metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C44.56D4, Q8⋊2Dic11, D4⋊2Dic11, C11⋊3C4≀C2, (D4×C11)⋊2C4, C44.9(C2×C4), (Q8×C11)⋊2C4, (C2×C22).3D4, C4○D4.1D11, (C2×C4).41D22, C44.C4⋊4C2, (C4×Dic11)⋊2C2, C4.3(C2×Dic11), C4.31(C11⋊D4), (C2×C44).20C22, C22.18(C22⋊C4), C22.3(C11⋊D4), C2.8(C23.D11), (C11×C4○D4).1C2, SmallGroup(352,43)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C44.56D4
G = < a,b,c | a44=b4=1, c2=a33, bab-1=cac-1=a21, cbc-1=a33b-1 >
Smallest permutation representation of C44.56D4
►On 88 points
Generators in S88
(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)
(2 22)(3 43)(4 20)(5 41)(6 18)(7 39)(8 16)(9 37)(10 14)(11 35)(13 33)(15 31)(17 29)(19 27)(21 25)(24 44)(26 42)(28 40)(30 38)(32 36)(45 54 67 76)(46 75 68 53)(47 52 69 74)(48 73 70 51)(49 50 71 72)(55 88 77 66)(56 65 78 87)(57 86 79 64)(58 63 80 85)(59 84 81 62)(60 61 82 83)
(1 88 34 77 23 66 12 55)(2 65 35 54 24 87 13 76)(3 86 36 75 25 64 14 53)(4 63 37 52 26 85 15 74)(5 84 38 73 27 62 16 51)(6 61 39 50 28 83 17 72)(7 82 40 71 29 60 18 49)(8 59 41 48 30 81 19 70)(9 80 42 69 31 58 20 47)(10 57 43 46 32 79 21 68)(11 78 44 67 33 56 22 45)
G:=sub<Sym(88)| (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), (2,22)(3,43)(4,20)(5,41)(6,18)(7,39)(8,16)(9,37)(10,14)(11,35)(13,33)(15,31)(17,29)(19,27)(21,25)(24,44)(26,42)(28,40)(30,38)(32,36)(45,54,67,76)(46,75,68,53)(47,52,69,74)(48,73,70,51)(49,50,71,72)(55,88,77,66)(56,65,78,87)(57,86,79,64)(58,63,80,85)(59,84,81,62)(60,61,82,83), (1,88,34,77,23,66,12,55)(2,65,35,54,24,87,13,76)(3,86,36,75,25,64,14,53)(4,63,37,52,26,85,15,74)(5,84,38,73,27,62,16,51)(6,61,39,50,28,83,17,72)(7,82,40,71,29,60,18,49)(8,59,41,48,30,81,19,70)(9,80,42,69,31,58,20,47)(10,57,43,46,32,79,21,68)(11,78,44,67,33,56,22,45)>;
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), (2,22)(3,43)(4,20)(5,41)(6,18)(7,39)(8,16)(9,37)(10,14)(11,35)(13,33)(15,31)(17,29)(19,27)(21,25)(24,44)(26,42)(28,40)(30,38)(32,36)(45,54,67,76)(46,75,68,53)(47,52,69,74)(48,73,70,51)(49,50,71,72)(55,88,77,66)(56,65,78,87)(57,86,79,64)(58,63,80,85)(59,84,81,62)(60,61,82,83), (1,88,34,77,23,66,12,55)(2,65,35,54,24,87,13,76)(3,86,36,75,25,64,14,53)(4,63,37,52,26,85,15,74)(5,84,38,73,27,62,16,51)(6,61,39,50,28,83,17,72)(7,82,40,71,29,60,18,49)(8,59,41,48,30,81,19,70)(9,80,42,69,31,58,20,47)(10,57,43,46,32,79,21,68)(11,78,44,67,33,56,22,45) );
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)], [(2,22),(3,43),(4,20),(5,41),(6,18),(7,39),(8,16),(9,37),(10,14),(11,35),(13,33),(15,31),(17,29),(19,27),(21,25),(24,44),(26,42),(28,40),(30,38),(32,36),(45,54,67,76),(46,75,68,53),(47,52,69,74),(48,73,70,51),(49,50,71,72),(55,88,77,66),(56,65,78,87),(57,86,79,64),(58,63,80,85),(59,84,81,62),(60,61,82,83)], [(1,88,34,77,23,66,12,55),(2,65,35,54,24,87,13,76),(3,86,36,75,25,64,14,53),(4,63,37,52,26,85,15,74),(5,84,38,73,27,62,16,51),(6,61,39,50,28,83,17,72),(7,82,40,71,29,60,18,49),(8,59,41,48,30,81,19,70),(9,80,42,69,31,58,20,47),(10,57,43,46,32,79,21,68),(11,78,44,67,33,56,22,45)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 11A | ··· | 11E | 22A | ··· | 22E | 22F | ··· | 22T | 44A | ··· | 44J | 44K | ··· | 44Y |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 44 | ··· | 44 |
| size | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 4 | 22 | 22 | 22 | 22 | 44 | 44 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D11 | C4≀C2 | D22 | Dic11 | Dic11 | C11⋊D4 | C11⋊D4 | C44.56D4 |
| kernel | C44.56D4 | C44.C4 | C4×Dic11 | C11×C4○D4 | D4×C11 | Q8×C11 | C44 | C2×C22 | C4○D4 | C11 | C2×C4 | D4 | Q8 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 5 | 4 | 5 | 5 | 5 | 10 | 10 | 10 |
Matrix representation of C44.56D4 ►in GL4(𝔽89) generated by
| 7 | 48 | 0 | 0 |
| 83 | 48 | 0 | 0 |
| 0 | 0 | 55 | 0 |
| 0 | 0 | 0 | 55 |
| 67 | 27 | 0 | 0 |
| 81 | 22 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 69 | 55 |
| 67 | 27 | 0 | 0 |
| 81 | 22 | 0 | 0 |
| 0 | 0 | 62 | 64 |
| 0 | 0 | 59 | 27 |
G:=sub<GL(4,GF(89))| [7,83,0,0,48,48,0,0,0,0,55,0,0,0,0,55],[67,81,0,0,27,22,0,0,0,0,1,69,0,0,0,55],[67,81,0,0,27,22,0,0,0,0,62,59,0,0,64,27] >;
C44.56D4 in GAP, Magma, Sage, TeX
C_{44}._{56}D_4 % in TeX
G:=Group("C44.56D4"); // GroupNames label
G:=SmallGroup(352,43);
// by ID
G=gap.SmallGroup(352,43);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,24,121,86,579,297,69,11525]);
// Polycyclic
G:=Group<a,b,c|a^44=b^4=1,c^2=a^33,b*a*b^-1=c*a*c^-1=a^21,c*b*c^-1=a^33*b^-1>;
// generators/relations
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