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## G = C43.18C2order 128 = 27

### 18th 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 — C43.18C2
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C43 — C43.18C2
 Lower central C1 — C23 — C43.18C2
 Upper central C1 — C23 — C43.18C2
 Jennings C1 — C23 — C43.18C2

Generators and relations for C43.18C2
G = < a,b,c,d | a4=b4=c4=1, d2=b2c2, ab=ba, ac=ca, dad-1=ac2, bc=cb, dbd-1=b-1c2, dcd-1=a2c >

Subgroups: 308 in 200 conjugacy classes, 108 normal (8 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C42, C42, C4⋊C4, C22×C4, C22×C4, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C43, C428C4, C23.63C23, C23.65C23, C43.18C2
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C4○D4, C24, C42.C2, C22×Q8, C2×C4○D4, C2×C42.C2, C23.36C23, C23.37C23, C43.18C2

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

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

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

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

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 2 2 type + + + + + - image C1 C2 C2 C2 C2 Q8 C4○D4 kernel C43.18C2 C43 C42⋊8C4 C23.63C23 C23.65C23 C42 C2×C4 # reps 1 1 2 8 4 4 24

Matrix representation of C43.18C2 in GL6(𝔽5)

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

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

C43.18C2 in GAP, Magma, Sage, TeX

`C_4^3._{18}C_2`
`% in TeX`

`G:=Group("C4^3.18C2");`
`// GroupNames label`

`G:=SmallGroup(128,1596);`
`// by ID`

`G=gap.SmallGroup(128,1596);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,456,758,268,2019,80]);`
`// Polycyclic`

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

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