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## G = C2.D8⋊5C4order 128 = 27

### 3rd semidirect product of C2.D8 and C4 acting via C4/C2=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2.D8⋊5C4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4⋊C4 — C4×C4⋊C4 — C2.D8⋊5C4
 Lower central C1 — C2 — C2×C4 — C2.D8⋊5C4
 Upper central C1 — C23 — C2×C42 — C2.D8⋊5C4
 Jennings C1 — C2 — C2 — C22×C4 — C2.D8⋊5C4

Generators and relations for C2.D85C4
G = < a,b,c,d | a2=b8=d4=1, c2=a, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=ab5, dcd-1=ab4c >

Subgroups: 220 in 115 conjugacy classes, 58 normal (44 characteristic)
C1, C2 [×7], C4 [×4], C4 [×10], C22 [×7], C8 [×3], C2×C4 [×6], C2×C4 [×2], C2×C4 [×20], C23, C42 [×4], C4⋊C4 [×6], C4⋊C4 [×7], C2×C8 [×2], C2×C8 [×5], C22×C4 [×3], C22×C4 [×4], C2.C42 [×2], C2.D8 [×4], C2×C42, C2×C42, C2×C4⋊C4 [×4], C2×C4⋊C4, C22×C8 [×2], C22.7C42, C22.4Q16 [×3], C4×C4⋊C4, C23.65C23, C2×C2.D8, C2.D85C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, Q16 [×2], C22×C4, C2×D4, C2×Q8, C4○D4 [×4], C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C422C2, C2×Q16, C4○D8, C8⋊C22 [×2], C23.63C23, C4×Q16, D8⋊C4, C4.Q16, D4.Q8, C23.19D4, C23.48D4, C2.D85C4

Smallest permutation representation of C2.D85C4
Regular action on 128 points
Generators in S128
```(1 60)(2 61)(3 62)(4 63)(5 64)(6 57)(7 58)(8 59)(9 109)(10 110)(11 111)(12 112)(13 105)(14 106)(15 107)(16 108)(17 102)(18 103)(19 104)(20 97)(21 98)(22 99)(23 100)(24 101)(25 94)(26 95)(27 96)(28 89)(29 90)(30 91)(31 92)(32 93)(33 80)(34 73)(35 74)(36 75)(37 76)(38 77)(39 78)(40 79)(41 88)(42 81)(43 82)(44 83)(45 84)(46 85)(47 86)(48 87)(49 71)(50 72)(51 65)(52 66)(53 67)(54 68)(55 69)(56 70)(113 128)(114 121)(115 122)(116 123)(117 124)(118 125)(119 126)(120 127)
(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)
(1 110 60 10)(2 109 61 9)(3 108 62 16)(4 107 63 15)(5 106 64 14)(6 105 57 13)(7 112 58 12)(8 111 59 11)(17 86 102 47)(18 85 103 46)(19 84 104 45)(20 83 97 44)(21 82 98 43)(22 81 99 42)(23 88 100 41)(24 87 101 48)(25 78 94 39)(26 77 95 38)(27 76 96 37)(28 75 89 36)(29 74 90 35)(30 73 91 34)(31 80 92 33)(32 79 93 40)(49 118 71 125)(50 117 72 124)(51 116 65 123)(52 115 66 122)(53 114 67 121)(54 113 68 128)(55 120 69 127)(56 119 70 126)
(1 78 127 103)(2 36 128 23)(3 80 121 97)(4 38 122 17)(5 74 123 99)(6 40 124 19)(7 76 125 101)(8 34 126 21)(9 32 68 84)(10 90 69 42)(11 26 70 86)(12 92 71 44)(13 28 72 88)(14 94 65 46)(15 30 66 82)(16 96 67 48)(18 60 39 120)(20 62 33 114)(22 64 35 116)(24 58 37 118)(25 51 85 106)(27 53 87 108)(29 55 81 110)(31 49 83 112)(41 105 89 50)(43 107 91 52)(45 109 93 54)(47 111 95 56)(57 79 117 104)(59 73 119 98)(61 75 113 100)(63 77 115 102)```

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4R 4S 4T 4U 4V 8A ··· 8H order 1 2 ··· 2 4 ··· 4 4 ··· 4 4 4 4 4 8 ··· 8 size 1 1 ··· 1 2 ··· 2 4 ··· 4 8 8 8 8 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + - + - + image C1 C2 C2 C2 C2 C2 C4 Q8 D4 Q16 C4○D4 C4○D8 C8⋊C22 kernel C2.D8⋊5C4 C22.7C42 C22.4Q16 C4×C4⋊C4 C23.65C23 C2×C2.D8 C2.D8 C4⋊C4 C22×C4 C2×C4 C2×C4 C22 C22 # reps 1 1 3 1 1 1 8 2 2 4 8 4 2

Matrix representation of C2.D85C4 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
,
 3 3 0 0 0 0 3 14 0 0 0 0 0 0 0 3 0 0 0 0 11 0 0 0 0 0 0 0 2 0 0 0 0 0 0 9
,
 12 12 0 0 0 0 12 5 0 0 0 0 0 0 11 12 0 0 0 0 7 6 0 0 0 0 0 0 0 8 0 0 0 0 2 0
,
 0 1 0 0 0 0 16 0 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 16 0 0 0 0 0 0 16

`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],[3,3,0,0,0,0,3,14,0,0,0,0,0,0,0,11,0,0,0,0,3,0,0,0,0,0,0,0,2,0,0,0,0,0,0,9],[12,12,0,0,0,0,12,5,0,0,0,0,0,0,11,7,0,0,0,0,12,6,0,0,0,0,0,0,0,2,0,0,0,0,8,0],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,0,16] >;`

C2.D85C4 in GAP, Magma, Sage, TeX

`C_2.D_8\rtimes_5C_4`
`% in TeX`

`G:=Group("C2.D8:5C4");`
`// GroupNames label`

`G:=SmallGroup(128,653);`
`// by ID`

`G=gap.SmallGroup(128,653);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,680,422,394,2804,718,172]);`
`// Polycyclic`

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

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