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G = C42.55Q8order 128 = 27

15th 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 — C42.55Q8
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×C8 — C2×C4×C8 — C42.55Q8
 Lower central C1 — C2 — C2×C4 — C42.55Q8
 Upper central C1 — C23 — C2×C42 — C42.55Q8
 Jennings C1 — C2 — C2 — C22×C4 — C42.55Q8

Generators and relations for C42.55Q8
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=a2b2c3 >

Subgroups: 236 in 124 conjugacy classes, 76 normal (16 characteristic)
C1, C2 [×3], C2 [×4], C4 [×12], C4 [×4], C22 [×3], C22 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×16], C2×C4 [×12], C23, C42 [×4], C4⋊C4 [×16], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4 [×4], C4×C8 [×4], C2×C42, C2×C4⋊C4 [×4], C2×C4⋊C4 [×4], C22×C8 [×2], C22.4Q16 [×4], C429C4 [×2], C2×C4×C8, C42.55Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], D8 [×2], SD16 [×4], Q16 [×2], C22×C4, C2×D4, C2×Q8, C4○D4 [×4], C4.Q8 [×4], C2.D8 [×4], C2×C4⋊C4, C42⋊C2 [×2], C4.4D4 [×2], C42.C2 [×2], C2×D8, C2×SD16 [×2], C2×Q16, C428C4, C2×C4.Q8, C2×C2.D8, C4.4D8 [×2], C4.SD16 [×2], C42.55Q8

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

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

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

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

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 D8 SD16 Q16 C4○D4 kernel C42.55Q8 C22.4Q16 C42⋊9C4 C2×C4×C8 C4×C8 C42 C22×C4 C2×C4 C2×C4 C2×C4 C2×C4 # reps 1 4 2 1 8 2 2 4 8 4 8

Matrix representation of C42.55Q8 in GL5(𝔽17)

 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 0 13 0 0 0 13 0
,
 16 0 0 0 0 0 16 15 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 16 0 0 0 0 0 0 6 0 0 0 14 11 0 0 0 0 0 4 0 0 0 0 0 4
,
 4 0 0 0 0 0 15 5 0 0 0 13 2 0 0 0 0 0 1 11 0 0 0 6 16

`G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,13,0],[16,0,0,0,0,0,16,1,0,0,0,15,1,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,0,14,0,0,0,6,11,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,15,13,0,0,0,5,2,0,0,0,0,0,1,6,0,0,0,11,16] >;`

C42.55Q8 in GAP, Magma, Sage, TeX

`C_4^2._{55}Q_8`
`% in TeX`

`G:=Group("C4^2.55Q8");`
`// GroupNames label`

`G:=SmallGroup(128,566);`
`// by ID`

`G=gap.SmallGroup(128,566);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,120,422,58,2019,248,2804,172]);`
`// 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,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=a^2*b^2*c^3>;`
`// generators/relations`

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