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## G = C42.37Q8order 128 = 27

### 37th non-split extension by C42 of Q8 acting via Q8/C2=C22

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C42.37Q8
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C4⋊C4 — C42.37Q8
 Lower central C1 — C23 — C42.37Q8
 Upper central C1 — C23 — C42.37Q8
 Jennings C1 — C23 — C42.37Q8

Generators and relations for C42.37Q8
G = < a,b,c,d | a4=b4=c4=1, d2=a2c2, ab=ba, cac-1=a-1b2, dad-1=ab2, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >

Subgroups: 308 in 190 conjugacy classes, 108 normal (12 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×14], C22 [×3], C22 [×4], C2×C4 [×14], C2×C4 [×38], C23, C42 [×4], C42 [×8], C4⋊C4 [×22], C22×C4, C22×C4 [×14], C2.C42 [×16], C2×C42, C2×C42 [×4], C2×C4⋊C4 [×14], C4×C4⋊C4 [×2], C428C4 [×4], C429C4, C23.65C23 [×4], C23.83C23 [×4], C42.37Q8
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×8], C24, C42.C2 [×8], C22×Q8, C2×C4○D4 [×4], 2+ 1+4, 2- 1+4, C2×C42.C2 [×2], C23.41C23, C22.49C24, C22.50C24 [×2], C22.53C24, C42.37Q8

Smallest permutation representation of C42.37Q8
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 41 55 16)(2 42 56 13)(3 43 53 14)(4 44 54 15)(5 110 39 87)(6 111 40 88)(7 112 37 85)(8 109 38 86)(9 29 18 58)(10 30 19 59)(11 31 20 60)(12 32 17 57)(21 98 46 65)(22 99 47 66)(23 100 48 67)(24 97 45 68)(25 69 50 64)(26 70 51 61)(27 71 52 62)(28 72 49 63)(33 116 126 83)(34 113 127 84)(35 114 128 81)(36 115 125 82)(73 93 106 120)(74 94 107 117)(75 95 108 118)(76 96 105 119)(77 124 104 91)(78 121 101 92)(79 122 102 89)(80 123 103 90)
(1 46 9 72)(2 24 10 62)(3 48 11 70)(4 22 12 64)(5 76 113 89)(6 108 114 121)(7 74 115 91)(8 106 116 123)(13 68 59 52)(14 100 60 26)(15 66 57 50)(16 98 58 28)(17 69 54 47)(18 63 55 21)(19 71 56 45)(20 61 53 23)(25 44 99 32)(27 42 97 30)(29 49 41 65)(31 51 43 67)(33 80 86 93)(34 102 87 119)(35 78 88 95)(36 104 85 117)(37 107 82 124)(38 73 83 90)(39 105 84 122)(40 75 81 92)(77 112 94 125)(79 110 96 127)(101 111 118 128)(103 109 120 126)
(1 104 11 119)(2 78 12 93)(3 102 9 117)(4 80 10 95)(5 65 115 51)(6 99 116 27)(7 67 113 49)(8 97 114 25)(13 121 57 106)(14 89 58 74)(15 123 59 108)(16 91 60 76)(17 120 56 101)(18 94 53 79)(19 118 54 103)(20 96 55 77)(21 36 61 87)(22 126 62 111)(23 34 63 85)(24 128 64 109)(26 39 98 82)(28 37 100 84)(29 107 43 122)(30 75 44 90)(31 105 41 124)(32 73 42 92)(33 71 88 47)(35 69 86 45)(38 68 81 50)(40 66 83 52)(46 125 70 110)(48 127 72 112)```

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4Z 4AA 4AB 4AC 4AD order 1 2 ··· 2 4 ··· 4 4 ··· 4 4 4 4 4 size 1 1 ··· 1 2 ··· 2 4 ··· 4 8 8 8 8

38 irreducible representations

 dim 1 1 1 1 1 1 2 2 4 4 type + + + + + + - + - image C1 C2 C2 C2 C2 C2 Q8 C4○D4 2+ 1+4 2- 1+4 kernel C42.37Q8 C4×C4⋊C4 C42⋊8C4 C42⋊9C4 C23.65C23 C23.83C23 C42 C2×C4 C22 C22 # reps 1 2 4 1 4 4 4 16 1 1

Matrix representation of C42.37Q8 in GL6(𝔽5)

 3 2 0 0 0 0 0 2 0 0 0 0 0 0 0 4 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 4 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 0 1 0 0 0 0 4 0
,
 3 0 0 0 0 0 1 2 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 2 0 0 0 0 0 0 2
,
 4 1 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 1 0 0 0 0 0 0 4

`G:=sub<GL(6,GF(5))| [3,0,0,0,0,0,2,2,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[3,1,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[4,0,0,0,0,0,1,1,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,4] >;`

C42.37Q8 in GAP, Magma, Sage, TeX

`C_4^2._{37}Q_8`
`% in TeX`

`G:=Group("C4^2.37Q8");`
`// GroupNames label`

`G:=SmallGroup(128,1303);`
`// by ID`

`G=gap.SmallGroup(128,1303);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,568,758,723,268,675,80]);`
`// Polycyclic`

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

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