Copied to
clipboard

## G = C2.(C8⋊Q8)  order 128 = 27

### 3rd central stem extension by C2 of C8⋊Q8

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — C2.(C8⋊Q8)
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4⋊C4 — C23.65C23 — C2.(C8⋊Q8)
 Lower central C1 — C2 — C22×C4 — C2.(C8⋊Q8)
 Upper central C1 — C23 — C2×C42 — C2.(C8⋊Q8)
 Jennings C1 — C2 — C2 — C22×C4 — C2.(C8⋊Q8)

Generators and relations for C2.(C8⋊Q8)
G = < a,b,c,d | a2=b8=c4=1, d2=ac2, ab=ba, ac=ca, ad=da, cbc-1=ab5, dbd-1=ab3, dcd-1=ac-1 >

Subgroups: 240 in 125 conjugacy classes, 54 normal (44 characteristic)
C1, C2 [×7], C4 [×4], C4 [×9], C22 [×7], C8 [×4], C2×C4 [×6], C2×C4 [×21], C23, C42 [×2], C4⋊C4 [×6], C4⋊C4 [×15], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4 [×4], C2.C42, C4.Q8 [×2], C2.D8 [×2], C2×C42, C2×C4⋊C4 [×4], C2×C4⋊C4 [×3], C42.C2 [×4], C22×C8 [×2], C22.7C42, C22.4Q16 [×2], C23.65C23, C2×C4.Q8, C2×C2.D8, C2×C42.C2, C2.(C8⋊Q8)
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×6], C23, C2×D4 [×3], C2×Q8 [×3], C4○D4, C22≀C2, C22⋊Q8 [×3], C4⋊Q8 [×3], C4○D8 [×2], C8⋊C22, C8.C22, C23.78C23, D4⋊D4, D4.7D4, D4.Q8, Q8.Q8, C8.5Q8, C8⋊Q8, C2.(C8⋊Q8)

Smallest permutation representation of C2.(C8⋊Q8)
Regular action on 128 points
Generators in S128
```(1 105)(2 106)(3 107)(4 108)(5 109)(6 110)(7 111)(8 112)(9 89)(10 90)(11 91)(12 92)(13 93)(14 94)(15 95)(16 96)(17 47)(18 48)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)(25 61)(26 62)(27 63)(28 64)(29 57)(30 58)(31 59)(32 60)(33 84)(34 85)(35 86)(36 87)(37 88)(38 81)(39 82)(40 83)(49 115)(50 116)(51 117)(52 118)(53 119)(54 120)(55 113)(56 114)(65 98)(66 99)(67 100)(68 101)(69 102)(70 103)(71 104)(72 97)(73 124)(74 125)(75 126)(76 127)(77 128)(78 121)(79 122)(80 123)
(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)
(1 127 21 96)(2 73 22 13)(3 121 23 90)(4 75 24 15)(5 123 17 92)(6 77 18 9)(7 125 19 94)(8 79 20 11)(10 107 78 45)(12 109 80 47)(14 111 74 41)(16 105 76 43)(25 83 67 52)(26 37 68 115)(27 85 69 54)(28 39 70 117)(29 87 71 56)(30 33 72 119)(31 81 65 50)(32 35 66 113)(34 102 120 63)(36 104 114 57)(38 98 116 59)(40 100 118 61)(42 91 112 122)(44 93 106 124)(46 95 108 126)(48 89 110 128)(49 62 88 101)(51 64 82 103)(53 58 84 97)(55 60 86 99)
(1 49 43 37)(2 118 44 83)(3 55 45 35)(4 116 46 81)(5 53 47 33)(6 114 48 87)(7 51 41 39)(8 120 42 85)(9 29 128 104)(10 60 121 66)(11 27 122 102)(12 58 123 72)(13 25 124 100)(14 64 125 70)(15 31 126 98)(16 62 127 68)(17 84 109 119)(18 36 110 56)(19 82 111 117)(20 34 112 54)(21 88 105 115)(22 40 106 52)(23 86 107 113)(24 38 108 50)(26 76 101 96)(28 74 103 94)(30 80 97 92)(32 78 99 90)(57 77 71 89)(59 75 65 95)(61 73 67 93)(63 79 69 91)```

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

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

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

32 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 8A ··· 8H order 1 2 ··· 2 4 4 4 4 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 2 2 2 4 4 4 4 8 ··· 8 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + - - + + - image C1 C2 C2 C2 C2 C2 C2 D4 Q8 Q8 D4 C4○D4 C4○D8 C8⋊C22 C8.C22 kernel C2.(C8⋊Q8) C22.7C42 C22.4Q16 C23.65C23 C2×C4.Q8 C2×C2.D8 C2×C42.C2 C4⋊C4 C4⋊C4 C2×C8 C22×C4 C2×C4 C22 C22 C22 # reps 1 1 2 1 1 1 1 4 2 4 2 2 8 1 1

Matrix representation of C2.(C8⋊Q8) in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 15 0 0 0 0 0 0 8 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 12 1
,
 16 0 0 0 0 0 0 1 0 0 0 0 0 0 7 16 0 0 0 0 16 10 0 0 0 0 0 0 6 1 0 0 0 0 16 11
,
 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 6 1 0 0 0 0 14 11

`G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[15,0,0,0,0,0,0,8,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,12,0,0,0,0,0,1],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,7,16,0,0,0,0,16,10,0,0,0,0,0,0,6,16,0,0,0,0,1,11],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,1,7,0,0,0,0,7,16,0,0,0,0,0,0,6,14,0,0,0,0,1,11] >;`

C2.(C8⋊Q8) in GAP, Magma, Sage, TeX

`C_2.(C_8\rtimes Q_8)`
`% in TeX`

`G:=Group("C2.(C8:Q8)");`
`// GroupNames label`

`G:=SmallGroup(128,791);`
`// by ID`

`G=gap.SmallGroup(128,791);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,512,422,387,58,2804,718,172]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^2=b^8=c^4=1,d^2=a*c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^5,d*b*d^-1=a*b^3,d*c*d^-1=a*c^-1>;`
`// generators/relations`

׿
×
𝔽