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## G = C42⋊19Q8order 128 = 27

### 6th semidirect product of C42 and Q8 acting via Q8/C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C42⋊19Q8
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C43 — C42⋊19Q8
 Lower central C1 — C23 — C42⋊19Q8
 Upper central C1 — C23 — C42⋊19Q8
 Jennings C1 — C23 — C42⋊19Q8

Generators and relations for C4219Q8
G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 516 in 324 conjugacy classes, 180 normal (6 characteristic)
C1, C2, C2, C4, C4, C22, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×C42, C2×C4⋊C4, C4⋊Q8, C22×Q8, C43, C429C4, C2×C4⋊Q8, C4219Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C24, C41D4, C4⋊Q8, C22×D4, C22×Q8, C2×C41D4, C2×C4⋊Q8, C4219Q8

Smallest permutation representation of C4219Q8
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 98 22 52)(2 99 23 49)(3 100 24 50)(4 97 21 51)(5 114 92 83)(6 115 89 84)(7 116 90 81)(8 113 91 82)(9 64 20 32)(10 61 17 29)(11 62 18 30)(12 63 19 31)(13 74 53 37)(14 75 54 38)(15 76 55 39)(16 73 56 40)(25 80 57 36)(26 77 58 33)(27 78 59 34)(28 79 60 35)(41 67 111 124)(42 68 112 121)(43 65 109 122)(44 66 110 123)(45 108 103 118)(46 105 104 119)(47 106 101 120)(48 107 102 117)(69 86 127 93)(70 87 128 94)(71 88 125 95)(72 85 126 96)
(1 27 12 16)(2 28 9 13)(3 25 10 14)(4 26 11 15)(5 45 127 42)(6 46 128 43)(7 47 125 44)(8 48 126 41)(17 54 24 57)(18 55 21 58)(19 56 22 59)(20 53 23 60)(29 38 50 36)(30 39 51 33)(31 40 52 34)(32 37 49 35)(61 75 100 80)(62 76 97 77)(63 73 98 78)(64 74 99 79)(65 115 105 94)(66 116 106 95)(67 113 107 96)(68 114 108 93)(69 112 92 103)(70 109 89 104)(71 110 90 101)(72 111 91 102)(81 120 88 123)(82 117 85 124)(83 118 86 121)(84 119 87 122)
(1 95 12 116)(2 94 9 115)(3 93 10 114)(4 96 11 113)(5 100 127 61)(6 99 128 64)(7 98 125 63)(8 97 126 62)(13 65 28 105)(14 68 25 108)(15 67 26 107)(16 66 27 106)(17 83 24 86)(18 82 21 85)(19 81 22 88)(20 84 23 87)(29 92 50 69)(30 91 51 72)(31 90 52 71)(32 89 49 70)(33 102 39 111)(34 101 40 110)(35 104 37 109)(36 103 38 112)(41 77 48 76)(42 80 45 75)(43 79 46 74)(44 78 47 73)(53 122 60 119)(54 121 57 118)(55 124 58 117)(56 123 59 120)```

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4AB 4AC ··· 4AJ order 1 2 ··· 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 2 ··· 2 8 ··· 8

44 irreducible representations

 dim 1 1 1 1 2 2 type + + + + + - image C1 C2 C2 C2 D4 Q8 kernel C42⋊19Q8 C43 C42⋊9C4 C2×C4⋊Q8 C42 C42 # reps 1 1 8 6 12 16

Matrix representation of C4219Q8 in GL6(𝔽5)

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

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

C4219Q8 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_{19}Q_8`
`% in TeX`

`G:=Group("C4^2:19Q8");`
`// GroupNames label`

`G:=SmallGroup(128,1600);`
`// by ID`

`G=gap.SmallGroup(128,1600);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,184,2019,248]);`
`// Polycyclic`

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

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