Copied to
clipboard

## G = C2.D8⋊4C4order 128 = 27

### 2nd 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⋊4C4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4⋊C4 — C4×C4⋊C4 — C2.D8⋊4C4
 Lower central C1 — C2 — C2×C4 — C2.D8⋊4C4
 Upper central C1 — C23 — C2×C42 — C2.D8⋊4C4
 Jennings C1 — C2 — C2 — C22×C4 — C2.D8⋊4C4

Generators and relations for C2.D84C4
G = < a,b,c,d | a2=b8=d4=1, c2=a, dbd-1=ab=ba, dcd-1=ac=ca, ad=da, cbc-1=b-1 >

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.D84C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D8 [×2], C22×C4, C2×D4, C2×Q8, C4○D4 [×4], C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C422C2, C2×D8, C4○D8, C8.C22 [×2], C23.63C23, C4×D8, Q16⋊C4, D4⋊Q8, Q8.Q8, C22.D8, C23.20D4, C2.D84C4

Smallest permutation representation of C2.D84C4
Regular action on 128 points
Generators in S128
```(1 60)(2 61)(3 62)(4 63)(5 64)(6 57)(7 58)(8 59)(9 105)(10 106)(11 107)(12 108)(13 109)(14 110)(15 111)(16 112)(17 97)(18 98)(19 99)(20 100)(21 101)(22 102)(23 103)(24 104)(25 89)(26 90)(27 91)(28 92)(29 93)(30 94)(31 95)(32 96)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 73)(40 74)(41 83)(42 84)(43 85)(44 86)(45 87)(46 88)(47 81)(48 82)(49 71)(50 72)(51 65)(52 66)(53 67)(54 68)(55 69)(56 70)(113 124)(114 125)(115 126)(116 127)(117 128)(118 121)(119 122)(120 123)
(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)
(1 110 60 14)(2 109 61 13)(3 108 62 12)(4 107 63 11)(5 106 64 10)(6 105 57 9)(7 112 58 16)(8 111 59 15)(17 87 97 45)(18 86 98 44)(19 85 99 43)(20 84 100 42)(21 83 101 41)(22 82 102 48)(23 81 103 47)(24 88 104 46)(25 79 89 37)(26 78 90 36)(27 77 91 35)(28 76 92 34)(29 75 93 33)(30 74 94 40)(31 73 95 39)(32 80 96 38)(49 118 71 121)(50 117 72 128)(51 116 65 127)(52 115 66 126)(53 114 67 125)(54 113 68 124)(55 120 69 123)(56 119 70 122)
(1 74 123 99)(2 33 124 20)(3 76 125 101)(4 35 126 22)(5 78 127 103)(6 37 128 24)(7 80 121 97)(8 39 122 18)(9 25 72 88)(10 90 65 47)(11 27 66 82)(12 92 67 41)(13 29 68 84)(14 94 69 43)(15 31 70 86)(16 96 71 45)(17 58 38 118)(19 60 40 120)(21 62 34 114)(23 64 36 116)(26 51 81 106)(28 53 83 108)(30 55 85 110)(32 49 87 112)(42 109 93 54)(44 111 95 56)(46 105 89 50)(48 107 91 52)(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)(9,105)(10,106)(11,107)(12,108)(13,109)(14,110)(15,111)(16,112)(17,97)(18,98)(19,99)(20,100)(21,101)(22,102)(23,103)(24,104)(25,89)(26,90)(27,91)(28,92)(29,93)(30,94)(31,95)(32,96)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,73)(40,74)(41,83)(42,84)(43,85)(44,86)(45,87)(46,88)(47,81)(48,82)(49,71)(50,72)(51,65)(52,66)(53,67)(54,68)(55,69)(56,70)(113,124)(114,125)(115,126)(116,127)(117,128)(118,121)(119,122)(120,123), (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), (1,110,60,14)(2,109,61,13)(3,108,62,12)(4,107,63,11)(5,106,64,10)(6,105,57,9)(7,112,58,16)(8,111,59,15)(17,87,97,45)(18,86,98,44)(19,85,99,43)(20,84,100,42)(21,83,101,41)(22,82,102,48)(23,81,103,47)(24,88,104,46)(25,79,89,37)(26,78,90,36)(27,77,91,35)(28,76,92,34)(29,75,93,33)(30,74,94,40)(31,73,95,39)(32,80,96,38)(49,118,71,121)(50,117,72,128)(51,116,65,127)(52,115,66,126)(53,114,67,125)(54,113,68,124)(55,120,69,123)(56,119,70,122), (1,74,123,99)(2,33,124,20)(3,76,125,101)(4,35,126,22)(5,78,127,103)(6,37,128,24)(7,80,121,97)(8,39,122,18)(9,25,72,88)(10,90,65,47)(11,27,66,82)(12,92,67,41)(13,29,68,84)(14,94,69,43)(15,31,70,86)(16,96,71,45)(17,58,38,118)(19,60,40,120)(21,62,34,114)(23,64,36,116)(26,51,81,106)(28,53,83,108)(30,55,85,110)(32,49,87,112)(42,109,93,54)(44,111,95,56)(46,105,89,50)(48,107,91,52)(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)(9,105)(10,106)(11,107