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## G = C42.201D4order 128 = 27

### 183rd non-split extension by C42 of D4 acting via D4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C42.201D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C4⋊C4 — C23.83C23 — C42.201D4
 Lower central C1 — C23 — C42.201D4
 Upper central C1 — C23 — C42.201D4
 Jennings C1 — C23 — C42.201D4

Generators and relations for C42.201D4
G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=a-1, dad-1=ab2, cbc-1=a2b-1, dbd-1=a2b, dcd-1=a2c-1 >

Subgroups: 324 in 190 conjugacy classes, 92 normal (10 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, C23, C42, C4⋊C4, C22×C4, C22×C4, C2.C42, C2×C42, C2×C4⋊C4, C42.C2, C428C4, C23.81C23, C23.83C23, C2×C42.C2, C42.201D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, 2- 1+4, C23.38C23, C22.31C24, C22.57C24, C22.58C24, C42.201D4

Character table of C42.201D4

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 4Q 4R size 1 1 1 1 1 1 1 1 4 4 4 4 4 4 8 8 8 8 8 8 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 1 1 -1 -1 1 -1 1 -1 1 1 -1 linear of order 2 ρ5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 -1 -1 1 1 linear of order 2 ρ6 1 1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ7 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 -1 1 -1 -1 1 -1 1 1 -1 -1 1 linear of order 2 ρ8 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 linear of order 2 ρ9 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ10 1 1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 linear of order 2 ρ11 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 1 -1 linear of order 2 ρ12 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 1 -1 1 -1 1 1 -1 1 -1 -1 1 linear of order 2 ρ13 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 -1 -1 linear of order 2 ρ14 1 1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 1 1 1 1 linear of order 2 ρ15 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 linear of order 2 ρ16 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 1 linear of order 2 ρ17 2 -2 2 -2 2 -2 2 -2 -2 2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 -2 2 -2 2 -2 2 -2 2 -2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 -2 2 -2 2 -2 2 -2 -2 2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 -2 2 -2 2 -2 2 -2 2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 4 4 -4 -4 -4 -4 4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from 2+ 1+4 ρ22 4 -4 -4 4 -4 4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from 2- 1+4, Schur index 2 ρ23 4 -4 4 4 -4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from 2- 1+4, Schur index 2 ρ24 4 4 4 -4 -4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from 2- 1+4, Schur index 2 ρ25 4 -4 -4 -4 4 4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from 2- 1+4, Schur index 2 ρ26 4 4 -4 4 4 -4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from 2- 1+4, Schur index 2

Smallest permutation representation of C42.201D4
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 76 103 13)(2 73 104 14)(3 74 101 15)(4 75 102 16)(5 36 70 95)(6 33 71 96)(7 34 72 93)(8 35 69 94)(9 107 40 48)(10 108 37 45)(11 105 38 46)(12 106 39 47)(17 56 80 115)(18 53 77 116)(19 54 78 113)(20 55 79 114)(21 111 84 52)(22 112 81 49)(23 109 82 50)(24 110 83 51)(25 121 88 62)(26 122 85 63)(27 123 86 64)(28 124 87 61)(29 58 92 117)(30 59 89 118)(31 60 90 119)(32 57 91 120)(41 126 97 68)(42 127 98 65)(43 128 99 66)(44 125 100 67)
(1 33 9 65)(2 36 10 68)(3 35 11 67)(4 34 12 66)(5 106 97 75)(6 105 98 74)(7 108 99 73)(8 107 100 76)(13 69 48 44)(14 72 45 43)(15 71 46 42)(16 70 47 41)(17 117 52 86)(18 120 49 85)(19 119 50 88)(20 118 51 87)(21 121 56 90)(22 124 53 89)(23 123 54 92)(24 122 55 91)(25 78 60 109)(26 77 57 112)(27 80 58 111)(28 79 59 110)(29 82 64 113)(30 81 61 116)(31 84 62 115)(32 83 63 114)(37 126 104 95)(38 125 101 94)(39 128 102 93)(40 127 103 96)
(1 119 103 60)(2 57 104 120)(3 117 101 58)(4 59 102 118)(5 53 70 116)(6 113 71 54)(7 55 72 114)(8 115 69 56)(9 88 40 25)(10 26 37 85)(11 86 38 27)(12 28 39 87)(13 92 76 29)(14 30 73 89)(15 90 74 31)(16 32 75 91)(17 96 80 33)(18 34 77 93)(19 94 78 35)(20 36 79 95)(21 100 84 44)(22 41 81 97)(23 98 82 42)(24 43 83 99)(45 61 108 124)(46 121 105 62)(47 63 106 122)(48 123 107 64)(49 66 112 128)(50 125 109 67)(51 68 110 126)(52 127 111 65)```

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

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

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

Matrix representation of C42.201D4 in GL12(𝔽5)

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

`G:=sub<GL(12,GF(5))| [0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,1,1,0,0,0,0,0,0,0,0,1,3,4,4,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,4,0,0,0,0,0,0,0,0,0,4,4,2,0,0,0,0,0,0,0,0,3,1,2,1,0,0,0,0,0,0,0,0,4,3,3,4],[1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,2,1,3,3,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,1,3,4,0,0,0,0,0,0,0,0,0,1,4,2,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0],[0,3,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,4,0,0,0,0,0,0,0,0,0,4,2,1,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,3,2,0,1,0,0,0,0,0,0,0,0,4,4,4,3,0,0,0,0,0,0,0,0,0,1,2,4,0,0,0,0,0,0,0,0,4,1,2,1],[0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,2,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,4,2,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,4,0,2,0,0,0,0,0,0,0,0,4,1,2,1,0,0,0,0,0,0,0,0,3,3,1,2,0,0,0,0,0,0,0,0,1,1,1,2] >;`

C42.201D4 in GAP, Magma, Sage, TeX

`C_4^2._{201}D_4`
`% in TeX`

`G:=Group("C4^2.201D4");`
`// GroupNames label`

`G:=SmallGroup(128,1554);`
`// by ID`

`G=gap.SmallGroup(128,1554);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,120,758,723,184,794,185,80]);`
`// Polycyclic`

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

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