metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.264D6, (C4×C8)⋊6S3, (C4×C24)⋊6C2, C6.5(C4○D8), (C2×C4).63D12, (C2×C8).288D6, C2.8(C4○D24), (C2×C12).353D4, C2.Dic12⋊1C2, C12.6Q8⋊1C2, C2.D24.1C2, C42⋊7S3.1C2, C6.6(C4.4D4), (C2×D12).5C22, C22.93(C2×D12), C4⋊Dic3.6C22, C4.103(C4○D12), C12.219(C4○D4), (C2×C24).348C22, (C4×C12).310C22, (C2×C12).726C23, (C2×Dic6).4C22, C2.11(C42⋊7S3), C3⋊1(C42.78C22), (C2×C6).109(C2×D4), (C2×C4).669(C22×S3), SmallGroup(192,256)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.264D6
G = < a,b,c,d | a4=b4=1, c6=a2b, d2=a2, ab=ba, ac=ca, dad-1=a-1b2, bc=cb, dbd-1=b-1, dcd-1=b-1c5 >
Subgroups: 312 in 96 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×D4, C2×Q8, C24, Dic6, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C4×C8, D4⋊C4, Q8⋊C4, C4.4D4, C42.C2, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C4×C12, C2×C24, C2×Dic6, C2×D12, C42.78C22, C2.Dic12, C2.D24, C4×C24, C12.6Q8, C42⋊7S3, C42.264D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C22×S3, C4.4D4, C4○D8, C2×D12, C4○D12, C42.78C22, C42⋊7S3, C4○D24, C42.264D6
►On 96 points
Generators in S96
(1 61 31 85)(2 62 32 86)(3 63 33 87)(4 64 34 88)(5 65 35 89)(6 66 36 90)(7 67 37 91)(8 68 38 92)(9 69 39 93)(10 70 40 94)(11 71 41 95)(12 72 42 96)(13 49 43 73)(14 50 44 74)(15 51 45 75)(16 52 46 76)(17 53 47 77)(18 54 48 78)(19 55 25 79)(20 56 26 80)(21 57 27 81)(22 58 28 82)(23 59 29 83)(24 60 30 84)
(1 37 13 25)(2 38 14 26)(3 39 15 27)(4 40 16 28)(5 41 17 29)(6 42 18 30)(7 43 19 31)(8 44 20 32)(9 45 21 33)(10 46 22 34)(11 47 23 35)(12 48 24 36)(49 79 61 91)(50 80 62 92)(51 81 63 93)(52 82 64 94)(53 83 65 95)(54 84 66 96)(55 85 67 73)(56 86 68 74)(57 87 69 75)(58 88 70 76)(59 89 71 77)(60 90 72 78)
(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 18 31 48)(2 47 32 17)(3 16 33 46)(4 45 34 15)(5 14 35 44)(6 43 36 13)(7 12 37 42)(8 41 38 11)(9 10 39 40)(19 24 25 30)(20 29 26 23)(21 22 27 28)(49 78 73 54)(50 53 74 77)(51 76 75 52)(55 96 79 72)(56 71 80 95)(57 94 81 70)(58 69 82 93)(59 92 83 68)(60 67 84 91)(61 90 85 66)(62 65 86 89)(63 88 87 64)
G:=sub<Sym(96)| (1,61,31,85)(2,62,32,86)(3,63,33,87)(4,64,34,88)(5,65,35,89)(6,66,36,90)(7,67,37,91)(8,68,38,92)(9,69,39,93)(10,70,40,94)(11,71,41,95)(12,72,42,96)(13,49,43,73)(14,50,44,74)(15,51,45,75)(16,52,46,76)(17,53,47,77)(18,54,48,78)(19,55,25,79)(20,56,26,80)(21,57,27,81)(22,58,28,82)(23,59,29,83)(24,60,30,84), (1,37,13,25)(2,38,14,26)(3,39,15,27)(4,40,16,28)(5,41,17,29)(6,42,18,30)(7,43,19,31)(8,44,20,32)(9,45,21,33)(10,46,22,34)(11,47,23,35)(12,48,24,36)(49,79,61,91)(50,80,62,92)(51,81,63,93)(52,82,64,94)(53,83,65,95)(54,84,66,96)(55,85,67,73)(56,86,68,74)(57,87,69,75)(58,88,70,76)(59,89,71,77)(60,90,72,78), (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), (1,18,31,48)(2,47,32,17)(3,16,33,46)(4,45,34,15)(5,14,35,44)(6,43,36,13)(7,12,37,42)(8,41,38,11)(9,10,39,40)(19,24,25,30)(20,29,26,23)(21,22,27,28)(49,78,73,54)(50,53,74,77)(51,76,75,52)(55,96,79,72)(56,71,80,95)(57,94,81,70)(58,69,82,93)(59,92,83,68)(60,67,84,91)(61,90,85,66)(62,65,86,89)(63,88,87,64)>;
