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## G = C42.45Q8order 128 = 27

### 5th non-split extension by C42 of Q8 acting via Q8/C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C42.45Q8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C2×C4×C8 — C42.45Q8
 Lower central C1 — C22 — C42.45Q8
 Upper central C1 — C22×C4 — C42.45Q8
 Jennings C1 — C2 — C2 — C22×C4 — C42.45Q8

Generators and relations for C42.45Q8
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, dad-1=ab2, bc=cb, bd=db, dcd-1=a2b-1c3 >

Subgroups: 180 in 130 conjugacy classes, 84 normal (16 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×10], C22 [×3], C22 [×4], C8 [×8], C2×C4 [×2], C2×C4 [×12], C2×C4 [×14], C23, C42 [×8], C42 [×2], C2×C8 [×8], C2×C8 [×8], C22×C4, C22×C4 [×6], C2.C42 [×4], C4×C8 [×2], C8⋊C4 [×2], C4⋊C8 [×8], C2×C42 [×3], C22×C8 [×4], C22.7C42 [×2], C424C4, C2×C4×C8, C2×C8⋊C4, C2×C4⋊C8 [×2], C42.45Q8
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×2], Q8 [×2], C23, C42 [×4], C4⋊C4 [×4], C22×C4 [×3], C2×D4, C2×Q8, C4○D4 [×2], C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C4×Q8 [×2], C8○D4 [×4], C4×C4⋊C4, C82M4(2) [×2], C42.6C22 [×2], C42.7C22 [×2], C42.45Q8

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

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4P 4Q ··· 4X 8A ··· 8P 8Q ··· 8X order 1 2 ··· 2 4 ··· 4 4 ··· 4 4 ··· 4 8 ··· 8 8 ··· 8 size 1 1 ··· 1 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 C4 D4 Q8 C4○D4 C8○D4 kernel C42.45Q8 C22.7C42 C42⋊4C4 C2×C4×C8 C2×C8⋊C4 C2×C4⋊C8 C2.C42 C4⋊C8 C42 C42 C2×C4 C22 # reps 1 2 1 1 1 2 8 16 2 2 4 16

Matrix representation of C42.45Q8 in GL5(𝔽17)

 13 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 4 15 0 0 0 16 13
,
 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 4 0 0 0 0 0 4
,
 16 0 0 0 0 0 0 1 0 0 0 16 0 0 0 0 0 0 2 16 0 0 0 8 15
,
 13 0 0 0 0 0 4 11 0 0 0 11 13 0 0 0 0 0 9 9 0 0 0 6 8

`G:=sub<GL(5,GF(17))| [13,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,16,0,0,0,15,13],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,4,0,0,0,0,0,4],[16,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,2,8,0,0,0,16,15],[13,0,0,0,0,0,4,11,0,0,0,11,13,0,0,0,0,0,9,6,0,0,0,9,8] >;`

C42.45Q8 in GAP, Magma, Sage, TeX

`C_4^2._{45}Q_8`
`% in TeX`

`G:=Group("C4^2.45Q8");`
`// GroupNames label`

`G:=SmallGroup(128,500);`
`// by ID`

`G=gap.SmallGroup(128,500);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,1430,142,124]);`
`// 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*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;`
`// generators/relations`

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