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## G = C42.121D4order 128 = 27

### 103rd 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 — C42.121D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C4×C4⋊C4 — C42.121D4
 Lower central C1 — C2 — C2×C4 — C42.121D4
 Upper central C1 — C23 — C2×C42 — C42.121D4
 Jennings C1 — C2 — C2 — C22×C4 — C42.121D4

Generators and relations for C42.121D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=a2bc-1 >

Subgroups: 284 in 146 conjugacy classes, 72 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×6], C4 [×10], C22 [×3], C22 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×12], C2×C4 [×18], Q8 [×8], C23, C42 [×4], C42 [×2], C4⋊C4 [×8], C4⋊C4 [×8], C2×C8 [×6], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×8], C2.C42, C4⋊C8 [×2], C2×C42, C2×C42, C2×C4⋊C4 [×4], C2×C4⋊C4, C4⋊Q8 [×4], C4⋊Q8 [×2], C22×C8 [×2], C22×Q8, C22.4Q16 [×4], C4×C4⋊C4, C2×C4⋊C8, C2×C4⋊Q8, C42.121D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, C22⋊C4 [×4], SD16 [×2], Q16 [×2], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], Q8⋊C4 [×4], C2×C22⋊C4, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4, C4⋊Q8, C2×SD16, C2×Q16, C8⋊C22 [×2], C23.67C23, C2×Q8⋊C4, C23.37D4, D42Q8 [×2], C4.Q16 [×2], C42.121D4

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

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4R 4S 4T 4U 4V 8A ··· 8H order 1 2 ··· 2 4 ··· 4 4 ··· 4 4 4 4 4 8 ··· 8 size 1 1 ··· 1 2 ··· 2 4 ··· 4 8 8 8 8 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + - + - + image C1 C2 C2 C2 C2 C4 D4 Q8 D4 SD16 Q16 C4○D4 C8⋊C22 kernel C42.121D4 C22.4Q16 C4×C4⋊C4 C2×C4⋊C8 C2×C4⋊Q8 C4⋊Q8 C42 C4⋊C4 C22×C4 C2×C4 C2×C4 C2×C4 C22 # reps 1 4 1 1 1 8 2 4 2 4 4 4 2

Matrix representation of C42.121D4 in GL5(𝔽17)

 16 0 0 0 0 0 1 15 0 0 0 1 16 0 0 0 0 0 1 0 0 0 0 0 1
,
 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 16 0
,
 13 0 0 0 0 0 16 2 0 0 0 16 1 0 0 0 0 0 13 11 0 0 0 11 4
,
 16 0 0 0 0 0 13 0 0 0 0 13 4 0 0 0 0 0 7 16 0 0 0 16 10

`G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,1,0,0,0,15,16,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,1,0],[13,0,0,0,0,0,16,16,0,0,0,2,1,0,0,0,0,0,13,11,0,0,0,11,4],[16,0,0,0,0,0,13,13,0,0,0,0,4,0,0,0,0,0,7,16,0,0,0,16,10] >;`

C42.121D4 in GAP, Magma, Sage, TeX

`C_4^2._{121}D_4`
`% in TeX`

`G:=Group("C4^2.121D4");`
`// GroupNames label`

`G:=SmallGroup(128,719);`
`// by ID`

`G=gap.SmallGroup(128,719);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,400,422,100,1018,248,2028,124]);`
`// Polycyclic`

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

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