metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C44.1C4, C4.Dic11, C4.15D22, C11⋊2M4(2), C22.Dic11, C44.15C22, C11⋊C8⋊5C2, C22.7(C2×C4), (C2×C44).5C2, (C2×C22).3C4, (C2×C4).2D11, C2.3(C2×Dic11), SmallGroup(176,9)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C44.C4
G = < a,b | a44=1, b4=a22, bab-1=a-1 >
Smallest permutation representation of C44.C4
►On 88 points
(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 82 34 49 23 60 12 71)(2 81 35 48 24 59 13 70)(3 80 36 47 25 58 14 69)(4 79 37 46 26 57 15 68)(5 78 38 45 27 56 16 67)(6 77 39 88 28 55 17 66)(7 76 40 87 29 54 18 65)(8 75 41 86 30 53 19 64)(9 74 42 85 31 52 20 63)(10 73 43 84 32 51 21 62)(11 72 44 83 33 50 22 61)
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), (1,82,34,49,23,60,12,71)(2,81,35,48,24,59,13,70)(3,80,36,47,25,58,14,69)(4,79,37,46,26,57,15,68)(5,78,38,45,27,56,16,67)(6,77,39,88,28,55,17,66)(7,76,40,87,29,54,18,65)(8,75,41,86,30,53,19,64)(9,74,42,85,31,52,20,63)(10,73,43,84,32,51,21,62)(11,72,44,83,33,50,22,61)>;
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), (1,82,34,49,23,60,12,71)(2,81,35,48,24,59,13,70)(3,80,36,47,25,58,14,69)(4,79,37,46,26,57,15,68)(5,78,38,45,27,56,16,67)(6,77,39,88,28,55,17,66)(7,76,40,87,29,54,18,65)(8,75,41,86,30,53,19,64)(9,74,42,85,31,52,20,63)(10,73,43,84,32,51,21,62)(11,72,44,83,33,50,22,61) );
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)], [(1,82,34,49,23,60,12,71),(2,81,35,48,24,59,13,70),(3,80,36,47,25,58,14,69),(4,79,37,46,26,57,15,68),(5,78,38,45,27,56,16,67),(6,77,39,88,28,55,17,66),(7,76,40,87,29,54,18,65),(8,75,41,86,30,53,19,64),(9,74,42,85,31,52,20,63),(10,73,43,84,32,51,21,62),(11,72,44,83,33,50,22,61)]])
C44.C4 is a maximal subgroup of
D44⋊1C4 C88.C4 C44.53D4 C44.46D4 C44.47D4 C44.D4 C44.10D4 C44.56D4 D44.2C4 M4(2)×D11 D44⋊6C22 C44.C23 Q8.Dic11 Q8⋊D22 D4.9D22
C44.C4 is a maximal quotient of
C42.D11 C44⋊C8 C44.55D4
50 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 11A | ··· | 11E | 22A | ··· | 22O | 44A | ··· | 44T |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 22 | 22 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C4 | C4 | M4(2) | D11 | Dic11 | D22 | Dic11 | C44.C4 |
| kernel | C44.C4 | C11⋊C8 | C2×C44 | C44 | C2×C22 | C11 | C2×C4 | C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 20 |
Matrix representation of C44.C4 ►in GL2(𝔽89) generated by
| 9 | 0 |
| 67 | 10 |
| 36 | 55 |
| 81 | 53 |
G:=sub<GL(2,GF(89))| [9,67,0,10],[36,81,55,53] >;
C44.C4 in GAP, Magma, Sage, TeX
C_{44}.C_4 % in TeX
G:=Group("C44.C4"); // GroupNames label
G:=SmallGroup(176,9);
// by ID
G=gap.SmallGroup(176,9);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-11,20,101,42,4004]);
// Polycyclic
G:=Group<a,b|a^44=1,b^4=a^22,b*a*b^-1=a^-1>;
// generators/relations
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