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## G = C42.25Q8order 128 = 27

### 25th 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 — C42.25Q8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C42⋊4C4 — C42.25Q8
 Lower central C1 — C23 — C42.25Q8
 Upper central C1 — C22×C4 — C42.25Q8
 Jennings C1 — C2 — C2 — C22×C4 — C42.25Q8

Generators and relations for C42.25Q8
G = < a,b,c,d | a4=b4=c4=1, d2=a2bc2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 188 in 128 conjugacy classes, 76 normal (7 characteristic)
C1, C2, C2 [×6], C4, C4 [×3], C4 [×12], C22 [×7], C8 [×4], C2×C4 [×18], C2×C4 [×12], C23, C42 [×12], C2×C8 [×12], C22×C4, C22×C4 [×6], C2.C42 [×4], C4⋊C8 [×12], C2×C42 [×3], C22×C8 [×4], C424C4, C2×C4⋊C8 [×6], C42.25Q8
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], C8○D4 [×4], C429C4, C42.6C22 [×6], C42.25Q8

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 2 2 2 type + + + + - image C1 C2 C2 C4 D4 Q8 C8○D4 kernel C42.25Q8 C42⋊4C4 C2×C4⋊C8 C2.C42 C42 C42 C22 # reps 1 1 6 8 6 6 16

Matrix representation of C42.25Q8 in GL6(𝔽17)

 0 1 0 0 0 0 1 0 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 4 0 0 0 0 0 11 13
,
 13 0 0 0 0 0 0 13 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 16 0 0 0 0 0 0 0 4 15 0 0 0 0 0 13 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 0 15 0 0 0 0 15 0 0 0 0 0 0 0 1 8 0 0 0 0 4 16 0 0 0 0 0 0 10 2 0 0 0 0 9 7

`G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,11,0,0,0,0,0,13],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,15,13,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,15,0,0,0,0,15,0,0,0,0,0,0,0,1,4,0,0,0,0,8,16,0,0,0,0,0,0,10,9,0,0,0,0,2,7] >;`

C42.25Q8 in GAP, Magma, Sage, TeX

`C_4^2._{25}Q_8`
`% in TeX`

`G:=Group("C4^2.25Q8");`
`// GroupNames label`

`G:=SmallGroup(128,575);`
`// by ID`

`G=gap.SmallGroup(128,575);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,723,100,124]);`
`// Polycyclic`

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

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