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## G = C42.26Q8order 128 = 27

### 26th 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 — C42.26Q8
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C8⋊C4 — C42.26Q8
 Lower central C1 — C2 — C2×C4 — C42.26Q8
 Upper central C1 — C23 — C2×C42 — C42.26Q8
 Jennings C1 — C2 — C2 — C22×C4 — C42.26Q8

Generators and relations for C42.26Q8
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1c2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c3 >

Subgroups: 236 in 128 conjugacy classes, 76 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×8], C22 [×3], C22 [×4], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×16], C23, C42 [×4], C4⋊C4 [×12], C2×C8 [×12], C22×C4, C22×C4 [×2], C22×C4 [×4], C2.C42 [×2], C8⋊C4 [×4], C4.Q8 [×4], C2.D8 [×4], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C22×C8 [×2], C428C4, C429C4, C2×C8⋊C4, C2×C4.Q8 [×2], C2×C2.D8 [×2], C42.26Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×6], C23, C4⋊C4 [×12], C22×C4, C2×D4 [×3], C2×Q8 [×3], C2×C4⋊C4 [×3], C41D4, C4⋊Q8 [×3], C8⋊C22 [×2], C8.C22 [×2], C429C4, M4(2)⋊C4 [×2], C83D4, C8.2D4, C8⋊Q8 [×2], C42.26Q8

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

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

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

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

32 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 8A ··· 8H order 1 2 ··· 2 4 4 4 4 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 2 2 2 4 4 4 4 8 ··· 8 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + - + - + + - image C1 C2 C2 C2 C2 C2 C4 Q8 D4 Q8 D4 C8⋊C22 C8.C22 kernel C42.26Q8 C42⋊8C4 C42⋊9C4 C2×C8⋊C4 C2×C4.Q8 C2×C2.D8 C8⋊C4 C42 C2×C8 C2×C8 C22×C4 C22 C22 # reps 1 1 1 1 2 2 8 2 4 4 2 2 2

Matrix representation of C42.26Q8 in GL8(𝔽17)

 4 1 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 13 13 0 0 0 0 0 16 0 0 0 0 0 0 0 0 15 15 4
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 16 4 4 0 0 0 0 0 8 2 4 13
,
 4 1 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 9 15 13 4 0 0 0 0 0 2 2 13 0 0 0 0 0 1 0 0
,
 15 5 0 0 0 0 0 0 16 2 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 13 12 13 16 0 0 0 0 8 3 14 4 0 0 0 0 1 12 16 4 0 0 0 0 7 8 1 2

`G:=sub<GL(8,GF(17))| [4,0,0,0,0,0,0,0,1,13,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,13,0,15,0,0,0,0,1,13,0,15,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,16,8,0,0,0,0,16,0,4,2,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,13],[4,0,0,0,0,0,0,0,1,13,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,15,2,1,0,0,0,0,0,13,2,0,0,0,0,0,1,4,13,0],[15,16,0,0,0,0,0,0,5,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,13,8,1,7,0,0,0,0,12,3,12,8,0,0,0,0,13,14,16,1,0,0,0,0,16,4,4,2] >;`

C42.26Q8 in GAP, Magma, Sage, TeX

`C_4^2._{26}Q_8`
`% in TeX`

`G:=Group("C4^2.26Q8");`
`// GroupNames label`

`G:=SmallGroup(128,579);`
`// by ID`

`G=gap.SmallGroup(128,579);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,723,100,2019,248]);`
`// 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=a*b^2,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;`
`// generators/relations`

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