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## G = C42.176D4order 128 = 27

### 158th non-split extension by C42 of D4 acting via D4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C42.176D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C4⋊C4 — C42.176D4
 Lower central C1 — C23 — C42.176D4
 Upper central C1 — C23 — C42.176D4
 Jennings C1 — C23 — C42.176D4

Generators and relations for C42.176D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, cac-1=a-1b2, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 420 in 250 conjugacy classes, 132 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×20], C22 [×3], C22 [×4], C2×C4 [×26], C2×C4 [×32], Q8 [×16], C23, C42 [×4], C42 [×8], C4⋊C4 [×8], C4⋊C4 [×22], C22×C4 [×3], C22×C4 [×12], C2×Q8 [×4], C2×Q8 [×12], C2.C42 [×10], C2×C42, C2×C42 [×4], C2×C4⋊C4, C2×C4⋊C4 [×16], C4×Q8 [×4], C4⋊Q8 [×4], C22×Q8, C22×Q8 [×2], C4×C4⋊C4, C429C4 [×2], C23.65C23 [×4], C23.67C23 [×2], C23.78C23 [×4], C2×C4×Q8, C2×C4⋊Q8, C42.176D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×12], C23 [×15], C2×D4 [×6], C2×Q8 [×18], C4○D4 [×2], C24, C22⋊Q8 [×4], C4⋊Q8 [×4], C22×D4, C22×Q8 [×3], C2×C4○D4, 2+ 1+4 [×2], C2×C22⋊Q8, C2×C4⋊Q8, C22.29C24, D43Q8 [×2], Q82 [×2], C42.176D4

Smallest permutation representation of C42.176D4
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 15 44 50)(2 16 41 51)(3 13 42 52)(4 14 43 49)(5 36 37 127)(6 33 38 128)(7 34 39 125)(8 35 40 126)(9 22 17 46)(10 23 18 47)(11 24 19 48)(12 21 20 45)(25 57 54 30)(26 58 55 31)(27 59 56 32)(28 60 53 29)(61 65 72 100)(62 66 69 97)(63 67 70 98)(64 68 71 99)(73 80 111 114)(74 77 112 115)(75 78 109 116)(76 79 110 113)(81 101 108 88)(82 102 105 85)(83 103 106 86)(84 104 107 87)(89 117 124 96)(90 118 121 93)(91 119 122 94)(92 120 123 95)
(1 70 26 10)(2 62 27 17)(3 72 28 12)(4 64 25 19)(5 106 114 122)(6 82 115 90)(7 108 116 124)(8 84 113 92)(9 41 69 56)(11 43 71 54)(13 100 60 21)(14 68 57 48)(15 98 58 23)(16 66 59 46)(18 44 63 55)(20 42 61 53)(22 51 97 32)(24 49 99 30)(29 45 52 65)(31 47 50 67)(33 102 74 118)(34 88 75 96)(35 104 76 120)(36 86 73 94)(37 83 80 91)(38 105 77 121)(39 81 78 89)(40 107 79 123)(85 112 93 128)(87 110 95 126)(101 109 117 125)(103 111 119 127)
(1 125 3 127)(2 128 4 126)(5 50 7 52)(6 49 8 51)(9 102 11 104)(10 101 12 103)(13 37 15 39)(14 40 16 38)(17 85 19 87)(18 88 20 86)(21 83 23 81)(22 82 24 84)(25 110 27 112)(26 109 28 111)(29 114 31 116)(30 113 32 115)(33 43 35 41)(34 42 36 44)(45 106 47 108)(46 105 48 107)(53 73 55 75)(54 76 56 74)(57 79 59 77)(58 78 60 80)(61 94 63 96)(62 93 64 95)(65 122 67 124)(66 121 68 123)(69 118 71 120)(70 117 72 119)(89 100 91 98)(90 99 92 97)```

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4Z 4AA 4AB 4AC 4AD order 1 2 ··· 2 4 ··· 4 4 ··· 4 4 4 4 4 size 1 1 ··· 1 2 ··· 2 4 ··· 4 8 8 8 8

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + - - + image C1 C2 C2 C2 C2 C2 C2 C2 D4 Q8 Q8 C4○D4 2+ 1+4 kernel C42.176D4 C4×C4⋊C4 C42⋊9C4 C23.65C23 C23.67C23 C23.78C23 C2×C4×Q8 C2×C4⋊Q8 C42 C4⋊C4 C2×Q8 C2×C4 C22 # reps 1 1 2 4 2 4 1 1 4 8 4 4 2

Matrix representation of C42.176D4 in GL6(𝔽5)

 3 0 0 0 0 0 0 2 0 0 0 0 0 0 0 1 0 0 0 0 4 0 0 0 0 0 0 0 2 0 0 0 0 0 0 3
,
 2 0 0 0 0 0 0 3 0 0 0 0 0 0 0 1 0 0 0 0 4 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 3 0 0 0 0 0 0 2 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 0 1 0 0 0 0 4 0 0 0 0 0 0 0 0 3 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 3 0

`G:=sub<GL(6,GF(5))| [3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,3],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,3,0] >;`

C42.176D4 in GAP, Magma, Sage, TeX

`C_4^2._{176}D_4`
`% in TeX`

`G:=Group("C4^2.176D4");`
`// GroupNames label`

`G:=SmallGroup(128,1299);`
`// by ID`

`G=gap.SmallGroup(128,1299);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,568,758,723,184,675,80]);`
`// 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=a^-1*b^2,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;`
`// generators/relations`

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