Copied to
clipboard

## G = C42.46Q8order 128 = 27

### 6th 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 — C4 — C42.46Q8
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C2×C4×C8 — C42.46Q8
 Lower central C1 — C2 — C4 — C42.46Q8
 Upper central C1 — C22×C4 — C2×C42 — C42.46Q8
 Jennings C1 — C22 — C22 — C2×C42 — C42.46Q8

Generators and relations for C42.46Q8
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=a2b-1c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=a-1b2c3 >

Subgroups: 176 in 104 conjugacy classes, 60 normal (36 characteristic)
C1, C2 [×3], C2 [×4], C4 [×6], C4 [×2], C4 [×6], C22 [×3], C22 [×4], C8 [×6], C2×C4 [×10], C2×C4 [×4], C2×C4 [×10], C23, C42 [×4], C42 [×2], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×10], C22×C4 [×3], C22×C4 [×2], C2.C42, C4×C8 [×2], C4×C8, C4⋊C8 [×2], C4⋊C8, C2×C42, C2×C42, C2×C4⋊C4 [×2], C22×C8 [×2], C4×C4⋊C4, C2×C4×C8, C2×C4⋊C8, C42.46Q8
Quotients: C1, C2 [×3], C4 [×6], C22, C8 [×4], C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2×C8 [×2], M4(2) [×2], D8, SD16 [×2], Q16, C2.C42, C4×C8, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C4≀C2 [×2], C4⋊C8 [×2], C4.Q8, C2.D8, D4⋊C8 [×2], Q8⋊C8 [×2], C22.7C42, C426C4, C22.4Q16, C42.46Q8

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

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + - + + - image C1 C2 C2 C2 C4 C4 C4 C8 D4 Q8 D4 M4(2) D8 SD16 Q16 C4≀C2 kernel C42.46Q8 C4×C4⋊C4 C2×C4×C8 C2×C4⋊C8 C4×C8 C4⋊C8 C2×C4⋊C4 C4⋊C4 C42 C42 C22×C4 C2×C4 C2×C4 C2×C4 C2×C4 C22 # reps 1 1 1 1 4 4 4 16 1 1 2 4 2 4 2 8

Matrix representation of C42.46Q8 in GL4(𝔽17) generated by

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

C42.46Q8 in GAP, Magma, Sage, TeX

`C_4^2._{46}Q_8`
`% in TeX`

`G:=Group("C4^2.46Q8");`
`// GroupNames label`

`G:=SmallGroup(128,11);`
`// by ID`

`G=gap.SmallGroup(128,11);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,184,248,1684,242]);`
`// Polycyclic`

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

׿
×
𝔽