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## G = C4⋊C4.95D4order 128 = 27

### 50th non-split extension by C4⋊C4 of D4 acting via D4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — C4⋊C4.95D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — C2×Q8⋊C4 — C4⋊C4.95D4
 Lower central C1 — C2 — C22×C4 — C4⋊C4.95D4
 Upper central C1 — C23 — C2×C42 — C4⋊C4.95D4
 Jennings C1 — C2 — C2 — C22×C4 — C4⋊C4.95D4

Generators and relations for C4⋊C4.95D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=b-1, dbd-1=ab, dcd-1=c-1 >

Subgroups: 296 in 149 conjugacy classes, 54 normal (28 characteristic)
C1, C2 [×7], C4 [×4], C4 [×11], C22 [×7], C8 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×19], Q8 [×12], C23, C42 [×2], C4⋊C4 [×2], C4⋊C4 [×13], C2×C8 [×6], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×4], C2×Q8 [×10], C2.C42, Q8⋊C4 [×8], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×3], C4⋊Q8 [×4], C22×C8 [×2], C22×Q8 [×2], C22.7C42, C23.65C23, C2×Q8⋊C4 [×4], C2×C4⋊Q8, C4⋊C4.95D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×8], C23, SD16 [×2], Q16 [×2], C2×D4 [×4], C4○D4 [×3], C22≀C2, C4⋊D4 [×3], C22.D4 [×2], C4.4D4, C2×SD16, C2×Q16, C8.C22 [×2], C23.10D4, Q8⋊D4, C22⋊Q16, D4.D4, C42Q16, C4.SD16, C42.30C22, C4⋊C4.95D4

Smallest permutation representation of C4⋊C4.95D4
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)
(1 115 11 15)(2 114 12 14)(3 113 9 13)(4 116 10 16)(5 124 21 125)(6 123 22 128)(7 122 23 127)(8 121 24 126)(17 99 39 118)(18 98 40 117)(19 97 37 120)(20 100 38 119)(25 106 30 111)(26 105 31 110)(27 108 32 109)(28 107 29 112)(33 102 54 84)(34 101 55 83)(35 104 56 82)(36 103 53 81)(41 96 46 91)(42 95 47 90)(43 94 48 89)(44 93 45 92)(49 79 72 74)(50 78 69 73)(51 77 70 76)(52 80 71 75)(57 86 62 65)(58 85 63 68)(59 88 64 67)(60 87 61 66)
(1 97 8 106)(2 98 5 107)(3 99 6 108)(4 100 7 105)(9 118 22 109)(10 119 23 110)(11 120 24 111)(12 117 21 112)(13 39 128 32)(14 40 125 29)(15 37 126 30)(16 38 127 31)(17 123 27 113)(18 124 28 114)(19 121 25 115)(20 122 26 116)(33 73 43 67)(34 74 44 68)(35 75 41 65)(36 76 42 66)(45 85 55 79)(46 86 56 80)(47 87 53 77)(48 88 54 78)(49 92 58 83)(50 89 59 84)(51 90 60 81)(52 91 57 82)(61 103 70 95)(62 104 71 96)(63 101 72 93)(64 102 69 94)
(1 74 3 76)(2 73 4 75)(5 67 7 65)(6 66 8 68)(9 77 11 79)(10 80 12 78)(13 69 15 71)(14 72 16 70)(17 84 19 82)(18 83 20 81)(21 88 23 86)(22 87 24 85)(25 91 27 89)(26 90 28 92)(29 93 31 95)(30 96 32 94)(33 100 35 98)(34 99 36 97)(37 104 39 102)(38 103 40 101)(41 107 43 105)(42 106 44 108)(45 109 47 111)(46 112 48 110)(49 116 51 114)(50 115 52 113)(53 120 55 118)(54 119 56 117)(57 123 59 121)(58 122 60 124)(61 125 63 127)(62 128 64 126)```

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

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

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

32 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 8A ··· 8H order 1 2 ··· 2 4 4 4 4 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 2 2 2 4 4 4 4 8 ··· 8 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + - - image C1 C2 C2 C2 C2 D4 D4 D4 SD16 Q16 C4○D4 C8.C22 kernel C4⋊C4.95D4 C22.7C42 C23.65C23 C2×Q8⋊C4 C2×C4⋊Q8 C4⋊C4 C22×C4 C2×Q8 C2×C4 C2×C4 C2×C4 C22 # reps 1 1 1 4 1 2 2 4 4 4 6 2

Matrix representation of C4⋊C4.95D4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 15 0 0 0 0 1 16
,
 10 13 0 0 0 0 4 7 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 13 4
,
 4 7 0 0 0 0 10 13 0 0 0 0 0 0 0 4 0 0 0 0 4 0 0 0 0 0 0 0 16 2 0 0 0 0 16 1
,
 13 10 0 0 0 0 7 4 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 7 10 0 0 0 0 12 10

`G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,15,16],[10,4,0,0,0,0,13,7,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,13,13,0,0,0,0,0,4],[4,10,0,0,0,0,7,13,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,16,16,0,0,0,0,2,1],[13,7,0,0,0,0,10,4,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,7,12,0,0,0,0,10,10] >;`

C4⋊C4.95D4 in GAP, Magma, Sage, TeX

`C_4\rtimes C_4._{95}D_4`
`% in TeX`

`G:=Group("C4:C4.95D4");`
`// GroupNames label`

`G:=SmallGroup(128,775);`
`// by ID`

`G=gap.SmallGroup(128,775);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,448,141,456,422,387,58,2804,1411,718,172,4037,2028,1027,124]);`
`// Polycyclic`

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

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