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## G = C2.C43order 128 = 27

### 3rd central stem extension by C2 of C43

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C2.C43
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C42⋊4C4 — C2.C43
 Lower central C1 — C2 — C2.C43
 Upper central C1 — C22×C8 — C2.C43
 Jennings C1 — C2 — C2 — C22×C4 — C2.C43

Generators and relations for C2.C43
G = < a,b,c,d | a2=c4=d4=1, b4=a, ab=ba, dcd-1=ac=ca, ad=da, bc=cb, bd=db >

Subgroups: 188 in 164 conjugacy classes, 140 normal (8 characteristic)
C1, C2, C2, C4, C4, C4, C22, C8, C2×C4, C2×C4, C23, C42, C2×C8, C22×C4, C22×C4, C2.C42, C4×C8, C8⋊C4, C2×C42, C22×C8, C22×C8, C424C4, C2×C4×C8, C2×C8⋊C4, C2.C43
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, C2×C42, C8○D4, C43, C82M4(2), C2.C43

Smallest permutation representation of C2.C43
Regular action on 128 points
Generators in S128
```(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)(113 117)(114 118)(115 119)(116 120)(121 125)(122 126)(123 127)(124 128)
(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)
(1 125 109 37)(2 126 110 38)(3 127 111 39)(4 128 112 40)(5 121 105 33)(6 122 106 34)(7 123 107 35)(8 124 108 36)(9 119 47 31)(10 120 48 32)(11 113 41 25)(12 114 42 26)(13 115 43 27)(14 116 44 28)(15 117 45 29)(16 118 46 30)(17 93 79 61)(18 94 80 62)(19 95 73 63)(20 96 74 64)(21 89 75 57)(22 90 76 58)(23 91 77 59)(24 92 78 60)(49 97 81 67)(50 98 82 68)(51 99 83 69)(52 100 84 70)(53 101 85 71)(54 102 86 72)(55 103 87 65)(56 104 88 66)
(1 95 87 29)(2 96 88 30)(3 89 81 31)(4 90 82 32)(5 91 83 25)(6 92 84 26)(7 93 85 27)(8 94 86 28)(9 123 75 71)(10 124 76 72)(11 125 77 65)(12 126 78 66)(13 127 79 67)(14 128 80 68)(15 121 73 69)(16 122 74 70)(17 97 43 39)(18 98 44 40)(19 99 45 33)(20 100 46 34)(21 101 47 35)(22 102 48 36)(23 103 41 37)(24 104 42 38)(49 119 111 57)(50 120 112 58)(51 113 105 59)(52 114 106 60)(53 115 107 61)(54 116 108 62)(55 117 109 63)(56 118 110 64)```

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

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

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

80 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4AF 8A ··· 8P 8Q ··· 8AN order 1 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8 8 ··· 8 size 1 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

80 irreducible representations

 dim 1 1 1 1 1 1 1 2 type + + + + image C1 C2 C2 C2 C4 C4 C4 C8○D4 kernel C2.C43 C42⋊4C4 C2×C4×C8 C2×C8⋊C4 C2.C42 C4×C8 C8⋊C4 C22 # reps 1 1 3 3 8 24 24 16

Matrix representation of C2.C43 in GL4(𝔽17) generated by

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

C2.C43 in GAP, Magma, Sage, TeX

`C_2.C_4^3`
`% in TeX`

`G:=Group("C2.C4^3");`
`// GroupNames label`

`G:=SmallGroup(128,458);`
`// by ID`

`G=gap.SmallGroup(128,458);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,120,723,184,248]);`
`// Polycyclic`

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

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