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## G = C42.111D4order 128 = 27

### 93rd 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 — C42.111D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C8⋊C4 — C42.111D4
 Lower central C1 — C2 — C2×C4 — C42.111D4
 Upper central C1 — C23 — C2×C42 — C42.111D4
 Jennings C1 — C2 — C2 — C22×C4 — C42.111D4

Generators and relations for C42.111D4
G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=bc-1 >

Subgroups: 292 in 154 conjugacy classes, 64 normal (14 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×10], C22 [×3], C22 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×18], Q8 [×12], C23, C42 [×4], C4⋊C4 [×12], C2×C8 [×4], C2×C8 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Q8 [×4], C2×Q8 [×10], C2.C42 [×2], C8⋊C4 [×2], Q8⋊C4 [×8], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C4⋊Q8 [×4], C4⋊Q8 [×2], C22×C8 [×2], C22×Q8 [×2], C428C4, C2×C8⋊C4, C2×Q8⋊C4 [×4], C2×C4⋊Q8, C42.111D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C41D4, C8.C22 [×4], C24.3C22, C23.38D4 [×2], C42.30C22 [×2], C8.2D4 [×2], C42.111D4

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

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

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

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

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 type + + + + + + + + - image C1 C2 C2 C2 C2 C4 D4 D4 D4 C4○D4 C8.C22 kernel C42.111D4 C42⋊8C4 C2×C8⋊C4 C2×Q8⋊C4 C2×C4⋊Q8 C4⋊Q8 C42 C2×C8 C22×C4 C2×C4 C22 # reps 1 1 1 4 1 8 2 4 2 4 4

Matrix representation of C42.111D4 in GL8(𝔽17)

 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 2 0 0 0 0 0 0 16 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0
,
 1 0 0 0 0 0 0 0 0 1 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 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0
,
 14 16 0 0 0 0 0 0 10 3 0 0 0 0 0 0 0 0 2 5 0 0 0 0 0 0 13 15 0 0 0 0 0 0 0 0 3 10 12 10 0 0 0 0 10 14 10 5 0 0 0 0 5 7 3 10 0 0 0 0 7 12 10 14
,
 3 11 0 0 0 0 0 0 7 14 0 0 0 0 0 0 0 0 7 8 0 0 0 0 0 0 11 10 0 0 0 0 0 0 0 0 1 3 15 11 0 0 0 0 3 16 11 2 0 0 0 0 15 11 16 14 0 0 0 0 11 2 14 1

`G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,16,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[14,10,0,0,0,0,0,0,16,3,0,0,0,0,0,0,0,0,2,13,0,0,0,0,0,0,5,15,0,0,0,0,0,0,0,0,3,10,5,7,0,0,0,0,10,14,7,12,0,0,0,0,12,10,3,10,0,0,0,0,10,5,10,14],[3,7,0,0,0,0,0,0,11,14,0,0,0,0,0,0,0,0,7,11,0,0,0,0,0,0,8,10,0,0,0,0,0,0,0,0,1,3,15,11,0,0,0,0,3,16,11,2,0,0,0,0,15,11,16,14,0,0,0,0,11,2,14,1] >;`

C42.111D4 in GAP, Magma, Sage, TeX

`C_4^2._{111}D_4`
`% in TeX`

`G:=Group("C4^2.111D4");`
`// GroupNames label`

`G:=SmallGroup(128,692);`
`// by ID`

`G=gap.SmallGroup(128,692);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,723,436,2019,248]);`
`// 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*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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