Copied to
clipboard

## G = C42.58Q8order 128 = 27

### 18th 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 — C42.58Q8
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C4×C8 — C42.58Q8
 Lower central C1 — C2 — C2×C4 — C42.58Q8
 Upper central C1 — C23 — C2×C42 — C42.58Q8
 Jennings C1 — C2 — C2 — C22×C4 — C42.58Q8

Generators and relations for C42.58Q8
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=c3 >

Subgroups: 252 in 140 conjugacy classes, 92 normal (12 characteristic)
C1, C2, C2 [×6], C4 [×12], C4 [×4], C22 [×3], C22 [×4], C8 [×8], C2×C4 [×2], C2×C4 [×16], C2×C4 [×12], C23, C42 [×4], C4⋊C4 [×16], C2×C8 [×12], C22×C4, C22×C4 [×2], C22×C4 [×4], C4×C8 [×4], C4.Q8 [×8], C2×C42, C2×C4⋊C4 [×4], C2×C4⋊C4 [×4], C22×C8 [×2], C429C4 [×2], C2×C4×C8, C2×C4.Q8 [×4], C42.58Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×6], C23, C4⋊C4 [×12], SD16 [×8], C22×C4, C2×D4 [×3], C2×Q8 [×3], C4.Q8 [×8], C2×C4⋊C4 [×3], C41D4, C4⋊Q8 [×3], C2×SD16 [×4], C429C4, C2×C4.Q8 [×2], C85D4 [×2], C83Q8 [×2], C42.58Q8

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

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

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

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

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 type + + + + - + - + image C1 C2 C2 C2 C4 Q8 D4 Q8 D4 SD16 kernel C42.58Q8 C42⋊9C4 C2×C4×C8 C2×C4.Q8 C4×C8 C42 C2×C8 C2×C8 C22×C4 C2×C4 # reps 1 2 1 4 8 2 4 4 2 16

Matrix representation of C42.58Q8 in GL5(𝔽17)

 1 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16 15 0 0 0 1 1
,
 16 0 0 0 0 0 16 2 0 0 0 16 1 0 0 0 0 0 16 0 0 0 0 0 16
,
 16 0 0 0 0 0 0 7 0 0 0 5 7 0 0 0 0 0 16 0 0 0 0 0 16
,
 13 0 0 0 0 0 6 3 0 0 0 16 11 0 0 0 0 0 9 3 0 0 0 1 8

`G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,1,0,0,0,15,1],[16,0,0,0,0,0,16,16,0,0,0,2,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,0,5,0,0,0,7,7,0,0,0,0,0,16,0,0,0,0,0,16],[13,0,0,0,0,0,6,16,0,0,0,3,11,0,0,0,0,0,9,1,0,0,0,3,8] >;`

C42.58Q8 in GAP, Magma, Sage, TeX

`C_4^2._{58}Q_8`
`% in TeX`

`G:=Group("C4^2.58Q8");`
`// GroupNames label`

`G:=SmallGroup(128,576);`
`// by ID`

`G=gap.SmallGroup(128,576);`
`# by ID`

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

׿
×
𝔽