metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C44.46D4, C4.11D44, M4(2)⋊3D11, (C2×C4).1D22, (C2×D44).6C2, (C22×D11).C4, C44.C4⋊2C2, C2.9(D22⋊C4), C11⋊1(C4.D4), C22.4(C4×D11), C4.21(C11⋊D4), C22.8(C22⋊C4), (C11×M4(2))⋊7C2, (C2×C44).13C22, (C2×C22).2(C2×C4), SmallGroup(352,29)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C44.46D4
G = < a,b,c | a44=c2=1, b4=a22, bab-1=cac=a-1, cbc=a11b3 >
2C2
44C2
44C2
22C22
22C22
44C22
44C22
2C22
4D11
4D11
2C8
11C23
11C23
22C8
22D4
22D4
2D22
2D22
4D22
4D22
11M4(2)
11C2×D4
2C88
2D44
2D44
11C4.D4
Smallest permutation representation of C44.46D4
►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 81 12 70 23 59 34 48)(2 80 13 69 24 58 35 47)(3 79 14 68 25 57 36 46)(4 78 15 67 26 56 37 45)(5 77 16 66 27 55 38 88)(6 76 17 65 28 54 39 87)(7 75 18 64 29 53 40 86)(8 74 19 63 30 52 41 85)(9 73 20 62 31 51 42 84)(10 72 21 61 32 50 43 83)(11 71 22 60 33 49 44 82)
(1 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 44)(14 43)(15 42)(16 41)(17 40)(18 39)(19 38)(20 37)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 30)(28 29)(45 84)(46 83)(47 82)(48 81)(49 80)(50 79)(51 78)(52 77)(53 76)(54 75)(55 74)(56 73)(57 72)(58 71)(59 70)(60 69)(61 68)(62 67)(63 66)(64 65)(85 88)(86 87)
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), (1,81,12,70,23,59,34,48)(2,80,13,69,24,58,35,47)(3,79,14,68,25,57,36,46)(4,78,15,67,26,56,37,45)(5,77,16,66,27,55,38,88)(6,76,17,65,28,54,39,87)(7,75,18,64,29,53,40,86)(8,74,19,63,30,52,41,85)(9,73,20,62,31,51,42,84)(10,72,21,61,32,50,43,83)(11,71,22,60,33,49,44,82), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(45,84)(46,83)(47,82)(48,81)(49,80)(50,79)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,72)(58,71)(59,70)(60,69)(61,68)(62,67)(63,66)(64,65)(85,88)(86,87)>;
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), (1,81,12,70,23,59,34,48)(2,80,13,69,24,58,35,47)(3,79,14,68,25,57,36,46)(4,78,15,67,26,56,37,45)(5,77,16,66,27,55,38,88)(6,76,17,65,28,54,39,87)(7,75,18,64,29,53,40,86)(8,74,19,63,30,52,41,85)(9,73,20,62,31,51,42,84)(10,72,21,61,32,50,43,83)(11,71,22,60,33,49,44,82), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(45,84)(46,83)(47,82)(48,81)(49,80)(50,79)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,72)(58,71)(59,70)(60,69)(61,68)(62,67)(63,66)(64,65)(85,88)(86,87) );
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)], [(1,81,12,70,23,59,34,48),(2,80,13,69,24,58,35,47),(3,79,14,68,25,57,36,46),(4,78,15,67,26,56,37,45),(5,77,16,66,27,55,38,88),(6,76,17,65,28,54,39,87),(7,75,18,64,29,53,40,86),(8,74,19,63,30,52,41,85),(9,73,20,62,31,51,42,84),(10,72,21,61,32,50,43,83),(11,71,22,60,33,49,44,82)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,44),(14,43),(15,42),(16,41),(17,40),(18,39),(19,38),(20,37),(21,36),(22,35),(23,34),(24,33),(25,32),(26,31),(27,30),(28,29),(45,84),(46,83),(47,82),(48,81),(49,80),(50,79),(51,78),(52,77),(53,76),(54,75),(55,74),(56,73),(57,72),(58,71),(59,70),(60,69),(61,68),(62,67),(63,66),(64,65),(85,88),(86,87)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 8A | 8B | 8C | 8D | 11A | ··· | 11E | 22A | ··· | 22E | 22F | ··· | 22J | 44A | ··· | 44J | 44K | ··· | 44O | 88A | ··· | 88T |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 2 | 44 | 44 | 2 | 2 | 4 | 4 | 44 | 44 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D11 | D22 | D44 | C11⋊D4 | C4×D11 | C4.D4 | C44.46D4 |
| kernel | C44.46D4 | C44.C4 | C11×M4(2) | C2×D44 | C22×D11 | C44 | M4(2) | C2×C4 | C4 | C4 | C22 | C11 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 2 | 5 | 5 | 10 | 10 | 10 | 1 | 10 |
Matrix representation of C44.46D4 ►in GL4(𝔽89) generated by
| 16 | 73 | 0 | 0 |
| 75 | 53 | 0 | 0 |
| 0 | 0 | 24 | 73 |
| 0 | 0 | 16 | 45 |
| 54 | 10 | 25 | 7 |
| 11 | 35 | 34 | 16 |
| 6 | 40 | 20 | 9 |
| 32 | 14 | 5 | 69 |
| 60 | 45 | 0 | 0 |
| 11 | 29 | 0 | 0 |
| 69 | 15 | 86 | 59 |
| 12 | 72 | 24 | 3 |
G:=sub<GL(4,GF(89))| [16,75,0,0,73,53,0,0,0,0,24,16,0,0,73,45],[54,11,6,32,10,35,40,14,25,34,20,5,7,16,9,69],[60,11,69,12,45,29,15,72,0,0,86,24,0,0,59,3] >;
C44.46D4 in GAP, Magma, Sage, TeX
C_{44}._{46}D_4 % in TeX
G:=Group("C44.46D4"); // GroupNames label
G:=SmallGroup(352,29);
// by ID
G=gap.SmallGroup(352,29);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,121,31,362,86,297,11525]);
// Polycyclic
G:=Group<a,b,c|a^44=c^2=1,b^4=a^22,b*a*b^-1=c*a*c=a^-1,c*b*c=a^11*b^3>;
// generators/relations
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