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## G = C42.56Q8order 128 = 27

### 16th 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 — C42.56Q8
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×C8 — C2×C4×C8 — C42.56Q8
 Lower central C1 — C2 — C2×C4 — C42.56Q8
 Upper central C1 — C23 — C2×C42 — C42.56Q8
 Jennings C1 — C2 — C2 — C22×C4 — C42.56Q8

Generators and relations for C42.56Q8
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, dad-1=a-1b2, bc=cb, dbd-1=b-1, dcd-1=a2b2c3 >

Subgroups: 204 in 108 conjugacy classes, 60 normal (10 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×8], C22, C22 [×6], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×16], C23, C42 [×4], C4⋊C4 [×8], C2×C8 [×4], C2×C8 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2.C42 [×4], C4×C8 [×4], C2×C42, C2×C4⋊C4 [×4], C22×C8 [×2], C22.4Q16 [×4], C428C4 [×2], C2×C4×C8, C42.56Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C4○D4 [×4], C2×C4⋊C4, C42⋊C2 [×2], C4.4D4 [×2], C42.C2 [×2], C4○D8 [×4], C428C4, C23.25D4 [×2], C42.78C22 [×4], C42.56Q8

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

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

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

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

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 type + + + + - + image C1 C2 C2 C2 C4 Q8 D4 C4○D4 C4○D8 kernel C42.56Q8 C22.4Q16 C42⋊8C4 C2×C4×C8 C4×C8 C42 C22×C4 C2×C4 C22 # reps 1 4 2 1 8 2 2 8 16

Matrix representation of C42.56Q8 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 13 0 0 0 0 4 0 0 0 0 0 0 0 13 0 0 0 0 0 0 13
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 1 0
,
 4 0 0 0 0 0 0 13 0 0 0 0 0 0 14 3 0 0 0 0 14 14 0 0 0 0 0 0 5 5 0 0 0 0 12 5
,
 0 1 0 0 0 0 16 0 0 0 0 0 0 0 1 7 0 0 0 0 7 16 0 0 0 0 0 0 11 13 0 0 0 0 13 6

`G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,13,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,14,14,0,0,0,0,3,14,0,0,0,0,0,0,5,12,0,0,0,0,5,5],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,1,7,0,0,0,0,7,16,0,0,0,0,0,0,11,13,0,0,0,0,13,6] >;`

C42.56Q8 in GAP, Magma, Sage, TeX

`C_4^2._{56}Q_8`
`% in TeX`

`G:=Group("C4^2.56Q8");`
`// GroupNames label`

`G:=SmallGroup(128,567);`
`// by ID`

`G=gap.SmallGroup(128,567);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,568,422,58,2019,248,2804,172]);`
`// 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^2,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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