metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.56D6, Q8⋊5(C4×S3), (C4×Q8)⋊6S3, (Q8×C12)⋊2C2, C6.74(C4×D4), C4⋊C4.252D6, Q8⋊2S3⋊5C4, (C4×D12).15C2, D12.17(C2×C4), (C2×C12).258D4, C6.Q16⋊33C2, (C2×Q8).183D6, C4.41(C4○D12), C12.59(C4○D4), C2.4(D4⋊D6), Q8⋊2Dic3⋊11C2, C3⋊6(SD16⋊C4), C42.S3⋊7C2, C12.26(C22×C4), (C4×C12).97C22, C6.D8.10C2, C6.111(C8⋊C22), (C2×C12).346C23, C6.87(C8.C22), (C6×Q8).194C22, C2.3(Q8.11D6), (C2×D12).239C22, C4⋊Dic3.331C22, C3⋊C8⋊10(C2×C4), C4.26(S3×C2×C4), (C3×Q8)⋊9(C2×C4), C2.20(C4×C3⋊D4), (C2×C6).477(C2×D4), (C2×C3⋊C8).100C22, (C2×Q8⋊2S3).4C2, C22.80(C2×C3⋊D4), (C2×C4).221(C3⋊D4), (C3×C4⋊C4).283C22, (C2×C4).446(C22×S3), SmallGroup(192,585)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.56D6
G = < a,b,c,d | a4=b4=1, c6=b2, d2=b, ab=ba, cac-1=dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=b-1c5 >
Subgroups: 328 in 120 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C3⋊C8, C3⋊C8, C4×S3, D12, D12, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, C22×S3, C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C4×D4, C4×Q8, C2×SD16, C2×C3⋊C8, C4⋊Dic3, D6⋊C4, Q8⋊2S3, C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C6×Q8, SD16⋊C4, C42.S3, C6.Q16, C6.D8, Q8⋊2Dic3, C4×D12, C2×Q8⋊2S3, Q8×C12, C42.56D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C4×S3, C3⋊D4, C22×S3, C4×D4, C8⋊C22, C8.C22, S3×C2×C4, C4○D12, C2×C3⋊D4, SD16⋊C4, C4×C3⋊D4, Q8.11D6, D4⋊D6, C42.56D6
►On 96 points
(1 55 84 86)(2 50 73 93)(3 57 74 88)(4 52 75 95)(5 59 76 90)(6 54 77 85)(7 49 78 92)(8 56 79 87)(9 51 80 94)(10 58 81 89)(11 53 82 96)(12 60 83 91)(13 61 40 26)(14 68 41 33)(15 63 42 28)(16 70 43 35)(17 65 44 30)(18 72 45 25)(19 67 46 32)(20 62 47 27)(21 69 48 34)(22 64 37 29)(23 71 38 36)(24 66 39 31)
(1 39 7 45)(2 46 8 40)(3 41 9 47)(4 48 10 42)(5 43 11 37)(6 38 12 44)(13 73 19 79)(14 80 20 74)(15 75 21 81)(16 82 22 76)(17 77 23 83)(18 84 24 78)(25 55 31 49)(26 50 32 56)(27 57 33 51)(28 52 34 58)(29 59 35 53)(30 54 36 60)(61 93 67 87)(62 88 68 94)(63 95 69 89)(64 90 70 96)(65 85 71 91)(66 92 72 86)
(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 6 39 38 7 12 45 44)(2 43 46 11 8 37 40 5)(3 4 41 48 9 10 47 42)(13 76 73 16 19 82 79 22)(14 21 80 81 20 15 74 75)(17 84 77 24 23 78 83 18)(25 36 55 60 31 30 49 54)(26 53 50 29 32 59 56 35)(27 34 57 58 33 28 51 52)(61 96 93 64 67 90 87 70)(62 69 88 89 68 63 94 95)(65 92 85 72 71 86 91 66)
