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## G = C42.125D4order 128 = 27

### 107th non-split extension by C42 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 — C2×C4 — C42.125D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C8⋊C4 — C42.125D4
 Lower central C1 — C2 — C2×C4 — C42.125D4
 Upper central C1 — C23 — C2×C42 — C42.125D4
 Jennings C1 — C2 — C2 — C22×C4 — C42.125D4

Generators and relations for C42.125D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=bc-1 >

Subgroups: 276 in 138 conjugacy classes, 64 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×10], C22 [×3], C22 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×18], Q8 [×8], C23, C42 [×4], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×8], C2.C42 [×2], C8⋊C4 [×2], C2×C42, C2×C4⋊C4 [×4], C2×C4⋊C4, C4⋊Q8 [×4], C4⋊Q8 [×2], C22×C8 [×2], C22×Q8, C22.4Q16 [×4], C428C4, C2×C8⋊C4, C2×C4⋊Q8, C42.125D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], C2×C22⋊C4, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4, C4⋊Q8, C8⋊C22 [×2], C8.C22 [×2], C23.67C23, C23.37D4, C23.38D4, C42.28C22 [×2], C8⋊Q8 [×2], C42.125D4

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

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

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

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

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 1 2 2 2 2 4 4 type + + + + + + - + + - image C1 C2 C2 C2 C2 C4 D4 Q8 D4 C4○D4 C8⋊C22 C8.C22 kernel C42.125D4 C22.4Q16 C42⋊8C4 C2×C8⋊C4 C2×C4⋊Q8 C4⋊Q8 C42 C2×C8 C22×C4 C2×C4 C22 C22 # reps 1 4 1 1 1 8 2 4 2 4 2 2

Matrix representation of C42.125D4 in GL8(𝔽17)

 1 15 0 0 0 0 0 0 1 16 0 0 0 0 0 0 0 0 16 15 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 4 1 13 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 3 16
,
 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 13 16 4 0 0 0 0 0 6 14 6 13
,
 7 12 0 0 0 0 0 0 13 10 0 0 0 0 0 0 0 0 9 15 0 0 0 0 0 0 7 8 0 0 0 0 0 0 0 0 16 7 8 7 0 0 0 0 0 6 6 6 0 0 0 0 15 11 8 7 0 0 0 0 2 14 3 4
,
 8 15 0 0 0 0 0 0 7 9 0 0 0 0 0 0 0 0 9 15 0 0 0 0 0 0 7 8 0 0 0 0 0 0 0 0 2 14 3 4 0 0 0 0 9 0 2 1 0 0 0 0 8 11 9 13 0 0 0 0 0 6 6 6

`G:=sub<GL(8,GF(17))| [1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,1,0,12,0,0,0,0,1,13,0,3,0,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,13,6,0,0,0,0,16,0,16,14,0,0,0,0,0,0,4,6,0,0,0,0,0,0,0,13],[7,13,0,0,0,0,0,0,12,10,0,0,0,0,0,0,0,0,9,7,0,0,0,0,0,0,15,8,0,0,0,0,0,0,0,0,16,0,15,2,0,0,0,0,7,6,11,14,0,0,0,0,8,6,8,3,0,0,0,0,7,6,7,4],[8,7,0,0,0,0,0,0,15,9,0,0,0,0,0,0,0,0,9,7,0,0,0,0,0,0,15,8,0,0,0,0,0,0,0,0,2,9,8,0,0,0,0,0,14,0,11,6,0,0,0,0,3,2,9,6,0,0,0,0,4,1,13,6] >;`

C42.125D4 in GAP, Magma, Sage, TeX

`C_4^2._{125}D_4`
`% in TeX`

`G:=Group("C4^2.125D4");`
`// GroupNames label`

`G:=SmallGroup(128,725);`
`// by ID`

`G=gap.SmallGroup(128,725);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,400,422,723,436,2019,248]);`
`// 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,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b*c^-1>;`
`// generators/relations`

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