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## G = C42.60Q8order 128 = 27

### 20th non-split extension by C42 of Q8 acting via Q8/C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.60Q8
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C4×C8 — C42.60Q8
 Lower central C1 — C2 — C2×C4 — C42.60Q8
 Upper central C1 — C23 — C2×C42 — C42.60Q8
 Jennings C1 — C2 — C2 — C22×C4 — C42.60Q8

Generators and relations for C42.60Q8
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, dad-1=a-1b2, bc=cb, dbd-1=b-1, dcd-1=c3 >

Subgroups: 220 in 124 conjugacy classes, 76 normal (22 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×8], C22 [×3], C22 [×4], C8 [×8], C2×C4 [×6], C2×C4 [×4], C2×C4 [×16], C23, C42 [×4], C4⋊C4 [×8], C2×C8 [×12], C22×C4, C22×C4 [×2], C22×C4 [×4], C2.C42 [×4], C4×C8 [×4], C4.Q8 [×4], C2.D8 [×4], C2×C42, C2×C4⋊C4 [×4], C22×C8 [×2], C428C4 [×2], C2×C4×C8, C2×C4.Q8 [×2], C2×C2.D8 [×2], C42.60Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×6], C23, C4⋊C4 [×12], C22×C4, C2×D4 [×3], C2×Q8 [×3], C2×C4⋊C4 [×3], C41D4, C4⋊Q8 [×3], C4○D8 [×4], C429C4, C23.25D4 [×2], C8.12D4 [×2], C8.5Q8 [×2], C42.60Q8

Smallest permutation representation of C42.60Q8
Regular action on 128 points
Generators in S128
```(1 101 61 52)(2 102 62 53)(3 103 63 54)(4 104 64 55)(5 97 57 56)(6 98 58 49)(7 99 59 50)(8 100 60 51)(9 95 84 124)(10 96 85 125)(11 89 86 126)(12 90 87 127)(13 91 88 128)(14 92 81 121)(15 93 82 122)(16 94 83 123)(17 27 67 74)(18 28 68 75)(19 29 69 76)(20 30 70 77)(21 31 71 78)(22 32 72 79)(23 25 65 80)(24 26 66 73)(33 46 120 105)(34 47 113 106)(35 48 114 107)(36 41 115 108)(37 42 116 109)(38 43 117 110)(39 44 118 111)(40 45 119 112)
(1 29 5 25)(2 30 6 26)(3 31 7 27)(4 32 8 28)(9 42 13 46)(10 43 14 47)(11 44 15 48)(12 45 16 41)(17 54 21 50)(18 55 22 51)(19 56 23 52)(20 49 24 53)(33 124 37 128)(34 125 38 121)(35 126 39 122)(36 127 40 123)(57 80 61 76)(58 73 62 77)(59 74 63 78)(60 75 64 79)(65 101 69 97)(66 102 70 98)(67 103 71 99)(68 104 72 100)(81 106 85 110)(82 107 86 111)(83 108 87 112)(84 109 88 105)(89 118 93 114)(90 119 94 115)(91 120 95 116)(92 113 96 117)
(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)
(1 113 27 94)(2 116 28 89)(3 119 29 92)(4 114 30 95)(5 117 31 90)(6 120 32 93)(7 115 25 96)(8 118 26 91)(9 100 48 66)(10 103 41 69)(11 98 42 72)(12 101 43 67)(13 104 44 70)(14 99 45 65)(15 102 46 68)(16 97 47 71)(17 87 52 110)(18 82 53 105)(19 85 54 108)(20 88 55 111)(21 83 56 106)(22 86 49 109)(23 81 50 112)(24 84 51 107)(33 79 122 58)(34 74 123 61)(35 77 124 64)(36 80 125 59)(37 75 126 62)(38 78 127 57)(39 73 128 60)(40 76 121 63)```

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

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

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

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 type + + + + + - + - + image C1 C2 C2 C2 C2 C4 Q8 D4 Q8 D4 C4○D8 kernel C42.60Q8 C42⋊8C4 C2×C4×C8 C2×C4.Q8 C2×C2.D8 C4×C8 C42 C2×C8 C2×C8 C22×C4 C22 # reps 1 2 1 2 2 8 2 4 4 2 16

Matrix representation of C42.60Q8 in GL5(𝔽17)

 16 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 0 13 0 0 0 4 0
,
 16 0 0 0 0 0 0 1 0 0 0 16 0 0 0 0 0 0 0 1 0 0 0 16 0
,
 1 0 0 0 0 0 5 12 0 0 0 5 5 0 0 0 0 0 5 12 0 0 0 5 5
,
 13 0 0 0 0 0 11 13 0 0 0 13 6 0 0 0 0 0 6 4 0 0 0 4 11

`G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,13,0],[16,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,1,0],[1,0,0,0,0,0,5,5,0,0,0,12,5,0,0,0,0,0,5,5,0,0,0,12,5],[13,0,0,0,0,0,11,13,0,0,0,13,6,0,0,0,0,0,6,4,0,0,0,4,11] >;`

C42.60Q8 in GAP, Magma, Sage, TeX

`C_4^2._{60}Q_8`
`% in TeX`

`G:=Group("C4^2.60Q8");`
`// GroupNames label`

`G:=SmallGroup(128,578);`
`// by ID`

`G=gap.SmallGroup(128,578);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,436,2019,248]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b^-1*c^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;`
`// generators/relations`

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