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## G = C42.436D4order 128 = 27

### 69th non-split extension by C42 of D4 acting via D4/C22=C2

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.436D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C4×C8 — C42.436D4
 Lower central C1 — C2 — C2×C4 — C42.436D4
 Upper central C1 — C23 — C2×C42 — C42.436D4
 Jennings C1 — C2 — C2 — C22×C4 — C42.436D4

Generators and relations for C42.436D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=bc-1 >

Subgroups: 292 in 150 conjugacy classes, 80 normal (26 characteristic)
C1, C2 [×7], C4 [×12], C4 [×6], C22 [×7], C8 [×4], C2×C4 [×2], C2×C4 [×16], C2×C4 [×14], Q8 [×8], C23, C42 [×4], C4⋊C4 [×4], C4⋊C4 [×14], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×8], C4×C8 [×2], C2×C42, C2×C4⋊C4 [×4], C2×C4⋊C4 [×3], C4⋊Q8 [×4], C4⋊Q8 [×2], C22×C8 [×2], C22×Q8, C22.4Q16 [×4], C429C4, C2×C4×C8, C2×C4⋊Q8, C42.436D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, C22⋊C4 [×4], D8 [×2], SD16 [×4], Q16 [×2], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C2×C22⋊C4, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4, C4⋊Q8, C2×D8, C2×SD16 [×2], C2×Q16, C23.67C23, C2×D4⋊C4, C2×Q8⋊C4, C4.4D8, C4.SD16, C83Q8, C82Q8, C42.436D4

Smallest permutation representation of C42.436D4
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 39 29 13)(2 40 30 14)(3 37 31 15)(4 38 32 16)(5 120 108 19)(6 117 105 20)(7 118 106 17)(8 119 107 18)(9 49 35 25)(10 50 36 26)(11 51 33 27)(12 52 34 28)(21 110 122 126)(22 111 123 127)(23 112 124 128)(24 109 121 125)(41 81 71 57)(42 82 72 58)(43 83 69 59)(44 84 70 60)(45 53 67 61)(46 54 68 62)(47 55 65 63)(48 56 66 64)(73 103 113 85)(74 104 114 86)(75 101 115 87)(76 102 116 88)(77 89 93 99)(78 90 94 100)(79 91 95 97)(80 92 96 98)
(1 87 9 79)(2 86 10 78)(3 85 11 77)(4 88 12 80)(5 84 23 66)(6 83 24 65)(7 82 21 68)(8 81 22 67)(13 75 25 91)(14 74 26 90)(15 73 27 89)(16 76 28 92)(17 72 126 62)(18 71 127 61)(19 70 128 64)(20 69 125 63)(29 101 35 95)(30 104 36 94)(31 103 33 93)(32 102 34 96)(37 113 51 99)(38 116 52 98)(39 115 49 97)(40 114 50 100)(41 111 53 119)(42 110 54 118)(43 109 55 117)(44 112 56 120)(45 107 57 123)(46 106 58 122)(47 105 59 121)(48 108 60 124)
(1 125 31 111)(2 128 32 110)(3 127 29 109)(4 126 30 112)(5 52 106 26)(6 51 107 25)(7 50 108 28)(8 49 105 27)(9 20 33 119)(10 19 34 118)(11 18 35 117)(12 17 36 120)(13 24 37 123)(14 23 38 122)(15 22 39 121)(16 21 40 124)(41 115 69 73)(42 114 70 76)(43 113 71 75)(44 116 72 74)(45 79 65 93)(46 78 66 96)(47 77 67 95)(48 80 68 94)(53 97 63 89)(54 100 64 92)(55 99 61 91)(56 98 62 90)(57 87 83 103)(58 86 84 102)(59 85 81 101)(60 88 82 104)```

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4L 4M ··· 4T 8A ··· 8P order 1 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 ··· 2 8 ··· 8 2 ··· 2

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + - + + - image C1 C2 C2 C2 C2 C4 D4 Q8 D4 D8 SD16 Q16 C4○D4 kernel C42.436D4 C22.4Q16 C42⋊9C4 C2×C4×C8 C2×C4⋊Q8 C4⋊Q8 C42 C2×C8 C22×C4 C2×C4 C2×C4 C2×C4 C2×C4 # reps 1 4 1 1 1 8 2 4 2 4 8 4 4

Matrix representation of C42.436D4 in GL5(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 16 0 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 16 2 0 0 0 16 1
,
 13 0 0 0 0 0 4 0 0 0 0 0 13 0 0 0 0 0 4 9 0 0 0 0 13
,
 1 0 0 0 0 0 4 0 0 0 0 0 13 0 0 0 0 0 0 10 0 0 0 5 0

`G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,16,0,0,0,2,1],[13,0,0,0,0,0,4,0,0,0,0,0,13,0,0,0,0,0,4,0,0,0,0,9,13],[1,0,0,0,0,0,4,0,0,0,0,0,13,0,0,0,0,0,0,5,0,0,0,10,0] >;`

C42.436D4 in GAP, Magma, Sage, TeX

`C_4^2._{436}D_4`
`% in TeX`

`G:=Group("C4^2.436D4");`
`// GroupNames label`

`G:=SmallGroup(128,722);`
`// by ID`

`G=gap.SmallGroup(128,722);`
`# by ID`

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