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## G = C42.39Q8order 128 = 27

### 39th non-split extension by C42 of Q8 acting via Q8/C2=C22

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C42.39Q8
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C4⋊C4 — C23.65C23 — C42.39Q8
 Lower central C1 — C23 — C42.39Q8
 Upper central C1 — C23 — C42.39Q8
 Jennings C1 — C23 — C42.39Q8

Generators and relations for C42.39Q8
G = < a,b,c,d | a4=b4=c4=1, d2=a2b2c2, ab=ba, cac-1=a-1b2, dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=c-1 >

Subgroups: 308 in 180 conjugacy classes, 100 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×16], C22 [×3], C22 [×4], C2×C4 [×10], C2×C4 [×40], C23, C42 [×4], C42 [×4], C4⋊C4 [×20], C22×C4 [×3], C22×C4 [×12], C2.C42 [×16], C2×C42 [×3], C2×C4⋊C4 [×16], C424C4, C428C4, C429C4, C23.65C23 [×4], C23.81C23 [×4], C23.83C23 [×4], C42.39Q8
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×4], C24, C42.C2 [×4], C22×Q8, C2×C4○D4 [×2], 2+ 1+4 [×2], 2- 1+4 [×2], C2×C42.C2, C22.34C24, C22.35C24, C22.36C24 [×2], C23.41C23 [×2], C42.39Q8

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

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

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

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

32 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E ··· 4P 4Q ··· 4X order 1 2 ··· 2 4 4 4 4 4 ··· 4 4 ··· 4 size 1 1 ··· 1 2 2 2 2 4 ··· 4 8 ··· 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 4 4 type + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 Q8 C4○D4 2+ 1+4 2- 1+4 kernel C42.39Q8 C42⋊4C4 C42⋊8C4 C42⋊9C4 C23.65C23 C23.81C23 C23.83C23 C42 C2×C4 C22 C22 # reps 1 1 1 1 4 4 4 4 8 2 2

Matrix representation of C42.39Q8 in GL8(𝔽5)

 4 4 0 0 0 0 0 0 2 1 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 2 2 0 0 0 0 0 0 1 2 4 0 0 0 0 4 0 4 1 0 0 0 0 3 2 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 2 2 0 0 0 0 0 0 3 1 3 0 0 0 0 0 0 3 3 2
,
 2 0 0 0 0 0 0 0 1 3 0 0 0 0 0 0 0 0 3 2 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 3 1 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 2 4 2 0 0 0 0 0 4 4 1
,
 2 2 0 0 0 0 0 0 1 3 0 0 0 0 0 0 0 0 1 4 0 0 0 0 0 0 2 4 0 0 0 0 0 0 0 0 2 1 1 0 0 0 0 0 3 1 3 3 0 0 0 0 2 2 0 2 0 0 0 0 0 3 3 2

`G:=sub<GL(8,GF(5))| [4,2,0,0,0,0,0,0,4,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,4,3,0,0,0,0,2,1,0,2,0,0,0,0,2,2,4,0,0,0,0,0,0,4,1,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,3,2,3,0,0,0,0,0,0,2,1,3,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,2],[2,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,2,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,2,2,4,0,0,0,0,0,0,4,4,0,0,0,0,0,0,2,1],[2,1,0,0,0,0,0,0,2,3,0,0,0,0,0,0,0,0,1,2,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,0,2,3,2,0,0,0,0,0,1,1,2,3,0,0,0,0,1,3,0,3,0,0,0,0,0,3,2,2] >;`

C42.39Q8 in GAP, Magma, Sage, TeX

`C_4^2._{39}Q_8`
`% in TeX`

`G:=Group("C4^2.39Q8");`
`// GroupNames label`

`G:=SmallGroup(128,1379);`
`// by ID`

`G=gap.SmallGroup(128,1379);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,723,184,185,80]);`
`// Polycyclic`

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

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