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## G = C42.180D4order 128 = 27

### 162nd 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 — C42.180D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C4×Q8 — C42.180D4
 Lower central C1 — C23 — C42.180D4
 Upper central C1 — C23 — C42.180D4
 Jennings C1 — C23 — C42.180D4

Generators and relations for C42.180D4
G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=a-1, dad-1=ab2, cbc-1=dbd-1=b-1, dcd-1=a2c-1 >

Subgroups: 388 in 242 conjugacy classes, 124 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, Q8, C23, C42, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C4×Q8, C22×Q8, C429C4, C23.65C23, C23.67C23, C23.81C23, C2×C4×Q8, C42.180D4
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C22⋊Q8, C22×D4, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C22⋊Q8, C22.31C24, Q83Q8, Q82, C22.53C24, C42.180D4

Smallest permutation representation of C42.180D4
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 20 45 13)(2 17 46 14)(3 18 47 15)(4 19 48 16)(5 37 127 36)(6 38 128 33)(7 39 125 34)(8 40 126 35)(9 42 23 50)(10 43 24 51)(11 44 21 52)(12 41 22 49)(25 54 30 57)(26 55 31 58)(27 56 32 59)(28 53 29 60)(61 65 100 72)(62 66 97 69)(63 67 98 70)(64 68 99 71)(73 108 111 86)(74 105 112 87)(75 106 109 88)(76 107 110 85)(77 113 82 102)(78 114 83 103)(79 115 84 104)(80 116 81 101)(89 117 96 124)(90 118 93 121)(91 119 94 122)(92 120 95 123)
(1 125 60 75)(2 128 57 74)(3 127 58 73)(4 126 59 76)(5 55 111 47)(6 54 112 46)(7 53 109 45)(8 56 110 48)(9 96 70 81)(10 95 71 84)(11 94 72 83)(12 93 69 82)(13 34 29 106)(14 33 30 105)(15 36 31 108)(16 35 32 107)(17 38 25 87)(18 37 26 86)(19 40 27 85)(20 39 28 88)(21 91 65 78)(22 90 66 77)(23 89 67 80)(24 92 68 79)(41 118 62 113)(42 117 63 116)(43 120 64 115)(44 119 61 114)(49 121 97 102)(50 124 98 101)(51 123 99 104)(52 122 100 103)
(1 101 45 116)(2 113 46 102)(3 103 47 114)(4 115 48 104)(5 63 127 98)(6 99 128 64)(7 61 125 100)(8 97 126 62)(9 108 23 86)(10 87 24 105)(11 106 21 88)(12 85 22 107)(13 80 20 81)(14 82 17 77)(15 78 18 83)(16 84 19 79)(25 90 30 93)(26 94 31 91)(27 92 32 95)(28 96 29 89)(33 71 38 68)(34 65 39 72)(35 69 40 66)(36 67 37 70)(41 110 49 76)(42 73 50 111)(43 112 51 74)(44 75 52 109)(53 117 60 124)(54 121 57 118)(55 119 58 122)(56 123 59 120)```

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

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

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

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 2 2 2 4 4 type + + + + + + + - + - image C1 C2 C2 C2 C2 C2 D4 Q8 C4○D4 2+ 1+4 2- 1+4 kernel C42.180D4 C42⋊9C4 C23.65C23 C23.67C23 C23.81C23 C2×C4×Q8 C42 C2×Q8 C2×C4 C22 C22 # reps 1 1 4 4 4 2 4 8 8 1 1

Matrix representation of C42.180D4 in GL6(𝔽5)

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

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

C42.180D4 in GAP, Magma, Sage, TeX

`C_4^2._{180}D_4`
`% in TeX`

`G:=Group("C4^2.180D4");`
`// GroupNames label`

`G:=SmallGroup(128,1314);`
`// by ID`

`G=gap.SmallGroup(128,1314);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,568,758,723,352,675,80]);`
`// Polycyclic`

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

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