G:=sub<Sym(96)| (1,55,84,86)(2,50,73,93)(3,57,74,88)(4,52,75,95)(5,59,76,90)(6,54,77,85)(7,49,78,92)(8,56,79,87)(9,51,80,94)(10,58,81,89)(11,53,82,96)(12,60,83,91)(13,61,40,26)(14,68,41,33)(15,63,42,28)(16,70,43,35)(17,65,44,30)(18,72,45,25)(19,67,46,32)(20,62,47,27)(21,69,48,34)(22,64,37,29)(23,71,38,36)(24,66,39,31), (1,39,7,45)(2,46,8,40)(3,41,9,47)(4,48,10,42)(5,43,11,37)(6,38,12,44)(13,73,19,79)(14,80,20,74)(15,75,21,81)(16,82,22,76)(17,77,23,83)(18,84,24,78)(25,55,31,49)(26,50,32,56)(27,57,33,51)(28,52,34,58)(29,59,35,53)(30,54,36,60)(61,93,67,87)(62,88,68,94)(63,95,69,89)(64,90,70,96)(65,85,71,91)(66,92,72,86), (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), (1,6,39,38,7,12,45,44)(2,43,46,11,8,37,40,5)(3,4,41,48,9,10,47,42)(13,76,73,16,19,82,79,22)(14,21,80,81,20,15,74,75)(17,84,77,24,23,78,83,18)(25,36,55,60,31,30,49,54)(26,53,50,29,32,59,56,35)(27,34,57,58,33,28,51,52)(61,96,93,64,67,90,87,70)(62,69,88,89,68,63,94,95)(65,92,85,72,71,86,91,66)>;
G:=Group( (1,55,84,86)(2,50,73,93)(3,57,74,88)(4,52,75,95)(5,59,76,90)(6,54,77,85)(7,49,78,92)(8,56,79,87)(9,51,80,94)(10,58,81,89)(11,53,82,96)(12,60,83,91)(13,61,40,26)(14,68,41,33)(15,63,42,28)(16,70,43,35)(17,65,44,30)(18,72,45,25)(19,67,46,32)(20,62,47,27)(21,69,48,34)(22,64,37,29)(23,71,38,36)(24,66,39,31), (1,39,7,45)(2,46,8,40)(3,41,9,47)(4,48,10,42)(5,43,11,37)(6,38,12,44)(13,73,19,79)(14,80,20,74)(15,75,21,81)(16,82,22,76)(17,77,23,83)(18,84,24,78)(25,55,31,49)(26,50,32,56)(27,57,33,51)(28,52,34,58)(29,59,35,53)(30,54,36,60)(61,93,67,87)(62,88,68,94)(63,95,69,89)(64,90,70,96)(65,85,71,91)(66,92,72,86), (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), (1,6,39,38,7,12,45,44)(2,43,46,11,8,37,40,5)(3,4,41,48,9,10,47,42)(13,76,73,16,19,82,79,22)(14,21,80,81,20,15,74,75)(17,84,77,24,23,78,83,18)(25,36,55,60,31,30,49,54)(26,53,50,29,32,59,56,35)(27,34,57,58,33,28,51,52)(61,96,93,64,67,90,87,70)(62,69,88,89,68,63,94,95)(65,92,85,72,71,86,91,66) );
G=PermutationGroup([[(1,55,84,86),(2,50,73,93),(3,57,74,88),(4,52,75,95),(5,59,76,90),(6,54,77,85),(7,49,78,92),(8,56,79,87),(9,51,80,94),(10,58,81,89),(11,53,82,96),(12,60,83,91),(13,61,40,26),(14,68,41,33),(15,63,42,28),(16,70,43,35),(17,65,44,30),(18,72,45,25),(19,67,46,32),(20,62,47,27),(21,69,48,34),(22,64,37,29),(23,71,38,36),(24,66,39,31)], [(1,39,7,45),(2,46,8,40),(3,41,9,47),(4,48,10,42),(5,43,11,37),(6,38,12,44),(13,73,19,79),(14,80,20,74),(15,75,21,81),(16,82,22,76),(17,77,23,83),(18,84,24,78),(25,55,31,49),(26,50,32,56),(27,57,33,51),(28,52,34,58),(29,59,35,53),(30,54,36,60),(61,93,67,87),(62,88,68,94),(63,95,69,89),(64,90,70,96),(65,85,71,91),(66,92,72,86)], [(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)], [(1,6,39,38,7,12,45,44),(2,43,46,11,8,37,40,5),(3,4,41,48,9,10,47,42),(13,76,73,16,19,82,79,22),(14,21,80,81,20,15,74,75),(17,84,77,24,23,78,83,18),(25,36,55,60,31,30,49,54),(26,53,50,29,32,59,56,35),(27,34,57,58,33,28,51,52),(61,96,93,64,67,90,87,70),(62,69,88,89,68,63,94,95),(65,92,85,72,71,86,91,66)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D6 | D6 | C4○D4 | C3⋊D4 | C4×S3 | C4○D12 | C8⋊C22 | C8.C22 | Q8.11D6 | D4⋊D6 |
| kernel | C42.56D6 | C42.S3 | C6.Q16 | C6.D8 | Q8⋊2Dic3 | C4×D12 | C2×Q8⋊2S3 | Q8×C12 | Q8⋊2S3 | C4×Q8 | C2×C12 | C42 | C4⋊C4 | C2×Q8 | C12 | C2×C4 | Q8 | C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of C42.56D6 ►in GL8(𝔽73)
| 46 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 71 | 0 | 67 | 0 |
| 0 | 0 | 0 | 0 | 14 | 46 | 2 | 71 |
| 0 | 0 | 0 | 0 | 37 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 53 | 72 | 6 | 27 |
| 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 48 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 66 | 49 | 0 | 1 |
| 0 | 0 | 0 | 0 | 64 | 70 | 72 | 0 |
| 68 | 67 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 43 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 44 | 60 | 41 | 41 |
| 0 | 0 | 0 | 0 | 8 | 33 | 0 | 70 |
| 0 | 0 | 0 | 0 | 20 | 56 | 17 | 17 |
| 0 | 0 | 0 | 0 | 35 | 61 | 12 | 52 |
| 5 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 20 | 68 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 68 | 13 | 41 | 32 |
| 0 | 0 | 0 | 0 | 65 | 40 | 0 | 3 |
| 0 | 0 | 0 | 0 | 46 | 17 | 17 | 56 |
| 0 | 0 | 0 | 0 | 12 | 12 | 12 | 21 |
G:=sub<GL(8,GF(73))| [46,0,0,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,71,14,37,53,0,0,0,0,0,46,0,72,0,0,0,0,67,2,2,6,0,0,0,0,0,71,0,27],[72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,48,66,64,0,0,0,0,3,72,49,70,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0],[68,4,0,0,0,0,0,0,67,5,0,0,0,0,0,0,0,0,13,43,0,0,0,0,0,0,30,43,0,0,0,0,0,0,0,0,44,8,20,35,0,0,0,0,60,33,56,61,0,0,0,0,41,0,17,12,0,0,0,0,41,70,17,52],[5,20,0,0,0,0,0,0,6,68,0,0,0,0,0,0,0,0,30,60,0,0,0,0,0,0,30,43,0,0,0,0,0,0,0,0,68,65,46,12,0,0,0,0,13,40,17,12,0,0,0,0,41,0,17,12,0,0,0,0,32,3,56,21] >;
C42.56D6 in GAP, Magma, Sage, TeX
C_4^2._{56}D_6 % in TeX
G:=Group("C4^2.56D6"); // GroupNames label
G:=SmallGroup(192,585);
// by ID
G=gap.SmallGroup(192,585);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,232,387,58,1684,851,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=b^2,d^2=b,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^5>;
// generators/relations