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## G = C42⋊4C4.C2order 128 = 27

### 2nd non-split extension by C42⋊4C4 of C2 acting faithfully

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C42⋊4C4.C2
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C42⋊4C4 — C42⋊4C4.C2
 Lower central C1 — C23 — C42⋊4C4.C2
 Upper central C1 — C22×C4 — C42⋊4C4.C2
 Jennings C1 — C2 — C2 — C22×C4 — C42⋊4C4.C2

Generators and relations for C424C4.C2
G = < a,b,c,d | a4=b4=c4=1, d2=bc2, ab=ba, cac-1=ab2, dad-1=a-1b2c2, bc=cb, bd=db, dcd-1=a2b2c >

Subgroups: 164 in 98 conjugacy classes, 52 normal (6 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×6], C22 [×7], C8 [×4], C2×C4 [×6], C2×C4 [×18], C23, C42 [×6], C2×C8 [×12], C22×C4, C22×C4 [×6], C2.C42 [×4], C2×C42 [×3], C22×C8 [×4], C22.7C42 [×6], C424C4, C424C4.C2
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C22×C4, C4○D4 [×6], C42⋊C2 [×3], C422C2 [×4], C8○D4 [×4], C425C4, C42.7C22 [×6], C424C4.C2

Smallest permutation representation of C424C4.C2
Regular action on 128 points
Generators in S128
```(1 71 99 73)(2 9 100 128)(3 65 101 75)(4 11 102 122)(5 67 103 77)(6 13 104 124)(7 69 97 79)(8 15 98 126)(10 17 121 41)(12 19 123 43)(14 21 125 45)(16 23 127 47)(18 76 42 66)(20 78 44 68)(22 80 46 70)(24 74 48 72)(25 57 49 33)(26 94 50 118)(27 59 51 35)(28 96 52 120)(29 61 53 37)(30 90 54 114)(31 63 55 39)(32 92 56 116)(34 110 58 86)(36 112 60 88)(38 106 62 82)(40 108 64 84)(81 113 105 89)(83 115 107 91)(85 117 109 93)(87 119 111 95)
(1 21 5 17)(2 22 6 18)(3 23 7 19)(4 24 8 20)(9 80 13 76)(10 73 14 77)(11 74 15 78)(12 75 16 79)(25 83 29 87)(26 84 30 88)(27 85 31 81)(28 86 32 82)(33 91 37 95)(34 92 38 96)(35 93 39 89)(36 94 40 90)(41 99 45 103)(42 100 46 104)(43 101 47 97)(44 102 48 98)(49 107 53 111)(50 108 54 112)(51 109 55 105)(52 110 56 106)(57 115 61 119)(58 116 62 120)(59 117 63 113)(60 118 64 114)(65 127 69 123)(66 128 70 124)(67 121 71 125)(68 122 72 126)
(1 27 19 83)(2 56 20 112)(3 29 21 85)(4 50 22 106)(5 31 23 87)(6 52 24 108)(7 25 17 81)(8 54 18 110)(9 120 78 64)(10 93 79 37)(11 114 80 58)(12 95 73 39)(13 116 74 60)(14 89 75 33)(15 118 76 62)(16 91 77 35)(26 46 82 102)(28 48 84 104)(30 42 86 98)(32 44 88 100)(34 122 90 70)(36 124 92 72)(38 126 94 66)(40 128 96 68)(41 105 97 49)(43 107 99 51)(45 109 101 53)(47 111 103 55)(57 125 113 65)(59 127 115 67)(61 121 117 69)(63 123 119 71)
(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)```

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4T 8A ··· 8P order 1 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 1 ··· 1 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 2 2 type + + + image C1 C2 C2 C4 C4○D4 C8○D4 kernel C42⋊4C4.C2 C22.7C42 C42⋊4C4 C2.C42 C2×C4 C22 # reps 1 6 1 8 12 16

Matrix representation of C424C4.C2 in GL6(𝔽17)

 1 2 0 0 0 0 0 16 0 0 0 0 0 0 2 9 0 0 0 0 11 15 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 4 0 0 0 0 0 0 4 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 16 0 0 0 0 0 1 1 0 0 0 0 0 0 16 15 0 0 0 0 1 1 0 0 0 0 0 0 16 16 0 0 0 0 2 1
,
 9 1 0 0 0 0 8 8 0 0 0 0 0 0 13 16 0 0 0 0 12 4 0 0 0 0 0 0 0 5 0 0 0 0 10 0

`G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,2,16,0,0,0,0,0,0,2,11,0,0,0,0,9,15,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,1,0,0,0,0,0,1,0,0,0,0,0,0,16,1,0,0,0,0,15,1,0,0,0,0,0,0,16,2,0,0,0,0,16,1],[9,8,0,0,0,0,1,8,0,0,0,0,0,0,13,12,0,0,0,0,16,4,0,0,0,0,0,0,0,10,0,0,0,0,5,0] >;`

C424C4.C2 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_4C_4.C_2`
`% in TeX`

`G:=Group("C4^2:4C4.C2");`
`// GroupNames label`

`G:=SmallGroup(128,572);`
`// by ID`

`G=gap.SmallGroup(128,572);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,176,422,723,58,124]);`
`// Polycyclic`

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

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