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## G = C42.124D4order 128 = 27

### 106th 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 — C42.124D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C8⋊C4 — C42.124D4
 Lower central C1 — C2 — C2×C4 — C42.124D4
 Upper central C1 — C23 — C2×C42 — C42.124D4
 Jennings C1 — C2 — C2 — C22×C4 — C42.124D4

Generators and relations for C42.124D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2b2, ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=bc-1 >

Subgroups: 244 in 130 conjugacy classes, 64 normal (16 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], C23, C42 [×4], C4⋊C4 [×4], C4⋊C4 [×18], C2×C8 [×4], C2×C8 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C8⋊C4 [×2], C2×C42, C2×C4⋊C4 [×4], C2×C4⋊C4 [×4], C42.C2 [×4], C42.C2 [×2], C22×C8 [×2], C22.4Q16 [×4], C429C4, C2×C8⋊C4, C2×C42.C2, C42.124D4
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, C23.36D4 [×2], C42.29C22, C42.30C22, C8⋊Q8 [×2], C42.124D4

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

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

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

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

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 C42.124D4 C22.4Q16 C42⋊9C4 C2×C8⋊C4 C2×C42.C2 C42.C2 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 C42.124D4 in GL8(𝔽17)

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

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

C42.124D4 in GAP, Magma, Sage, TeX

`C_4^2._{124}D_4`
`% in TeX`

`G:=Group("C4^2.124D4");`
`// GroupNames label`

`G:=SmallGroup(128,724);`
`// by ID`

`G=gap.SmallGroup(128,724);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,400,422,723,100,2019,248]);`
`// Polycyclic`

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

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