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## G = C42.327D4order 128 = 27

### 23rd non-split extension by C42 of D4 acting via D4/C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C42.327D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C2×C4×C8 — C42.327D4
 Lower central C1 — C22 — C42.327D4
 Upper central C1 — C22×C4 — C42.327D4
 Jennings C1 — C2 — C2 — C22×C4 — C42.327D4

Generators and relations for C42.327D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2b2, ab=ba, ac=ca, dad-1=a-1, bc=cb, bd=db, dcd-1=a2b-1c3 >

Subgroups: 220 in 146 conjugacy classes, 80 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×6], C4 [×8], C22 [×3], C22 [×4], C8 [×6], C2×C4 [×2], C2×C4 [×16], C2×C4 [×12], Q8 [×8], C23, C42 [×4], C42 [×4], C4⋊C4 [×6], C2×C8 [×4], C2×C8 [×10], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×4], C2×Q8 [×4], C4×C8 [×2], C4⋊C8 [×2], C2×C42, C2×C42 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C4×Q8 [×4], C22×C8 [×4], C22×Q8, C22.7C42 [×4], C2×C4×C8, C2×C4⋊C8, C2×C4×Q8, C42.327D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×4], Q8 [×4], C23, C22⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], C22⋊C8 [×4], C2×C22⋊C4, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4, C4⋊Q8, C22×C8, C2×M4(2), C8○D4 [×2], C23.67C23, C2×C22⋊C8, (C22×C8)⋊C2, C8×Q8 [×2], C84Q8 [×2], C42.327D4

Smallest permutation representation of C42.327D4
Regular action on 128 points
Generators in S128
```(1 61 53 71)(2 62 54 72)(3 63 55 65)(4 64 56 66)(5 57 49 67)(6 58 50 68)(7 59 51 69)(8 60 52 70)(9 113 92 121)(10 114 93 122)(11 115 94 123)(12 116 95 124)(13 117 96 125)(14 118 89 126)(15 119 90 127)(16 120 91 128)(17 28 104 78)(18 29 97 79)(19 30 98 80)(20 31 99 73)(21 32 100 74)(22 25 101 75)(23 26 102 76)(24 27 103 77)(33 47 85 108)(34 48 86 109)(35 41 87 110)(36 42 88 111)(37 43 81 112)(38 44 82 105)(39 45 83 106)(40 46 84 107)
(1 31 5 27)(2 32 6 28)(3 25 7 29)(4 26 8 30)(9 112 13 108)(10 105 14 109)(11 106 15 110)(12 107 16 111)(17 72 21 68)(18 65 22 69)(19 66 23 70)(20 67 24 71)(33 113 37 117)(34 114 38 118)(35 115 39 119)(36 116 40 120)(41 94 45 90)(42 95 46 91)(43 96 47 92)(44 89 48 93)(49 77 53 73)(50 78 54 74)(51 79 55 75)(52 80 56 76)(57 103 61 99)(58 104 62 100)(59 97 63 101)(60 98 64 102)(81 125 85 121)(82 126 86 122)(83 127 87 123)(84 128 88 124)
(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)
(1 114 49 126)(2 81 50 33)(3 116 51 128)(4 83 52 35)(5 118 53 122)(6 85 54 37)(7 120 55 124)(8 87 56 39)(9 100 96 17)(10 67 89 61)(11 102 90 19)(12 69 91 63)(13 104 92 21)(14 71 93 57)(15 98 94 23)(16 65 95 59)(18 42 101 107)(20 44 103 109)(22 46 97 111)(24 48 99 105)(25 40 79 88)(26 127 80 115)(27 34 73 82)(28 121 74 117)(29 36 75 84)(30 123 76 119)(31 38 77 86)(32 125 78 113)(41 66 106 60)(43 68 108 62)(45 70 110 64)(47 72 112 58)```

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + - image C1 C2 C2 C2 C2 C4 C4 C8 D4 Q8 M4(2) C4○D4 C8○D4 kernel C42.327D4 C22.7C42 C2×C4×C8 C2×C4⋊C8 C2×C4×Q8 C2×C4⋊C4 C22×Q8 C2×Q8 C42 C2×C8 C2×C4 C2×C4 C22 # reps 1 4 1 1 1 6 2 16 4 4 4 4 8

Matrix representation of C42.327D4 in GL5(𝔽17)

 16 0 0 0 0 0 0 1 0 0 0 16 0 0 0 0 0 0 0 1 0 0 0 16 0
,
 13 0 0 0 0 0 13 0 0 0 0 0 13 0 0 0 0 0 4 0 0 0 0 0 4
,
 9 0 0 0 0 0 15 0 0 0 0 0 15 0 0 0 0 0 0 9 0 0 0 8 0
,
 4 0 0 0 0 0 3 14 0 0 0 14 14 0 0 0 0 0 1 0 0 0 0 0 16

`G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,1,0],[13,0,0,0,0,0,13,0,0,0,0,0,13,0,0,0,0,0,4,0,0,0,0,0,4],[9,0,0,0,0,0,15,0,0,0,0,0,15,0,0,0,0,0,0,8,0,0,0,9,0],[4,0,0,0,0,0,3,14,0,0,0,14,14,0,0,0,0,0,1,0,0,0,0,0,16] >;`

C42.327D4 in GAP, Magma, Sage, TeX

`C_4^2._{327}D_4`
`% in TeX`

`G:=Group("C4^2.327D4");`
`// GroupNames label`

`G:=SmallGroup(128,716);`
`// by ID`

`G=gap.SmallGroup(128,716);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,400,422,100,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=a^2*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;`
`// generators/relations`

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