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## G = C42.101D4order 128 = 27

### 83rd 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 — C42.101D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×Q8 — C2×C4×Q8 — C42.101D4
 Lower central C1 — C2 — C2×C4 — C42.101D4
 Upper central C1 — C23 — C2×C42 — C42.101D4
 Jennings C1 — C2 — C2 — C22×C4 — C42.101D4

Generators and relations for C42.101D4
G = < a,b,c,d | a4=b4=c4=1, d2=b, ab=ba, cac-1=dad-1=a-1b2, cbc-1=b-1, bd=db, dcd-1=b-1c-1 >

Subgroups: 260 in 146 conjugacy classes, 68 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×12], C22 [×3], C22 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×4], C2×C4 [×20], Q8 [×4], Q8 [×6], C23, C42 [×4], C42 [×4], C4⋊C4 [×2], C4⋊C4 [×9], C2×C8 [×6], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×6], C2×Q8 [×3], C2.C42 [×2], Q8⋊C4 [×4], C4⋊C8 [×2], C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4 [×2], C2×C4⋊C4, C4×Q8 [×4], C4×Q8 [×2], C22×C8 [×2], C22×Q8, C22.4Q16 [×2], C428C4, C2×Q8⋊C4 [×2], C2×C4⋊C8, C2×C4×Q8, C42.101D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], C4○D8 [×2], C8.C22 [×2], C23.7Q8, C23.24D4, C23.38D4, Q8.D4 [×2], Q8.Q8 [×2], C42.101D4

Smallest permutation representation of C42.101D4
Regular action on 128 points
Generators in S128
```(1 83 35 127)(2 124 36 88)(3 85 37 121)(4 126 38 82)(5 87 39 123)(6 128 40 84)(7 81 33 125)(8 122 34 86)(9 17 56 32)(10 29 49 22)(11 19 50 26)(12 31 51 24)(13 21 52 28)(14 25 53 18)(15 23 54 30)(16 27 55 20)(41 64 117 101)(42 98 118 61)(43 58 119 103)(44 100 120 63)(45 60 113 97)(46 102 114 57)(47 62 115 99)(48 104 116 59)(65 74 110 90)(66 95 111 79)(67 76 112 92)(68 89 105 73)(69 78 106 94)(70 91 107 75)(71 80 108 96)(72 93 109 77)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 91 93 95)(90 92 94 96)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)(113 115 117 119)(114 116 118 120)(121 123 125 127)(122 124 126 128)
(1 31 77 98)(2 101 78 26)(3 29 79 104)(4 99 80 32)(5 27 73 102)(6 97 74 30)(7 25 75 100)(8 103 76 28)(9 86 47 67)(10 70 48 81)(11 84 41 65)(12 68 42 87)(13 82 43 71)(14 66 44 85)(15 88 45 69)(16 72 46 83)(17 38 62 96)(18 91 63 33)(19 36 64 94)(20 89 57 39)(21 34 58 92)(22 95 59 37)(23 40 60 90)(24 93 61 35)(49 107 116 125)(50 128 117 110)(51 105 118 123)(52 126 119 108)(53 111 120 121)(54 124 113 106)(55 109 114 127)(56 122 115 112)
(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)```

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

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

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

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 1 2 2 2 2 2 2 4 type + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C4 D4 D4 D4 Q8 C4○D4 C4○D8 C8.C22 kernel C42.101D4 C22.4Q16 C42⋊8C4 C2×Q8⋊C4 C2×C4⋊C8 C2×C4×Q8 C4×Q8 C42 C22×C4 C2×Q8 C2×Q8 C2×C4 C22 C22 # reps 1 2 1 2 1 1 8 2 2 2 2 4 8 2

Matrix representation of C42.101D4 in GL6(𝔽17)

 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 15 0 0 0 0 1 16
,
 3 3 0 0 0 0 3 14 0 0 0 0 0 0 10 1 0 0 0 0 1 7 0 0 0 0 0 0 6 11 0 0 0 0 3 11
,
 3 3 0 0 0 0 3 14 0 0 0 0 0 0 10 1 0 0 0 0 1 7 0 0 0 0 0 0 0 10 0 0 0 0 12 10

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

C42.101D4 in GAP, Magma, Sage, TeX

`C_4^2._{101}D_4`
`% in TeX`

`G:=Group("C4^2.101D4");`
`// GroupNames label`

`G:=SmallGroup(128,537);`
`// by ID`

`G=gap.SmallGroup(128,537);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,512,422,352,2804,718,172]);`
`// Polycyclic`

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

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