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## G = C4.6Q32order 128 = 27

### 2nd non-split extension by C4 of Q32 acting via Q32/Q16=C2

p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C8 — C4.6Q32
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C4×C8 — C8⋊2Q8 — C4.6Q32
 Lower central C1 — C2 — C2×C4 — C2×C8 — C4.6Q32
 Upper central C1 — C22 — C42 — C4×C8 — C4.6Q32
 Jennings C1 — C2 — C2 — C2 — C2 — C2×C4 — C2×C4 — C4×C8 — C4.6Q32

Generators and relations for C4.6Q32
G = < a,b,c | a4=b16=1, c2=a2b8, bab-1=cac-1=a-1, cbc-1=a-1b-1 >

Character table of C4.6Q32

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 16A 16B 16C 16D 16E 16F 16G 16H size 1 1 1 1 2 2 2 2 4 16 16 2 2 2 2 4 4 8 8 8 8 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 1 1 1 1 -1 -1 1 1 -1 1 -1 -1 -1 -1 -1 1 1 i i -i -i i -i -i i i -i i -i linear of order 4 ρ6 1 1 1 1 -1 -1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 i i -i -i -i i i -i -i i -i i linear of order 4 ρ7 1 1 1 1 -1 -1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 -i -i i i i -i -i i i -i i -i linear of order 4 ρ8 1 1 1 1 -1 -1 1 1 -1 1 -1 -1 -1 -1 -1 1 1 -i -i i i -i i i -i -i i -i i linear of order 4 ρ9 2 2 2 2 -2 -2 2 2 -2 0 0 2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 2 2 2 2 0 0 -2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 2 2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 √2 √2 -√2 -√2 orthogonal lifted from D8 ρ12 2 2 2 2 2 2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 -√2 -√2 √2 √2 orthogonal lifted from D8 ρ13 2 2 -2 -2 -2 2 0 0 0 0 0 √2 -√2 √2 -√2 √2 -√2 0 0 0 0 ζ1615-ζ169 ζ165-ζ163 -ζ1615+ζ169 ζ165-ζ163 -ζ1615+ζ169 ζ1615-ζ169 -ζ165+ζ163 -ζ165+ζ163 symplectic lifted from Q32, Schur index 2 ρ14 2 -2 -2 2 0 0 2 -2 0 0 0 2 2 -2 -2 0 0 √2 -√2 √2 -√2 0 0 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ15 2 2 -2 -2 -2 2 0 0 0 0 0 -√2 √2 -√2 √2 -√2 √2 0 0 0 0 -ζ165+ζ163 ζ1615-ζ169 ζ165-ζ163 ζ1615-ζ169 ζ165-ζ163 -ζ165+ζ163 -ζ1615+ζ169 -ζ1615+ζ169 symplectic lifted from Q32, Schur index 2 ρ16 2 2 -2 -2 -2 2 0 0 0 0 0 -√2 √2 -√2 √2 -√2 √2 0 0 0 0 ζ165-ζ163 -ζ1615+ζ169 -ζ165+ζ163 -ζ1615+ζ169 -ζ165+ζ163 ζ165-ζ163 ζ1615-ζ169 ζ1615-ζ169 symplectic lifted from Q32, Schur index 2 ρ17 2 -2 -2 2 0 0 2 -2 0 0 0 2 2 -2 -2 0 0 -√2 √2 -√2 √2 0 0 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ18 2 2 -2 -2 -2 2 0 0 0 0 0 √2 -√2 √2 -√2 √2 -√2 0 0 0 0 -ζ1615+ζ169 -ζ165+ζ163 ζ1615-ζ169 -ζ165+ζ163 ζ1615-ζ169 -ζ1615+ζ169 ζ165-ζ163 ζ165-ζ163 symplectic lifted from Q32, Schur index 2 ρ19 2 2 2 2 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 -√-2 -√-2 √-2 √-2 -√-2 √-2 √-2 -√-2 complex lifted from SD16 ρ20 2 -2 -2 2 0 0 2 -2 0 0 0 -2 -2 2 2 0 0 -√-2 √-2 √-2 -√-2 0 0 0 0 0 0 0 0 complex lifted from SD16 ρ21 2 -2 -2 2 0 0 2 -2 0 0 0 -2 -2 2 2 0 0 √-2 -√-2 -√-2 √-2 0 0 0 0 0 0 0 0 complex lifted from SD16 ρ22 2 2 2 2 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 √-2 √-2 -√-2 -√-2 √-2 -√-2 -√-2 √-2 complex lifted from SD16 ρ23 2 2 -2 -2 2 -2 0 0 0 0 0 -√2 √2 -√2 √2 √2 -√2 0 0 0 0 ζ1613+ζ1611 ζ1615+ζ169 ζ1613+ζ1611 ζ167+ζ16 ζ165+ζ163 ζ165+ζ163 ζ1615+ζ169 ζ167+ζ16 complex lifted from SD32 ρ24 2 2 -2 -2 2 -2 0 0 0 0 0 √2 -√2 √2 -√2 -√2 √2 0 0 0 0 ζ1615+ζ169 ζ165+ζ163 ζ1615+ζ169 ζ1613+ζ1611 ζ167+ζ16 ζ167+ζ16 ζ165+ζ163 ζ1613+ζ1611 complex lifted from SD32 ρ25 2 2 -2 -2 2 -2 0 0 0 0 0 -√2 √2 -√2 √2 √2 -√2 0 0 0 0 ζ165+ζ163 ζ167+ζ16 ζ165+ζ163 ζ1615+ζ169 ζ1613+ζ1611 ζ1613+ζ1611 ζ167+ζ16 ζ1615+ζ169 complex lifted from SD32 ρ26 2 2 -2 -2 2 -2 0 0 0 0 0 √2 -√2 √2 -√2 -√2 √2 0 0 0 0 ζ167+ζ16 ζ1613+ζ1611 ζ167+ζ16 ζ165+ζ163 ζ1615+ζ169 ζ1615+ζ169 ζ1613+ζ1611 ζ165+ζ163 complex lifted from SD32 ρ27 4 -4 4 -4 0 0 0 0 0 0 0 -2√2 2√2 2√2 -2√2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from M5(2)⋊C2 ρ28 4 -4 4 -4 0 0 0 0 0 0 0 2√2 -2√2 -2√2 2√2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from M5(2)⋊C2 ρ29 4 -4 -4 4 0 0 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C4.10D4, Schur index 2