G:=Group( (1,61,31,85)(2,62,32,86)(3,63,33,87)(4,64,34,88)(5,65,35,89)(6,66,36,90)(7,67,37,91)(8,68,38,92)(9,69,39,93)(10,70,40,94)(11,71,41,95)(12,72,42,96)(13,49,43,73)(14,50,44,74)(15,51,45,75)(16,52,46,76)(17,53,47,77)(18,54,48,78)(19,55,25,79)(20,56,26,80)(21,57,27,81)(22,58,28,82)(23,59,29,83)(24,60,30,84), (1,37,13,25)(2,38,14,26)(3,39,15,27)(4,40,16,28)(5,41,17,29)(6,42,18,30)(7,43,19,31)(8,44,20,32)(9,45,21,33)(10,46,22,34)(11,47,23,35)(12,48,24,36)(49,79,61,91)(50,80,62,92)(51,81,63,93)(52,82,64,94)(53,83,65,95)(54,84,66,96)(55,85,67,73)(56,86,68,74)(57,87,69,75)(58,88,70,76)(59,89,71,77)(60,90,72,78), (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), (1,18,31,48)(2,47,32,17)(3,16,33,46)(4,45,34,15)(5,14,35,44)(6,43,36,13)(7,12,37,42)(8,41,38,11)(9,10,39,40)(19,24,25,30)(20,29,26,23)(21,22,27,28)(49,78,73,54)(50,53,74,77)(51,76,75,52)(55,96,79,72)(56,71,80,95)(57,94,81,70)(58,69,82,93)(59,92,83,68)(60,67,84,91)(61,90,85,66)(62,65,86,89)(63,88,87,64) );
G=PermutationGroup([[(1,61,31,85),(2,62,32,86),(3,63,33,87),(4,64,34,88),(5,65,35,89),(6,66,36,90),(7,67,37,91),(8,68,38,92),(9,69,39,93),(10,70,40,94),(11,71,41,95),(12,72,42,96),(13,49,43,73),(14,50,44,74),(15,51,45,75),(16,52,46,76),(17,53,47,77),(18,54,48,78),(19,55,25,79),(20,56,26,80),(21,57,27,81),(22,58,28,82),(23,59,29,83),(24,60,30,84)], [(1,37,13,25),(2,38,14,26),(3,39,15,27),(4,40,16,28),(5,41,17,29),(6,42,18,30),(7,43,19,31),(8,44,20,32),(9,45,21,33),(10,46,22,34),(11,47,23,35),(12,48,24,36),(49,79,61,91),(50,80,62,92),(51,81,63,93),(52,82,64,94),(53,83,65,95),(54,84,66,96),(55,85,67,73),(56,86,68,74),(57,87,69,75),(58,88,70,76),(59,89,71,77),(60,90,72,78)], [(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)], [(1,18,31,48),(2,47,32,17),(3,16,33,46),(4,45,34,15),(5,14,35,44),(6,43,36,13),(7,12,37,42),(8,41,38,11),(9,10,39,40),(19,24,25,30),(20,29,26,23),(21,22,27,28),(49,78,73,54),(50,53,74,77),(51,76,75,52),(55,96,79,72),(56,71,80,95),(57,94,81,70),(58,69,82,93),(59,92,83,68),(60,67,84,91),(61,90,85,66),(62,65,86,89),(63,88,87,64)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 8A | ··· | 8H | 12A | ··· | 12L | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 24 | 2 | 2 | ··· | 2 | 24 | 24 | 24 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | C4○D4 | D12 | C4○D8 | C4○D12 | C4○D24 |
| kernel | C42.264D6 | C2.Dic12 | C2.D24 | C4×C24 | C12.6Q8 | C42⋊7S3 | C4×C8 | C2×C12 | C42 | C2×C8 | C12 | C2×C4 | C6 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 8 | 8 | 16 |
Matrix representation of C42.264D6 ►in GL4(𝔽73) generated by
| 27 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 0 | 27 |
| 0 | 0 | 46 | 0 |
| 66 | 59 | 0 | 0 |
| 14 | 7 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 |
| 36 | 25 | 0 | 0 |
| 48 | 11 | 0 | 0 |
| 0 | 0 | 16 | 57 |
| 0 | 0 | 16 | 16 |
| 25 | 36 | 0 | 0 |
| 11 | 48 | 0 | 0 |
| 0 | 0 | 16 | 16 |
| 0 | 0 | 16 | 57 |
G:=sub<GL(4,GF(73))| [27,0,0,0,0,27,0,0,0,0,0,46,0,0,27,0],[66,14,0,0,59,7,0,0,0,0,0,72,0,0,1,0],[36,48,0,0,25,11,0,0,0,0,16,16,0,0,57,16],[25,11,0,0,36,48,0,0,0,0,16,16,0,0,16,57] >;
C42.264D6 in GAP, Magma, Sage, TeX
C_4^2._{264}D_6 % in TeX
G:=Group("C4^2.264D6"); // GroupNames label
G:=SmallGroup(192,256);
// by ID
G=gap.SmallGroup(192,256);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,142,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=a^2*b,d^2=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^-1*c^5>;
// generators/relations