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## G = C42.59Q8order 128 = 27

### 19th non-split extension by C42 of Q8 acting via Q8/C4=C2

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.59Q8
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C4×C8 — C42.59Q8
 Lower central C1 — C2 — C2×C4 — C42.59Q8
 Upper central C1 — C23 — C2×C42 — C42.59Q8
 Jennings C1 — C2 — C2 — C22×C4 — C42.59Q8

Generators and relations for C42.59Q8
G = < a,b,c,d | a4=b4=1, c4=b2, d2=bc2, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=b2c3 >

Subgroups: 252 in 140 conjugacy classes, 92 normal (14 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C4⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C4×C8, C2.D8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C429C4, C2×C4×C8, C2×C2.D8, C42.59Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, D8, Q16, C22×C4, C2×D4, C2×Q8, C2.D8, C2×C4⋊C4, C41D4, C4⋊Q8, C2×D8, C2×Q16, C429C4, C2×C2.D8, C84D4, C4⋊Q16, C82Q8, C42.59Q8

Smallest permutation representation of C42.59Q8
Regular action on 128 points
Generators in S128
```(1 25 56 24)(2 26 49 17)(3 27 50 18)(4 28 51 19)(5 29 52 20)(6 30 53 21)(7 31 54 22)(8 32 55 23)(9 41 106 89)(10 42 107 90)(11 43 108 91)(12 44 109 92)(13 45 110 93)(14 46 111 94)(15 47 112 95)(16 48 105 96)(33 119 87 127)(34 120 88 128)(35 113 81 121)(36 114 82 122)(37 115 83 123)(38 116 84 124)(39 117 85 125)(40 118 86 126)(57 67 99 80)(58 68 100 73)(59 69 101 74)(60 70 102 75)(61 71 103 76)(62 72 104 77)(63 65 97 78)(64 66 98 79)
(1 63 5 59)(2 64 6 60)(3 57 7 61)(4 58 8 62)(9 128 13 124)(10 121 14 125)(11 122 15 126)(12 123 16 127)(17 79 21 75)(18 80 22 76)(19 73 23 77)(20 74 24 78)(25 65 29 69)(26 66 30 70)(27 67 31 71)(28 68 32 72)(33 44 37 48)(34 45 38 41)(35 46 39 42)(36 47 40 43)(49 98 53 102)(50 99 54 103)(51 100 55 104)(52 101 56 97)(81 94 85 90)(82 95 86 91)(83 96 87 92)(84 89 88 93)(105 119 109 115)(106 120 110 116)(107 113 111 117)(108 114 112 118)
(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)(89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128)
(1 111 57 119)(2 110 58 118)(3 109 59 117)(4 108 60 116)(5 107 61 115)(6 106 62 114)(7 105 63 113)(8 112 64 120)(9 104 122 53)(10 103 123 52)(11 102 124 51)(12 101 125 50)(13 100 126 49)(14 99 127 56)(15 98 128 55)(16 97 121 54)(17 93 73 86)(18 92 74 85)(19 91 75 84)(20 90 76 83)(21 89 77 82)(22 96 78 81)(23 95 79 88)(24 94 80 87)(25 46 67 33)(26 45 68 40)(27 44 69 39)(28 43 70 38)(29 42 71 37)(30 41 72 36)(31 48 65 35)(32 47 66 34)```

