,94),(6,25,82,95),(7,26,83,96),(8,27,84,89),(9,66,36,73),(10,67,37,74),(11,68,38,75),(12,69,39,76),(13,70,40,77),(14,71,33,78),(15,72,34,79),(16,65,35,80),(17,47,52,62),(18,48,53,63),(19,41,54,64),(20,42,55,57),(21,43,56,58),(22,44,49,59),(23,45,50,60),(24,46,51,61)], [(1,7,5,3),(2,8,6,4),(9,15,13,11),(10,16,14,12),(17,23,21,19),(18,24,22,20),(25,31,29,27),(26,32,30,28),(33,39,37,35),(34,40,38,36),(41,47,45,43),(42,48,46,44),(49,55,53,51),(50,56,54,52),(57,63,61,59),(58,64,62,60),(65,71,69,67),(66,72,70,68),(73,79,77,75),(74,80,78,76),(81,87,85,83),(82,88,86,84),(89,95,93,91),(90,96,94,92)], [(1,24,67,68,23,2),(3,22,69,66,17,8),(4,7,18,65,70,21),(5,20,71,72,19,6),(9,43,89,26,59,35),(10,34,60,25,90,42),(11,41,91,32,61,33),(12,40,62,31,92,48),(13,47,93,30,63,39),(14,38,64,29,94,46),(15,45,95,28,57,37),(16,36,58,27,96,44),(49,76,73,52,84,87),(50,86,85,51,74,75),(53,80,77,56,88,83),(54,82,81,55,78,79)], [(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)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 8A | ··· | 8H | 12A | ··· | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 8 | 24 | 2 | 2 | ··· | 2 | 8 | 24 | 2 | 2 | 2 | 8 | 8 | 6 | ··· | 6 | 4 | ··· | 4 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | C3⋊D4 | C4○D8 | S3×D4 | Q8.13D6 |
| kernel | C42.214D6 | C4×C3⋊C8 | C42⋊7S3 | C2×D4⋊S3 | C2×D4.S3 | C2×Q8⋊2S3 | C2×C3⋊Q16 | C3×C4.4D4 | C4.4D4 | C3⋊C8 | C2×C12 | C42 | C2×D4 | C2×Q8 | C2×C4 | C6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 1 | 1 | 1 | 4 | 8 | 2 | 4 |
Matrix representation of C42.214D6 ►in GL6(𝔽73)
| 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 71 |
| 0 | 0 | 0 | 0 | 37 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 16 | 16 | 0 | 0 | 0 | 0 |
| 16 | 57 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 37 | 0 |
| 16 | 57 | 0 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 71 |
| 0 | 0 | 0 | 0 | 37 | 0 |
G:=sub<GL(6,GF(73))| [27,0,0,0,0,0,0,27,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,37,0,0,0,0,71,0],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[16,16,0,0,0,0,16,57,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,37,0,0,0,0,2,0],[16,16,0,0,0,0,57,16,0,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,0,0,0,0,37,0,0,0,0,71,0] >;
C42.214D6 in GAP, Magma, Sage, TeX
C_4^2._{214}D_6 % in TeX
G:=Group("C4^2.214D6"); // GroupNames label
G:=SmallGroup(192,618);
// by ID
G=gap.SmallGroup(192,618);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,219,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=c*b*c^-1=b^-1,a*b=b*a,c*a*c^-1=a^-1*b^2,a*d=d*a,b*d=d*b,d*c*d^-1=b^-1*c^-1>;
// generators/relations