Smallest permutation representation of C4.6Q32
Regular action on 128 points
Generators in S128
```(1 101 69 28)(2 29 70 102)(3 103 71 30)(4 31 72 104)(5 105 73 32)(6 17 74 106)(7 107 75 18)(8 19 76 108)(9 109 77 20)(10 21 78 110)(11 111 79 22)(12 23 80 112)(13 97 65 24)(14 25 66 98)(15 99 67 26)(16 27 68 100)(33 86 60 127)(34 128 61 87)(35 88 62 113)(36 114 63 89)(37 90 64 115)(38 116 49 91)(39 92 50 117)(40 118 51 93)(41 94 52 119)(42 120 53 95)(43 96 54 121)(44 122 55 81)(45 82 56 123)(46 124 57 83)(47 84 58 125)(48 126 59 85)
(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 58 77 39)(2 83 78 116)(3 56 79 37)(4 81 80 114)(5 54 65 35)(6 95 66 128)(7 52 67 33)(8 93 68 126)(9 50 69 47)(10 91 70 124)(11 64 71 45)(12 89 72 122)(13 62 73 43)(14 87 74 120)(15 60 75 41)(16 85 76 118)(17 53 98 34)(18 119 99 86)(19 51 100 48)(20 117 101 84)(21 49 102 46)(22 115 103 82)(23 63 104 44)(24 113 105 96)(25 61 106 42)(26 127 107 94)(27 59 108 40)(28 125 109 92)(29 57 110 38)(30 123 111 90)(31 55 112 36)(32 121 97 88)```

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

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

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

Matrix representation of C4.6Q32 in GL4(𝔽17) generated by

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

C4.6Q32 in GAP, Magma, Sage, TeX

`C_4._6Q_{32}`
`% in TeX`

`G:=Group("C4.6Q32");`
`// GroupNames label`

`G:=SmallGroup(128,97);`
`// by ID`

`G=gap.SmallGroup(128,97);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,568,422,387,520,794,416,2028,124]);`
`// Polycyclic`

`G:=Group<a,b,c|a^4=b^16=1,c^2=a^2*b^8,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^-1*b^-1>;`
`// generators/relations`

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