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## G = C44.C4order 176 = 24·11

### 1st non-split extension by C44 of C4 acting via C4/C2=C2

Aliases: C44.1C4, C4.Dic11, C4.15D22, C112M4(2), C22.Dic11, C44.15C22, C11⋊C85C2, 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

 Derived series C1 — C22 — C44.C4
 Chief series C1 — C11 — C22 — C44 — C11⋊C8 — C44.C4
 Lower central C11 — C22 — C44.C4
 Upper central C1 — C4 — C2×C4

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
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 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
D441C4  C88.C4  C44.53D4  C44.46D4  C44.47D4  C44.D4  C44.10D4  C44.56D4  D44.2C4  M4(2)×D11  D446C22  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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