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## G = C42.8Q8order 128 = 27

### 8th non-split extension by C42 of Q8 acting via Q8/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.8Q8
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C2×C4⋊C8 — C42.8Q8
 Lower central C1 — C22 — C2×C4 — C42.8Q8
 Upper central C1 — C23 — C2×C42 — C42.8Q8
 Jennings C1 — C22 — C22 — C2×C42 — C42.8Q8

Generators and relations for C42.8Q8
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1c2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=a-1b2c3 >

Subgroups: 184 in 94 conjugacy classes, 56 normal (16 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×4], C22 [×3], C22 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×12], C2×C4 [×8], C23, C42 [×2], C42 [×2], C4⋊C4 [×6], C2×C8 [×8], C22×C4, C22×C4 [×2], C22×C4 [×2], C4⋊C8 [×4], C4⋊C8 [×2], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C22×C8 [×2], C429C4, C2×C4⋊C8 [×2], C42.8Q8
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], D8 [×2], SD16 [×4], Q16 [×2], C2.C42, C4.D4, C4.10D4, D4⋊C4 [×4], Q8⋊C4 [×4], C4.Q8 [×2], C2.D8 [×2], C4.D8, C4.10D8 [×2], C4.6Q16, C22.4Q16 [×2], C22.C42, C42.8Q8

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

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I 4J 4K 4L 4M 4N 8A ··· 8P order 1 2 ··· 2 4 ··· 4 4 4 4 4 4 4 8 ··· 8 size 1 1 ··· 1 2 ··· 2 4 4 8 8 8 8 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + - + + - + - image C1 C2 C2 C4 C4 D4 Q8 D4 D8 SD16 Q16 C4.D4 C4.10D4 kernel C42.8Q8 C42⋊9C4 C2×C4⋊C8 C4⋊C8 C2×C4⋊C4 C42 C42 C22×C4 C2×C4 C2×C4 C2×C4 C22 C22 # reps 1 1 2 8 4 1 1 2 4 8 4 1 1

Matrix representation of C42.8Q8 in GL5(𝔽17)

 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 16 0 0 0 0 0 16
,
 13 0 0 0 0 0 12 5 0 0 0 12 12 0 0 0 0 0 8 2 0 0 0 10 9
,
 4 0 0 0 0 0 1 7 0 0 0 7 16 0 0 0 0 0 15 5 0 0 0 13 2

`G:=sub<GL(5,GF(17))| [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],[1,0,0,0,0,0,0,1,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,16],[13,0,0,0,0,0,12,12,0,0,0,5,12,0,0,0,0,0,8,10,0,0,0,2,9],[4,0,0,0,0,0,1,7,0,0,0,7,16,0,0,0,0,0,15,13,0,0,0,5,2] >;`

C42.8Q8 in GAP, Magma, Sage, TeX

`C_4^2._8Q_8`
`% in TeX`

`G:=Group("C4^2.8Q8");`
`// GroupNames label`

`G:=SmallGroup(128,28);`
`// by ID`

`G=gap.SmallGroup(128,28);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,723,184,3924,242]);`
`// Polycyclic`

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

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