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## G = C42.431D4order 128 = 27

### 64th non-split extension by C42 of D4 acting via D4/C22=C2

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.431D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C4×C8 — C42.431D4
 Lower central C1 — C2 — C2×C4 — C42.431D4
 Upper central C1 — C23 — C2×C42 — C42.431D4
 Jennings C1 — C2 — C2 — C22×C4 — C42.431D4

Generators and relations for C42.431D4
G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=bc-1 >

Subgroups: 308 in 166 conjugacy classes, 80 normal (16 characteristic)
C1, C2 [×3], C2 [×4], C4 [×12], C4 [×6], C22 [×3], C22 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×16], C2×C4 [×14], Q8 [×12], C23, C42 [×4], C4⋊C4 [×16], C2×C8 [×4], C2×C8 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Q8 [×4], C2×Q8 [×10], C4×C8 [×2], Q8⋊C4 [×8], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×4], C4⋊Q8 [×4], C4⋊Q8 [×2], C22×C8 [×2], C22×Q8 [×2], C429C4, C2×C4×C8, C2×Q8⋊C4 [×4], C2×C4⋊Q8, C42.431D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], SD16 [×4], Q16 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], Q8⋊C4 [×8], C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C41D4, C2×SD16 [×2], C2×Q16 [×2], C24.3C22, C2×Q8⋊C4 [×2], C4.SD16 [×2], C85D4, C4⋊Q16, C42.431D4

Smallest permutation representation of C42.431D4
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 39 29 13)(2 40 30 14)(3 37 31 15)(4 38 32 16)(5 120 108 19)(6 117 105 20)(7 118 106 17)(8 119 107 18)(9 49 35 25)(10 50 36 26)(11 51 33 27)(12 52 34 28)(21 110 122 126)(22 111 123 127)(23 112 124 128)(24 109 121 125)(41 81 71 57)(42 82 72 58)(43 83 69 59)(44 84 70 60)(45 53 67 61)(46 54 68 62)(47 55 65 63)(48 56 66 64)(73 103 113 85)(74 104 114 86)(75 101 115 87)(76 102 116 88)(77 89 93 99)(78 90 94 100)(79 91 95 97)(80 92 96 98)
(1 87 9 79)(2 86 10 78)(3 85 11 77)(4 88 12 80)(5 84 23 66)(6 83 24 65)(7 82 21 68)(8 81 22 67)(13 75 25 91)(14 74 26 90)(15 73 27 89)(16 76 28 92)(17 72 126 62)(18 71 127 61)(19 70 128 64)(20 69 125 63)(29 101 35 95)(30 104 36 94)(31 103 33 93)(32 102 34 96)(37 113 51 99)(38 116 52 98)(39 115 49 97)(40 114 50 100)(41 111 53 119)(42 110 54 118)(43 109 55 117)(44 112 56 120)(45 107 57 123)(46 106 58 122)(47 105 59 121)(48 108 60 124)
(1 127 29 111)(2 126 30 110)(3 125 31 109)(4 128 32 112)(5 52 108 28)(6 51 105 27)(7 50 106 26)(8 49 107 25)(9 18 35 119)(10 17 36 118)(11 20 33 117)(12 19 34 120)(13 22 39 123)(14 21 40 122)(15 24 37 121)(16 23 38 124)(41 115 71 75)(42 114 72 74)(43 113 69 73)(44 116 70 76)(45 79 67 95)(46 78 68 94)(47 77 65 93)(48 80 66 96)(53 97 61 91)(54 100 62 90)(55 99 63 89)(56 98 64 92)(57 87 81 101)(58 86 82 104)(59 85 83 103)(60 88 84 102)```

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4L 4M ··· 4T 8A ··· 8P order 1 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 ··· 2 8 ··· 8 2 ··· 2

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + - image C1 C2 C2 C2 C2 C4 D4 D4 D4 SD16 Q16 C4○D4 kernel C42.431D4 C42⋊9C4 C2×C4×C8 C2×Q8⋊C4 C2×C4⋊Q8 C4⋊Q8 C42 C2×C8 C22×C4 C2×C4 C2×C4 C2×C4 # reps 1 1 1 4 1 8 2 4 2 8 8 4

Matrix representation of C42.431D4 in GL5(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 16
,
 16 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 4 0 0 0 0 0 13
,
 4 0 0 0 0 0 13 6 0 0 0 6 4 0 0 0 0 0 0 15 0 0 0 8 0
,
 16 0 0 0 0 0 4 11 0 0 0 11 13 0 0 0 0 0 0 4 0 0 0 4 0

`G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,13],[4,0,0,0,0,0,13,6,0,0,0,6,4,0,0,0,0,0,0,8,0,0,0,15,0],[16,0,0,0,0,0,4,11,0,0,0,11,13,0,0,0,0,0,0,4,0,0,0,4,0] >;`

C42.431D4 in GAP, Magma, Sage, TeX

`C_4^2._{431}D_4`
`% in TeX`

`G:=Group("C4^2.431D4");`
`// GroupNames label`

`G:=SmallGroup(128,688);`
`// by ID`

`G=gap.SmallGroup(128,688);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,100,2019,248]);`
`// 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=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b*c^-1>;`
`// generators/relations`

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