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## G = C2.(C8⋊8D4)  order 128 = 27

### 4th central extension by C2 of C8⋊8D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2.(C8⋊8D4)
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×C8 — C2×C4×C8 — C2.(C8⋊8D4)
 Lower central C1 — C2 — C2×C4 — C2.(C8⋊8D4)
 Upper central C1 — C23 — C2×C42 — C2.(C8⋊8D4)
 Jennings C1 — C2 — C2 — C22×C4 — C2.(C8⋊8D4)

Generators and relations for C2.(C88D4)
G = < a,b,c,d | a2=b8=c4=1, d2=a, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=ab3, dcd-1=c-1 >

Subgroups: 244 in 128 conjugacy classes, 60 normal (44 characteristic)
C1, C2 [×7], C4 [×4], C4 [×10], C22 [×7], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×18], 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], C4×C8 [×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×C4×C8, C2×Q8⋊C4 [×2], C2.(C88D4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, SD16 [×2], Q16 [×2], C22×C4, C2×D4 [×2], C4○D4 [×4], C42⋊C2, C4×D4 [×2], C4⋊D4, C22.D4, C4.4D4, C422C2, C2×SD16, C2×Q16, C4○D8 [×2], C24.C22, C4×SD16, C4×Q16, C88D4, C8.18D4, C4.SD16, C42.78C22, C2.(C88D4)

Smallest permutation representation of C2.(C88D4)
Regular action on 128 points
Generators in S128
```(1 54)(2 55)(3 56)(4 49)(5 50)(6 51)(7 52)(8 53)(9 48)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)(33 122)(34 123)(35 124)(36 125)(37 126)(38 127)(39 128)(40 121)(57 80)(58 73)(59 74)(60 75)(61 76)(62 77)(63 78)(64 79)(65 101)(66 102)(67 103)(68 104)(69 97)(70 98)(71 99)(72 100)(81 119)(82 120)(83 113)(84 114)(85 115)(86 116)(87 117)(88 118)(89 108)(90 109)(91 110)(92 111)(93 112)(94 105)(95 106)(96 107)
(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 9 63 92)(2 43 64 106)(3 15 57 90)(4 41 58 112)(5 13 59 96)(6 47 60 110)(7 11 61 94)(8 45 62 108)(10 73 93 49)(12 79 95 55)(14 77 89 53)(16 75 91 51)(17 37 103 82)(18 121 104 115)(19 35 97 88)(20 127 98 113)(21 33 99 86)(22 125 100 119)(23 39 101 84)(24 123 102 117)(25 122 71 116)(26 36 72 81)(27 128 65 114)(28 34 66 87)(29 126 67 120)(30 40 68 85)(31 124 69 118)(32 38 70 83)(42 76 105 52)(44 74 107 50)(46 80 109 56)(48 78 111 54)
(1 83 54 113)(2 116 55 86)(3 81 56 119)(4 114 49 84)(5 87 50 117)(6 120 51 82)(7 85 52 115)(8 118 53 88)(9 70 48 98)(10 101 41 65)(11 68 42 104)(12 99 43 71)(13 66 44 102)(14 97 45 69)(15 72 46 100)(16 103 47 67)(17 110 29 91)(18 94 30 105)(19 108 31 89)(20 92 32 111)(21 106 25 95)(22 90 26 109)(23 112 27 93)(24 96 28 107)(33 64 122 79)(34 74 123 59)(35 62 124 77)(36 80 125 57)(37 60 126 75)(38 78 127 63)(39 58 128 73)(40 76 121 61)```

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

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

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

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 1 1 2 2 2 2 2 2 type + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 D4 D4 SD16 Q16 C4○D4 C4○D8 kernel C2.(C8⋊8D4) C22.4Q16 C23.65C23 C23.67C23 C2×C4×C8 C2×Q8⋊C4 Q8⋊C4 C2×C8 C22×C4 C2×C4 C2×C4 C2×C4 C22 # reps 1 2 1 1 1 2 8 2 2 4 4 8 8

Matrix representation of C2.(C88D4) in GL5(𝔽17)

 16 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 16
,
 13 0 0 0 0 0 5 12 0 0 0 5 5 0 0 0 0 0 14 14 0 0 0 3 14
,
 16 0 0 0 0 0 13 6 0 0 0 6 4 0 0 0 0 0 0 4 0 0 0 4 0
,
 4 0 0 0 0 0 13 6 0 0 0 6 4 0 0 0 0 0 13 0 0 0 0 0 4

`G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[13,0,0,0,0,0,5,5,0,0,0,12,5,0,0,0,0,0,14,3,0,0,0,14,14],[16,0,0,0,0,0,13,6,0,0,0,6,4,0,0,0,0,0,0,4,0,0,0,4,0],[4,0,0,0,0,0,13,6,0,0,0,6,4,0,0,0,0,0,13,0,0,0,0,0,4] >;`

C2.(C88D4) in GAP, Magma, Sage, TeX

`C_2.(C_8\rtimes_8D_4)`
`% in TeX`

`G:=Group("C2.(C8:8D4)");`
`// GroupNames label`

`G:=SmallGroup(128,665);`
`// by ID`

`G=gap.SmallGroup(128,665);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,58,2019,248,2804,172]);`
`// 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=d*b*d^-1=a*b^3,d*c*d^-1=c^-1>;`
`// generators/relations`

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