`G:=sub<Sym(128)| (1,25,56,24)(2,26,49,17)(3,27,50,18)(4,28,51,19)(5,29,52,20)(6,30,53,21)(7,31,54,22)(8,32,55,23)(9,41,106,89)(10,42,107,90)(11,43,108,91)(12,44,109,92)(13,45,110,93)(14,46,111,94)(15,47,112,95)(16,48,105,96)(33,119,87,127)(34,120,88,128)(35,113,81,121)(36,114,82,122)(37,115,83,123)(38,116,84,124)(39,117,85,125)(40,118,86,126)(57,67,99,80)(58,68,100,73)(59,69,101,74)(60,70,102,75)(61,71,103,76)(62,72,104,77)(63,65,97,78)(64,66,98,79), (1,63,5,59)(2,64,6,60)(3,57,7,61)(4,58,8,62)(9,128,13,124)(10,121,14,125)(11,122,15,126)(12,123,16,127)(17,79,21,75)(18,80,22,76)(19,73,23,77)(20,74,24,78)(25,65,29,69)(26,66,30,70)(27,67,31,71)(28,68,32,72)(33,44,37,48)(34,45,38,41)(35,46,39,42)(36,47,40,43)(49,98,53,102)(50,99,54,103)(51,100,55,104)(52,101,56,97)(81,94,85,90)(82,95,86,91)(83,96,87,92)(84,89,88,93)(105,119,109,115)(106,120,110,116)(107,113,111,117)(108,114,112,118), (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)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128), (1,111,57,119)(2,110,58,118)(3,109,59,117)(4,108,60,116)(5,107,61,115)(6,106,62,114)(7,105,63,113)(8,112,64,120)(9,104,122,53)(10,103,123,52)(11,102,124,51)(12,101,125,50)(13,100,126,49)(14,99,127,56)(15,98,128,55)(16,97,121,54)(17,93,73,86)(18,92,74,85)(19,91,75,84)(20,90,76,83)(21,89,77,82)(22,96,78,81)(23,95,79,88)(24,94,80,87)(25,46,67,33)(26,45,68,40)(27,44,69,39)(28,43,70,38)(29,42,71,37)(30,41,72,36)(31,48,65,35)(32,47,66,34)>;`

`G:=Group( (1,25,56,24)(2,26,49,17)(3,27,50,18)(4,28,51,19)(5,29,52,20)(6,30,53,21)(7,31,54,22)(8,32,55,23)(9,41,106,89)(10,42,107,90)(11,43,108,91)(12,44,109,92)(13,45,110,93)(14,46,111,94)(15,47,112,95)(16,48,105,96)(33,119,87,127)(34,120,88,128)(35,113,81,121)(36,114,82,122)(37,115,83,123)(38,116,84,124)(39,117,85,125)(40,118,86,126)(57,67,99,80)(58,68,100,73)(59,69,101,74)(60,70,102,75)(61,71,103,76)(62,72,104,77)(63,65,97,78)(64,66,98,79), (1,63,5,59)(2,64,6,60)(3,57,7,61)(4,58,8,62)(9,128,13,124)(10,121,14,125)(11,122,15,126)(12,123,16,127)(17,79,21,75)(18,80,22,76)(19,73,23,77)(20,74,24,78)(25,65,29,69)(26,66,30,70)(27,67,31,71)(28,68,32,72)(33,44,37,48)(34,45,38,41)(35,46,39,42)(36,47,40,43)(49,98,53,102)(50,99,54,103)(51,100,55,104)(52,101,56,97)(81,94,85,90)(82,95,86,91)(83,96,87,92)(84,89,88,93)(105,119,109,115)(106,120,110,116)(107,113,111,117)(108,114,112,118), (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)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128), (1,111,57,119)(2,110,58,118)(3,109,59,117)(4,108,60,116)(5,107,61,115)(6,106,62,114)(7,105,63,113)(8,112,64,120)(9,104,122,53)(10,103,123,52)(11,102,124,51)(12,101,125,50)(13,100,126,49)(14,99,127,56)(15,98,128,55)(16,97,121,54)(17,93,73,86)(18,92,74,85)(19,91,75,84)(20,90,76,83)(21,89,77,82)(22,96,78,81)(23,95,79,88)(24,94,80,87)(25,46,67,33)(26,45,68,40)(27,44,69,39)(28,43,70,38)(29,42,71,37)(30,41,72,36)(31,48,65,35)(32,47,66,34) );`

