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## G = C43.15C2order 128 = 27

### 15th non-split extension by C43 of C2 acting faithfully

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C43.15C2
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C43 — C43.15C2
 Lower central C1 — C23 — C43.15C2
 Upper central C1 — C23 — C43.15C2
 Jennings C1 — C23 — C43.15C2

Generators and relations for C43.15C2
G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, ac=ca, dad-1=ac2, bc=cb, dbd-1=b-1, dcd-1=a2c >

Subgroups: 356 in 236 conjugacy classes, 132 normal (12 characteristic)
C1, C2, C2 [×6], C4 [×12], C4 [×16], C22 [×3], C22 [×4], C2×C4 [×26], C2×C4 [×32], C23, C42 [×12], C42 [×8], C4⋊C4 [×40], C22×C4, C22×C4 [×14], C2.C42 [×8], C2×C42 [×3], C2×C42 [×4], C2×C4⋊C4 [×20], C42.C2 [×8], C43, C428C4 [×2], C429C4 [×2], C23.65C23 [×8], C2×C42.C2 [×2], C43.15C2
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], C2×D4 [×6], C2×Q8 [×12], C4○D4 [×8], C24, C42.C2 [×8], C4⋊Q8 [×4], C22×D4, C22×Q8 [×2], C2×C4○D4 [×4], C2×C42.C2 [×2], C2×C4⋊Q8, C22.26C24 [×2], C23.37C23 [×2], C43.15C2

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

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4AB 4AC ··· 4AJ order 1 2 ··· 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 2 ··· 2 8 ··· 8

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 type + + + + + + + - image C1 C2 C2 C2 C2 C2 D4 Q8 C4○D4 kernel C43.15C2 C43 C42⋊8C4 C42⋊9C4 C23.65C23 C2×C42.C2 C42 C42 C2×C4 # reps 1 1 2 2 8 2 4 8 16

Matrix representation of C43.15C2 in GL6(𝔽5)

 3 0 0 0 0 0 2 2 0 0 0 0 0 0 2 0 0 0 0 0 0 3 0 0 0 0 0 0 0 2 0 0 0 0 3 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 3 0 0 0 0 0 0 0 1 0 0 0 0 4 0
,
 3 0 0 0 0 0 2 2 0 0 0 0 0 0 3 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2
,
 2 4 0 0 0 0 0 3 0 0 0 0 0 0 0 3 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 3 0

`G:=sub<GL(6,GF(5))| [3,2,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,2,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[3,2,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[2,0,0,0,0,0,4,3,0,0,0,0,0,0,0,3,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,3,0] >;`

C43.15C2 in GAP, Magma, Sage, TeX

`C_4^3._{15}C_2`
`% in TeX`

`G:=Group("C4^3.15C2");`
`// GroupNames label`

`G:=SmallGroup(128,1591);`
`// by ID`

`G=gap.SmallGroup(128,1591);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,456,758,184,2019,80]);`
`// Polycyclic`

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

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