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## G = C42.27Q8order 128 = 27

### 27th 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 — C42.27Q8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C2×C8⋊C4 — C42.27Q8
 Lower central C1 — C23 — C42.27Q8
 Upper central C1 — C22×C4 — C42.27Q8
 Jennings C1 — C2 — C2 — C22×C4 — C42.27Q8

Generators and relations for C42.27Q8
G = < a,b,c,d | a4=b4=1, c4=b2, d2=c2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, bd=db, dcd-1=b-1c3 >

Subgroups: 188 in 124 conjugacy classes, 68 normal (30 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×10], C22 [×3], C22 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×8], C2×C4 [×14], C23, C42 [×4], C42 [×4], C4⋊C4 [×4], C2×C8 [×4], C2×C8 [×10], C22×C4 [×3], C22×C4 [×4], C2.C42 [×2], C8⋊C4 [×2], C4⋊C8 [×6], C2×C42, C2×C42 [×2], C2×C4⋊C4 [×2], C22×C8 [×2], C22×C8 [×2], C22.7C42 [×2], C4×C4⋊C4, C2×C8⋊C4, C2×C4⋊C8, C2×C4⋊C8 [×2], C42.27Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, C4⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C2×M4(2) [×2], C8○D4 [×2], C23.65C23, C4⋊M4(2), C42.6C22, C89D4 [×2], C84Q8 [×2], C42.27Q8

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

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4T 8A ··· 8P order 1 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 1 ··· 1 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + - + - image C1 C2 C2 C2 C2 C4 C4 D4 Q8 D4 Q8 M4(2) C4○D4 C8○D4 kernel C42.27Q8 C22.7C42 C4×C4⋊C4 C2×C8⋊C4 C2×C4⋊C8 C2.C42 C2×C4⋊C4 C42 C42 C2×C8 C2×C8 C2×C4 C2×C4 C22 # reps 1 2 1 1 3 4 4 2 2 2 2 8 4 8

Matrix representation of C42.27Q8 in GL6(𝔽17)

 1 15 0 0 0 0 1 16 0 0 0 0 0 0 1 15 0 0 0 0 1 16 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 13 0 0 0 0 0 0 13
,
 4 0 0 0 0 0 0 4 0 0 0 0 0 0 13 8 0 0 0 0 13 4 0 0 0 0 0 0 6 10 0 0 0 0 7 11
,
 7 10 0 0 0 0 12 10 0 0 0 0 0 0 6 11 0 0 0 0 3 11 0 0 0 0 0 0 0 15 0 0 0 0 15 0

`G:=sub<GL(6,GF(17))| [1,1,0,0,0,0,15,16,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,13,0,0,0,0,8,4,0,0,0,0,0,0,6,7,0,0,0,0,10,11],[7,12,0,0,0,0,10,10,0,0,0,0,0,0,6,3,0,0,0,0,11,11,0,0,0,0,0,0,0,15,0,0,0,0,15,0] >;`

C42.27Q8 in GAP, Magma, Sage, TeX

`C_4^2._{27}Q_8`
`% in TeX`

`G:=Group("C4^2.27Q8");`
`// GroupNames label`

`G:=SmallGroup(128,672);`
`// by ID`

`G=gap.SmallGroup(128,672);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,723,100,124]);`
`// Polycyclic`

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

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