`G=PermutationGroup([[(1,25,56,24),(2,26,49,17),(3,27,50,18),(4,28,51,19),(5,29,52,20),(6,30,53,21),(7,31,54,22),(8,32,55,23),(9,41,106,89),(10,42,107,90),(11,43,108,91),(12,44,109,92),(13,45,110,93),(14,46,111,94),(15,47,112,95),(16,48,105,96),(33,119,87,127),(34,120,88,128),(35,113,81,121),(36,114,82,122),(37,115,83,123),(38,116,84,124),(39,117,85,125),(40,118,86,126),(57,67,99,80),(58,68,100,73),(59,69,101,74),(60,70,102,75),(61,71,103,76),(62,72,104,77),(63,65,97,78),(64,66,98,79)], [(1,63,5,59),(2,64,6,60),(3,57,7,61),(4,58,8,62),(9,128,13,124),(10,121,14,125),(11,122,15,126),(12,123,16,127),(17,79,21,75),(18,80,22,76),(19,73,23,77),(20,74,24,78),(25,65,29,69),(26,66,30,70),(27,67,31,71),(28,68,32,72),(33,44,37,48),(34,45,38,41),(35,46,39,42),(36,47,40,43),(49,98,53,102),(50,99,54,103),(51,100,55,104),(52,101,56,97),(81,94,85,90),(82,95,86,91),(83,96,87,92),(84,89,88,93),(105,119,109,115),(106,120,110,116),(107,113,111,117),(108,114,112,118)], [(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),(89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128)], [(1,111,57,119),(2,110,58,118),(3,109,59,117),(4,108,60,116),(5,107,61,115),(6,106,62,114),(7,105,63,113),(8,112,64,120),(9,104,122,53),(10,103,123,52),(11,102,124,51),(12,101,125,50),(13,100,126,49),(14,99,127,56),(15,98,128,55),(16,97,121,54),(17,93,73,86),(18,92,74,85),(19,91,75,84),(20,90,76,83),(21,89,77,82),(22,96,78,81),(23,95,79,88),(24,94,80,87),(25,46,67,33),(26,45,68,40),(27,44,69,39),(28,43,70,38),(29,42,71,37),(30,41,72,36),(31,48,65,35),(32,47,66,34)]])`

44 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4L 4M ··· 4T 8A ··· 8P order 1 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 ··· 2 8 ··· 8 2 ··· 2

44 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 type + + + + - + - + + - image C1 C2 C2 C2 C4 Q8 D4 Q8 D4 D8 Q16 kernel C42.59Q8 C42⋊9C4 C2×C4×C8 C2×C2.D8 C4×C8 C42 C2×C8 C2×C8 C22×C4 C2×C4 C2×C4 # reps 1 2 1 4 8 2 4 4 2 8 8

Matrix representation of C42.59Q8 in GL5(𝔽17)

 16 0 0 0 0 0 0 1 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 1
,
 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 1 2 0 0 0 16 16
,
 1 0 0 0 0 0 0 16 0 0 0 1 0 0 0 0 0 0 6 6 0 0 0 14 0
,
 13 0 0 0 0 0 6 13 0 0 0 13 11 0 0 0 0 0 10 10 0 0 0 12 7

`G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,16,0,0,0,2,16],[1,0,0,0,0,0,0,1,0,0,0,16,0,0,0,0,0,0,6,14,0,0,0,6,0],[13,0,0,0,0,0,6,13,0,0,0,13,11,0,0,0,0,0,10,12,0,0,0,10,7] >;`

C42.59Q8 in GAP, Magma, Sage, TeX

`C_4^2._{59}Q_8`
`% in TeX`

`G:=Group("C4^2.59Q8");`
`// GroupNames label`

`G:=SmallGroup(128,577);`
`// by ID`

`G=gap.SmallGroup(128,577);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,512,422,100,2019,248]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b*c^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^2*c^3>;`
`// generators/relations`

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