metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C44.8D4, C23.Dic11, (C2×C4).3D22, (C2×D4).2D11, (D4×C22).2C2, C44.C4⋊3C2, C11⋊2(C4.D4), (C22×C22).2C4, C4.13(C11⋊D4), (C2×C44).17C22, C22.14(C22⋊C4), C2.4(C23.D11), C22.2(C2×Dic11), (C2×C22).28(C2×C4), SmallGroup(352,39)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C44.D4
G = < a,b,c | a44=1, b4=a22, c2=a33, bab-1=a-1, cac-1=a21, cbc-1=a11b3 >
Smallest permutation representation of C44.D4
►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)
(1 52 34 63 23 74 12 85)(2 51 35 62 24 73 13 84)(3 50 36 61 25 72 14 83)(4 49 37 60 26 71 15 82)(5 48 38 59 27 70 16 81)(6 47 39 58 28 69 17 80)(7 46 40 57 29 68 18 79)(8 45 41 56 30 67 19 78)(9 88 42 55 31 66 20 77)(10 87 43 54 32 65 21 76)(11 86 44 53 33 64 22 75)
(1 74 34 63 23 52 12 85)(2 51 35 84 24 73 13 62)(3 72 36 61 25 50 14 83)(4 49 37 82 26 71 15 60)(5 70 38 59 27 48 16 81)(6 47 39 80 28 69 17 58)(7 68 40 57 29 46 18 79)(8 45 41 78 30 67 19 56)(9 66 42 55 31 88 20 77)(10 87 43 76 32 65 21 54)(11 64 44 53 33 86 22 75)
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), (1,52,34,63,23,74,12,85)(2,51,35,62,24,73,13,84)(3,50,36,61,25,72,14,83)(4,49,37,60,26,71,15,82)(5,48,38,59,27,70,16,81)(6,47,39,58,28,69,17,80)(7,46,40,57,29,68,18,79)(8,45,41,56,30,67,19,78)(9,88,42,55,31,66,20,77)(10,87,43,54,32,65,21,76)(11,86,44,53,33,64,22,75), (1,74,34,63,23,52,12,85)(2,51,35,84,24,73,13,62)(3,72,36,61,25,50,14,83)(4,49,37,82,26,71,15,60)(5,70,38,59,27,48,16,81)(6,47,39,80,28,69,17,58)(7,68,40,57,29,46,18,79)(8,45,41,78,30,67,19,56)(9,66,42,55,31,88,20,77)(10,87,43,76,32,65,21,54)(11,64,44,53,33,86,22,75)>;
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), (1,52,34,63,23,74,12,85)(2,51,35,62,24,73,13,84)(3,50,36,61,25,72,14,83)(4,49,37,60,26,71,15,82)(5,48,38,59,27,70,16,81)(6,47,39,58,28,69,17,80)(7,46,40,57,29,68,18,79)(8,45,41,56,30,67,19,78)(9,88,42,55,31,66,20,77)(10,87,43,54,32,65,21,76)(11,86,44,53,33,64,22,75), (1,74,34,63,23,52,12,85)(2,51,35,84,24,73,13,62)(3,72,36,61,25,50,14,83)(4,49,37,82,26,71,15,60)(5,70,38,59,27,48,16,81)(6,47,39,80,28,69,17,58)(7,68,40,57,29,46,18,79)(8,45,41,78,30,67,19,56)(9,66,42,55,31,88,20,77)(10,87,43,76,32,65,21,54)(11,64,44,53,33,86,22,75) );
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)], [(1,52,34,63,23,74,12,85),(2,51,35,62,24,73,13,84),(3,50,36,61,25,72,14,83),(4,49,37,60,26,71,15,82),(5,48,38,59,27,70,16,81),(6,47,39,58,28,69,17,80),(7,46,40,57,29,68,18,79),(8,45,41,56,30,67,19,78),(9,88,42,55,31,66,20,77),(10,87,43,54,32,65,21,76),(11,86,44,53,33,64,22,75)], [(1,74,34,63,23,52,12,85),(2,51,35,84,24,73,13,62),(3,72,36,61,25,50,14,83),(4,49,37,82,26,71,15,60),(5,70,38,59,27,48,16,81),(6,47,39,80,28,69,17,58),(7,68,40,57,29,46,18,79),(8,45,41,78,30,67,19,56),(9,66,42,55,31,88,20,77),(10,87,43,76,32,65,21,54),(11,64,44,53,33,86,22,75)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 8A | 8B | 8C | 8D | 11A | ··· | 11E | 22A | ··· | 22O | 22P | ··· | 22AI | 44A | ··· | 44J |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 44 | 44 | 44 | 44 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C4 | D4 | D11 | D22 | Dic11 | C11⋊D4 | C4.D4 | C44.D4 |
| kernel | C44.D4 | C44.C4 | D4×C22 | C22×C22 | C44 | C2×D4 | C2×C4 | C23 | C4 | C11 | C1 |
| # reps | 1 | 2 | 1 | 4 | 2 | 5 | 5 | 10 | 20 | 1 | 10 |
Matrix representation of C44.D4 ►in GL4(𝔽89) generated by
| 16 | 44 | 0 | 0 |
| 45 | 73 | 0 | 0 |
| 3 | 87 | 39 | 30 |
| 12 | 39 | 41 | 50 |
| 61 | 28 | 18 | 0 |
| 77 | 22 | 84 | 46 |
| 79 | 75 | 16 | 57 |
| 80 | 32 | 63 | 79 |
| 61 | 28 | 18 | 0 |
| 0 | 79 | 0 | 43 |
| 1 | 16 | 28 | 32 |
| 76 | 8 | 0 | 10 |
G:=sub<GL(4,GF(89))| [16,45,3,12,44,73,87,39,0,0,39,41,0,0,30,50],[61,77,79,80,28,22,75,32,18,84,16,63,0,46,57,79],[61,0,1,76,28,79,16,8,18,0,28,0,0,43,32,10] >;
C44.D4 in GAP, Magma, Sage, TeX
C_{44}.D_4 % in TeX
G:=Group("C44.D4"); // GroupNames label
G:=SmallGroup(352,39);
// by ID
G=gap.SmallGroup(352,39);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,24,121,188,86,579,11525]);
// Polycyclic
G:=Group<a,b,c|a^44=1,b^4=a^22,c^2=a^33,b*a*b^-1=a^-1,c*a*c^-1=a^21,c*b*c^-1=a^11*b^3>;
// generators/relations
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