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## G = C42.24Q8order 128 = 27

### 24th non-split extension by C42 of Q8 acting via Q8/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.24Q8
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×C8 — C2×C8⋊C4 — C42.24Q8
 Lower central C1 — C2 — C2×C4 — C42.24Q8
 Upper central C1 — C23 — C2×C42 — C42.24Q8
 Jennings C1 — C2 — C2 — C22×C4 — C42.24Q8

Generators and relations for C42.24Q8
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1c2, ab=ba, cac-1=ab2, dad-1=a-1b2, bc=cb, dbd-1=b-1, dcd-1=a2c3 >

Subgroups: 220 in 112 conjugacy classes, 60 normal (16 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×8], C22 [×3], C22 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×16], C23, C42 [×4], C4⋊C4 [×12], C2×C8 [×4], C2×C8 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2.C42 [×2], C8⋊C4 [×4], C2×C42, C2×C4⋊C4 [×4], C2×C4⋊C4 [×2], C22×C8 [×2], C22.4Q16 [×4], C428C4, C429C4, C2×C8⋊C4, C42.24Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C4○D4 [×4], C2×C4⋊C4, C42⋊C2 [×2], C4.4D4 [×2], C42.C2 [×2], C8⋊C22 [×2], C8.C22 [×2], C428C4, M4(2)⋊C4 [×2], C42.28C22 [×2], C42.29C22, C42.30C22, C42.24Q8

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

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

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

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

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 4 4 type + + + + + - + + - image C1 C2 C2 C2 C2 C4 Q8 D4 C4○D4 C8⋊C22 C8.C22 kernel C42.24Q8 C22.4Q16 C42⋊8C4 C42⋊9C4 C2×C8⋊C4 C8⋊C4 C42 C22×C4 C2×C4 C22 C22 # reps 1 4 1 1 1 8 2 2 8 2 2

Matrix representation of C42.24Q8 in GL8(𝔽17)

 1 9 0 0 0 0 0 0 13 16 0 0 0 0 0 0 0 0 1 9 0 0 0 0 0 0 13 16 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 8 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 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 13
,
 4 2 0 0 0 0 0 0 1 13 0 0 0 0 0 0 0 0 4 2 0 0 0 0 0 0 1 13 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 16 0
,
 10 3 0 0 0 0 0 0 1 7 0 0 0 0 0 0 0 0 8 9 0 0 0 0 0 0 6 9 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0

`G:=sub<GL(8,GF(17))| [1,13,0,0,0,0,0,0,9,16,0,0,0,0,0,0,0,0,1,13,0,0,0,0,0,0,9,16,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,15,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13],[4,1,0,0,0,0,0,0,2,13,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,2,13,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0],[10,1,0,0,0,0,0,0,3,7,0,0,0,0,0,0,0,0,8,6,0,0,0,0,0,0,9,9,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;`

C42.24Q8 in GAP, Magma, Sage, TeX

`C_4^2._{24}Q_8`
`% in TeX`

`G:=Group("C4^2.24Q8");`
`// GroupNames label`

`G:=SmallGroup(128,568);`
`// by ID`

`G=gap.SmallGroup(128,568);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,120,422,723,58,2019,248,2804,172]);`
`// 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,c*a*c^-1=a*b^2,d*a*d^-1=a^-1*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=a^2*c^3>;`
`// generators/relations`

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