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## G = C2.(C8⋊D4)  order 128 = 27

### 4th central extension by C2 of C8⋊D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2.(C8⋊D4)
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×C8 — C2×C8⋊C4 — C2.(C8⋊D4)
 Lower central C1 — C2 — C2×C4 — C2.(C8⋊D4)
 Upper central C1 — C23 — C2×C42 — C2.(C8⋊D4)
 Jennings C1 — C2 — C2 — C22×C4 — C2.(C8⋊D4)

Generators and relations for C2.(C8⋊D4)
G = < a,b,c,d | a2=b8=c4=1, d2=a, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd-1=ab3, dcd-1=c-1 >

Subgroups: 244 in 124 conjugacy classes, 56 normal (44 characteristic)
C1, C2 [×7], C4 [×4], C4 [×8], C22 [×7], C8 [×4], C2×C4 [×6], C2×C4 [×20], Q8 [×6], C23, C42 [×2], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×2], C2×Q8 [×5], C2.C42 [×3], C8⋊C4 [×2], Q8⋊C4 [×4], Q8⋊C4 [×2], C2×C42, C2×C4⋊C4 [×3], C2×C4⋊C4, C22×C8 [×2], C22×Q8, C22.4Q16 [×2], C23.65C23, C23.67C23, C2×C8⋊C4, C2×Q8⋊C4 [×2], C2.(C8⋊D4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22×C4, C2×D4 [×2], C4○D4 [×4], C42⋊C2, C4×D4 [×2], C4⋊D4, C22.D4, C4.4D4, C422C2, C8⋊C22, C8.C22 [×3], C24.C22, SD16⋊C4, Q16⋊C4, C8⋊D4, C8.D4, C42.28C22, C42.30C22, C2.(C8⋊D4)

Smallest permutation representation of C2.(C8⋊D4)
Regular action on 128 points
Generators in S128
```(1 71)(2 72)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 33)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 57)(24 58)(25 55)(26 56)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(41 125)(42 126)(43 127)(44 128)(45 121)(46 122)(47 123)(48 124)(73 97)(74 98)(75 99)(76 100)(77 101)(78 102)(79 103)(80 104)(81 105)(82 106)(83 107)(84 108)(85 109)(86 110)(87 111)(88 112)(89 117)(90 118)(91 119)(92 120)(93 113)(94 114)(95 115)(96 116)
(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 33 29 118)(2 15 30 89)(3 39 31 116)(4 13 32 95)(5 37 25 114)(6 11 26 93)(7 35 27 120)(8 9 28 91)(10 49 92 69)(12 55 94 67)(14 53 96 65)(16 51 90 71)(17 83 104 126)(18 106 97 41)(19 81 98 124)(20 112 99 47)(21 87 100 122)(22 110 101 45)(23 85 102 128)(24 108 103 43)(34 50 119 70)(36 56 113 68)(38 54 115 66)(40 52 117 72)(42 59 107 80)(44 57 109 78)(46 63 111 76)(48 61 105 74)(58 84 79 127)(60 82 73 125)(62 88 75 123)(64 86 77 121)
(1 47 71 123)(2 126 72 42)(3 45 65 121)(4 124 66 48)(5 43 67 127)(6 122 68 46)(7 41 69 125)(8 128 70 44)(9 102 34 78)(10 73 35 97)(11 100 36 76)(12 79 37 103)(13 98 38 74)(14 77 39 101)(15 104 40 80)(16 75 33 99)(17 117 59 89)(18 92 60 120)(19 115 61 95)(20 90 62 118)(21 113 63 93)(22 96 64 116)(23 119 57 91)(24 94 58 114)(25 108 55 84)(26 87 56 111)(27 106 49 82)(28 85 50 109)(29 112 51 88)(30 83 52 107)(31 110 53 86)(32 81 54 105)```

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

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

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

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 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 D4 D4 C4○D4 C8⋊C22 C8.C22 kernel C2.(C8⋊D4) C22.4Q16 C23.65C23 C23.67C23 C2×C8⋊C4 C2×Q8⋊C4 Q8⋊C4 C2×C8 C22×C4 C2×C4 C22 C22 # reps 1 2 1 1 1 2 8 2 2 8 1 3

Matrix representation of C2.(C8⋊D4) in GL8(𝔽17)

 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16
,
 16 15 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 16 0 2 0 0 0 0 3 3 3 9 0 0 0 0 7 12 1 1 0 0 0 0 10 1 10 13
,
 13 0 0 0 0 0 0 0 4 4 0 0 0 0 0 0 0 0 10 4 0 0 0 0 0 0 13 7 0 0 0 0 0 0 0 0 8 15 14 5 0 0 0 0 4 3 13 3 0 0 0 0 6 10 12 7 0 0 0 0 16 7 7 11
,
 1 2 0 0 0 0 0 0 16 16 0 0 0 0 0 0 0 0 13 7 0 0 0 0 0 0 10 4 0 0 0 0 0 0 0 0 10 12 16 13 0 0 0 0 14 5 10 1 0 0 0 0 6 6 7 9 0 0 0 0 2 1 13 12

`G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,3,7,10,0,0,0,0,16,3,12,1,0,0,0,0,0,3,1,10,0,0,0,0,2,9,1,13],[13,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,10,13,0,0,0,0,0,0,4,7,0,0,0,0,0,0,0,0,8,4,6,16,0,0,0,0,15,3,10,7,0,0,0,0,14,13,12,7,0,0,0,0,5,3,7,11],[1,16,0,0,0,0,0,0,2,16,0,0,0,0,0,0,0,0,13,10,0,0,0,0,0,0,7,4,0,0,0,0,0,0,0,0,10,14,6,2,0,0,0,0,12,5,6,1,0,0,0,0,16,10,7,13,0,0,0,0,13,1,9,12] >;`

C2.(C8⋊D4) in GAP, Magma, Sage, TeX

`C_2.(C_8\rtimes D_4)`
`% in TeX`

`G:=Group("C2.(C8:D4)");`
`// GroupNames label`

`G:=SmallGroup(128,667);`
`// by ID`

`G=gap.SmallGroup(128,667);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,723,58,2019,248,2804,172]);`
`// Polycyclic`

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

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