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## G = C42.437D4order 128 = 27

### 70th 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 — C42.437D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C4×C8 — C42.437D4
 Lower central C1 — C2 — C2×C4 — C42.437D4
 Upper central C1 — C23 — C2×C42 — C42.437D4
 Jennings C1 — C2 — C2 — C22×C4 — C42.437D4

Generators and relations for C42.437D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=bc-1 >

Subgroups: 228 in 126 conjugacy classes, 64 normal (14 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×10], C22 [×3], C22 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×18], C23, C42 [×4], C4⋊C4 [×4], C4⋊C4 [×14], C2×C8 [×4], C2×C8 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2.C42 [×2], C4×C8 [×2], C2×C42, C2×C4⋊C4 [×4], C2×C4⋊C4 [×2], C42.C2 [×4], C42.C2 [×2], C22×C8 [×2], C22.4Q16 [×4], C428C4, C2×C4×C8, C2×C42.C2, C42.437D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], C2×C22⋊C4, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4, C4⋊Q8, C4○D8 [×4], C23.67C23, C23.24D4 [×2], C42.78C22 [×2], C8.5Q8 [×2], C42.437D4

Smallest permutation representation of C42.437D4
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 52 12 39)(2 49 9 40)(3 50 10 37)(4 51 11 38)(5 110 23 119)(6 111 24 120)(7 112 21 117)(8 109 22 118)(13 29 25 34)(14 30 26 35)(15 31 27 36)(16 32 28 33)(17 122 126 105)(18 123 127 106)(19 124 128 107)(20 121 125 108)(41 68 54 81)(42 65 55 82)(43 66 56 83)(44 67 53 84)(45 61 57 70)(46 62 58 71)(47 63 59 72)(48 64 60 69)(73 103 90 94)(74 104 91 95)(75 101 92 96)(76 102 89 93)(77 100 86 113)(78 97 87 114)(79 98 88 115)(80 99 85 116)
(1 95 34 87)(2 103 35 77)(3 93 36 85)(4 101 33 79)(5 67 106 57)(6 83 107 48)(7 65 108 59)(8 81 105 46)(9 94 30 86)(10 102 31 80)(11 96 32 88)(12 104 29 78)(13 97 52 91)(14 113 49 73)(15 99 50 89)(16 115 51 75)(17 71 109 54)(18 61 110 44)(19 69 111 56)(20 63 112 42)(21 82 121 47)(22 68 122 58)(23 84 123 45)(24 66 124 60)(25 114 39 74)(26 100 40 90)(27 116 37 76)(28 98 38 92)(41 126 62 118)(43 128 64 120)(53 127 70 119)(55 125 72 117)
(1 117 3 119)(2 111 4 109)(5 39 7 37)(6 51 8 49)(9 120 11 118)(10 110 12 112)(13 121 15 123)(14 107 16 105)(17 35 19 33)(18 29 20 31)(21 50 23 52)(22 40 24 38)(25 108 27 106)(26 124 28 122)(30 128 32 126)(34 125 36 127)(41 100 43 98)(42 116 44 114)(45 104 47 102)(46 94 48 96)(53 97 55 99)(54 113 56 115)(57 95 59 93)(58 103 60 101)(61 74 63 76)(62 90 64 92)(65 85 67 87)(66 79 68 77)(69 75 71 73)(70 91 72 89)(78 82 80 84)(81 86 83 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), (1,52,12,39)(2,49,9,40)(3,50,10,37)(4,51,11,38)(5,110,23,119)(6,111,24,120)(7,112,21,117)(8,109,22,118)(13,29,25,34)(14,30,26,35)(15,31,27,36)(16,32,28,33)(17,122,126,105)(18,123,127,106)(19,124,128,107)(20,121,125,108)(41,68,54,81)(42,65,55,82)(43,66,56,83)(44,67,53,84)(45,61,57,70)(46,62,58,71)(47,63,59,72)(48,64,60,69)(73,103,90,94)(74,104,91,95)(75,101,92,96)(76,102,89,93)(77,100,86,113)(78,97,87,114)(79,98,88,115)(80,99,85,116), (1,95,34,87)(2,103,35,77)(3,93,36,85)(4,101,33,79)(5,67,106,57)(6,83,107,48)(7,65,108,59)(8,81,105,46)(9,94,30,86)(10,102,31,80)(11,96,32,88)(12,104,29,78)(13,97,52,91)(14,113,49,73)(15,99,50,89)(16,115,51,75)(17,71,109,54)(18,61,110,44)(19,69,111,56)(20,63,112,42)(21,82,121,47)(22,68,122,58)(23,84,123,45)(24,66,124,60)(25,114,39,74)(26,100,40,90)(27,116,37,76)(28,98,38,92)(41,126,62,118)(43,128,64,120)(53,127,70,119)(55,125,72,117), (1,117,3,119)(2,111,4,109)(5,39,7,37)(6,51,8,49)(9,120,11,118)(10,110,12,112)(13,121,15,123)(14,107,16,105)(17,35,19,33)(18,29,20,31)(21,50,23,52)(22,40,24,38)(25,108,27,106)(26,124,28,122)(30,128,32,126)(34,125,36,127)(41,100,43,98)(42,116,44,114)(45,104,47,102)(46,94,48,96)(53,97,55,99)(54,113,56,115)(57,95,59,93)(58,103,60,101)(61,74,63,76)(62,90,64,92)(65,85,67,87)(66,79,68,77)(69,75,71,73)(70,91,72,89)(78,82,80,84)(81,86,83,88)>;`

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

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

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 type + + + + + + - + image C1 C2 C2 C2 C2 C4 D4 Q8 D4 C4○D4 C4○D8 kernel C42.437D4 C22.4Q16 C42⋊8C4 C2×C4×C8 C2×C42.C2 C42.C2 C42 C2×C8 C22×C4 C2×C4 C22 # reps 1 4 1 1 1 8 2 4 2 4 16

Matrix representation of C42.437D4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 16
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 4 13 0 0 0 0 0 0 13 0 0 0 0 0 0 4
,
 1 11 0 0 0 0 6 16 0 0 0 0 0 0 1 15 0 0 0 0 0 16 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 6 16 0 0 0 0 1 11 0 0 0 0 0 0 15 4 0 0 0 0 3 2 0 0 0 0 0 0 0 15 0 0 0 0 8 0

`G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,4,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,4],[1,6,0,0,0,0,11,16,0,0,0,0,0,0,1,0,0,0,0,0,15,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[6,1,0,0,0,0,16,11,0,0,0,0,0,0,15,3,0,0,0,0,4,2,0,0,0,0,0,0,0,8,0,0,0,0,15,0] >;`

C42.437D4 in GAP, Magma, Sage, TeX

`C_4^2._{437}D_4`
`% in TeX`

`G:=Group("C4^2.437D4");`
`// GroupNames label`

`G:=SmallGroup(128,723);`
`// by ID`

`G=gap.SmallGroup(128,723);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,400,422,436,2019,248]);`
`// Polycyclic`

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

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