)(12,108)(13,109)(14,110)(15,111)(16,112)(17,97)(18,98)(19,99)(20,100)(21,101)(22,102)(23,103)(24,104)(25,89)(26,90)(27,91)(28,92)(29,93)(30,94)(31,95)(32,96)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,73)(40,74)(41,83)(42,84)(43,85)(44,86)(45,87)(46,88)(47,81)(48,82)(49,71)(50,72)(51,65)(52,66)(53,67)(54,68)(55,69)(56,70)(113,124)(114,125)(115,126)(116,127)(117,128)(118,121)(119,122)(120,123), (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), (1,110,60,14)(2,109,61,13)(3,108,62,12)(4,107,63,11)(5,106,64,10)(6,105,57,9)(7,112,58,16)(8,111,59,15)(17,87,97,45)(18,86,98,44)(19,85,99,43)(20,84,100,42)(21,83,101,41)(22,82,102,48)(23,81,103,47)(24,88,104,46)(25,79,89,37)(26,78,90,36)(27,77,91,35)(28,76,92,34)(29,75,93,33)(30,74,94,40)(31,73,95,39)(32,80,96,38)(49,118,71,121)(50,117,72,128)(51,116,65,127)(52,115,66,126)(53,114,67,125)(54,113,68,124)(55,120,69,123)(56,119,70,122), (1,74,123,99)(2,33,124,20)(3,76,125,101)(4,35,126,22)(5,78,127,103)(6,37,128,24)(7,80,121,97)(8,39,122,18)(9,25,72,88)(10,90,65,47)(11,27,66,82)(12,92,67,41)(13,29,68,84)(14,94,69,43)(15,31,70,86)(16,96,71,45)(17,58,38,118)(19,60,40,120)(21,62,34,114)(23,64,36,116)(26,51,81,106)(28,53,83,108)(30,55,85,110)(32,49,87,112)(42,109,93,54)(44,111,95,56)(46,105,89,50)(48,107,91,52)(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),(9,105),(10,106),(11,107),(12,108),(13,109),(14,110),(15,111),(16,112),(17,97),(18,98),(19,99),(20,100),(21,101),(22,102),(23,103),(24,104),(25,89),(26,90),(27,91),(28,92),(29,93),(30,94),(31,95),(32,96),(33,75),(34,76),(35,77),(36,78),(37,79),(38,80),(39,73),(40,74),(41,83),(42,84),(43,85),(44,86),(45,87),(46,88),(47,81),(48,82),(49,71),(50,72),(51,65),(52,66),(53,67),(54,68),(55,69),(56,70),(113,124),(114,125),(115,126),(116,127),(117,128),(118,121),(119,122),(120,123)], [(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)], [(1,110,60,14),(2,109,61,13),(3,108,62,12),(4,107,63,11),(5,106,64,10),(6,105,57,9),(7,112,58,16),(8,111,59,15),(17,87,97,45),(18,86,98,44),(19,85,99,43),(20,84,100,42),(21,83,101,41),(22,82,102,48),(23,81,103,47),(24,88,104,46),(25,79,89,37),(26,78,90,36),(27,77,91,35),(28,76,92,34),(29,75,93,33),(30,74,94,40),(31,73,95,39),(32,80,96,38),(49,118,71,121),(50,117,72,128),(51,116,65,127),(52,115,66,126),(53,114,67,125),(54,113,68,124),(55,120,69,123),(56,119,70,122)], [(1,74,123,99),(2,33,124,20),(3,76,125,101),(4,35,126,22),(5,78,127,103),(6,37,128,24),(7,80,121,97),(8,39,122,18),(9,25,72,88),(10,90,65,47),(11,27,66,82),(12,92,67,41),(13,29,68,84),(14,94,69,43),(15,31,70,86),(16,96,71,45),(17,58,38,118),(19,60,40,120),(21,62,34,114),(23,64,36,116),(26,51,81,106),(28,53,83,108),(30,55,85,110),(32,49,87,112),(42,109,93,54),(44,111,95,56),(46,105,89,50),(48,107,91,52),(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 D8 C4○D4 C4○D8 C8.C22 kernel C2.D8⋊4C4 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.D84C4 in GL6(𝔽17)

 1 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 16 0 0 0 0 0 0 16
,
 2 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 2 0 0 0 0 0 0 11 16 0 0 0 0 1 6
,
 0 8 0 0 0 0 15 0 0 0 0 0 0 0 0 2 0 0 0 0 9 0 0 0 0 0 0 0 7 4 0 0 0 0 13 10
,
 13 0 0 0 0 0 0 13 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 1 0 0 0 0 1 0

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

C2.D84C4 in GAP, Magma, Sage, TeX

`C_2.D_8\rtimes_4C_4`
`% in TeX`

`G:=Group("C2.D8:4C4");`
`// GroupNames label`

`G:=SmallGroup(128,650);`
`// by ID`

`G=gap.SmallGroup(128,650);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,58,2804,718,172]);`
`// Polycyclic`

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

׿
×
𝔽