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## G = C2.(C8⋊3Q8)  order 128 = 27

### 2nd central stem extension by C2 of C8⋊3Q8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — C2.(C8⋊3Q8)
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — C2×C4.Q8 — C2.(C8⋊3Q8)
 Lower central C1 — C2 — C22×C4 — C2.(C8⋊3Q8)
 Upper central C1 — C23 — C2×C42 — C2.(C8⋊3Q8)
 Jennings C1 — C2 — C2 — C22×C4 — C2.(C8⋊3Q8)

Generators and relations for C2.(C83Q8)
G = < a,b,c,d | a2=b8=c4=1, d2=c2, cbc-1=ab=ba, ac=ca, ad=da, dbd-1=ab3, dcd-1=ac-1 >

Subgroups: 232 in 115 conjugacy classes, 54 normal (28 characteristic)
C1, C2 [×7], C4 [×4], C4 [×9], C22 [×7], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×17], C23, C42 [×2], C4⋊C4 [×2], C4⋊C4 [×13], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4 [×4], C2.C42, C4.Q8 [×4], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×3], C22×C8 [×2], C22.7C42, C22.4Q16 [×2], C429C4, C23.65C23, C2×C4.Q8 [×2], C2.(C83Q8)
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], Q8 [×4], C23, SD16 [×4], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×3], C4⋊D4, C22⋊Q8 [×2], C22.D4, C42.C2 [×2], C4⋊Q8, C2×SD16 [×2], C8⋊C22, C8.C22, C23.81C23, C4⋊SD16, D4.D4, C23.46D4, C23.47D4, C83Q8, C8⋊Q8, C2.(C83Q8)

Smallest permutation representation of C2.(C83Q8)
Regular action on 128 points
Generators in S128
```(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 29)(10 30)(11 31)(12 32)(13 25)(14 26)(15 27)(16 28)(17 94)(18 95)(19 96)(20 89)(21 90)(22 91)(23 92)(24 93)(33 48)(34 41)(35 42)(36 43)(37 44)(38 45)(39 46)(40 47)(57 97)(58 98)(59 99)(60 100)(61 101)(62 102)(63 103)(64 104)(65 76)(66 77)(67 78)(68 79)(69 80)(70 73)(71 74)(72 75)(81 127)(82 128)(83 121)(84 122)(85 123)(86 124)(87 125)(88 126)(105 116)(106 117)(107 118)(108 119)(109 120)(110 113)(111 114)(112 115)
(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 98 90 121)(2 59 91 84)(3 100 92 123)(4 61 93 86)(5 102 94 125)(6 63 95 88)(7 104 96 127)(8 57 89 82)(9 105 37 77)(10 117 38 67)(11 107 39 79)(12 119 40 69)(13 109 33 73)(14 113 34 71)(15 111 35 75)(16 115 36 65)(17 87 53 62)(18 126 54 103)(19 81 55 64)(20 128 56 97)(21 83 49 58)(22 122 50 99)(23 85 51 60)(24 124 52 101)(25 120 48 70)(26 110 41 74)(27 114 42 72)(28 112 43 76)(29 116 44 66)(30 106 45 78)(31 118 46 68)(32 108 47 80)
(1 75 90 111)(2 67 91 117)(3 73 92 109)(4 65 93 115)(5 79 94 107)(6 71 95 113)(7 77 96 105)(8 69 89 119)(9 64 37 81)(10 99 38 122)(11 62 39 87)(12 97 40 128)(13 60 33 85)(14 103 34 126)(15 58 35 83)(16 101 36 124)(17 118 53 68)(18 110 54 74)(19 116 55 66)(20 108 56 80)(21 114 49 72)(22 106 50 78)(23 120 51 70)(24 112 52 76)(25 100 48 123)(26 63 41 88)(27 98 42 121)(28 61 43 86)(29 104 44 127)(30 59 45 84)(31 102 46 125)(32 57 47 82)```

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

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

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

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 2 2 2 2 2 4 4 type + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 D4 Q8 D4 SD16 C4○D4 C8⋊C22 C8.C22 kernel C2.(C8⋊3Q8) C22.7C42 C22.4Q16 C42⋊9C4 C23.65C23 C2×C4.Q8 C4⋊C4 C2×C8 C22×C4 C2×C4 C2×C4 C22 C22 # reps 1 1 2 1 1 2 2 4 2 8 6 1 1

Matrix representation of C2.(C83Q8) 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 1 0 0 0 0 0 0 1
,
 10 4 0 0 0 0 13 7 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0 5 12
,
 1 11 0 0 0 0 6 16 0 0 0 0 0 0 8 2 0 0 0 0 10 9 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 13 0 0 0 0 0 0 13 0 0 0 0 0 0 15 8 0 0 0 0 10 2 0 0 0 0 0 0 4 11 0 0 0 0 11 13

`G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[10,13,0,0,0,0,4,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,5,0,0,0,0,12,12],[1,6,0,0,0,0,11,16,0,0,0,0,0,0,8,10,0,0,0,0,2,9,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,15,10,0,0,0,0,8,2,0,0,0,0,0,0,4,11,0,0,0,0,11,13] >;`

C2.(C83Q8) in GAP, Magma, Sage, TeX

`C_2.(C_8\rtimes_3Q_8)`
`% in TeX`

`G:=Group("C2.(C8:3Q8)");`
`// GroupNames label`

`G:=SmallGroup(128,816);`
`// by ID`

`G=gap.SmallGroup(128,816);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,64,422,387,394,2028,1027,124]);`
`// Polycyclic`

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

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