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## G = C42.92D6order 192 = 26·3

### 92nd non-split extension by C42 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C42.92D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C2×C4○D12 — C42.92D6
 Lower central C3 — C2×C6 — C42.92D6
 Upper central C1 — C22 — C42⋊C2

Generators and relations for C42.92D6
G = < a,b,c,d | a4=b4=1, c6=d2=a2, ab=ba, ac=ca, dad-1=a-1, cbc-1=a2b, dbd-1=b-1, dcd-1=c5 >

Subgroups: 680 in 270 conjugacy classes, 111 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×4], C4 [×10], C22, C22 [×2], C22 [×8], S3 [×2], C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×14], D4 [×6], Q8 [×10], C23, C23 [×2], Dic3 [×6], C12 [×4], C12 [×4], D6 [×6], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×8], C22×C4, C22×C4 [×4], C2×D4 [×3], C2×Q8 [×9], C4○D4 [×4], Dic6 [×10], C4×S3 [×4], D12 [×2], C2×Dic3 [×6], C2×Dic3 [×4], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×8], C22×S3 [×2], C22×C6, C42⋊C2, C22⋊Q8 [×4], C22.D4 [×4], C4.4D4 [×2], C4⋊Q8 [×2], C22×Q8, C2×C4○D4, C4⋊Dic3 [×8], D6⋊C4 [×8], C4×C12 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], C2×Dic6, C2×Dic6 [×4], C2×Dic6 [×4], S3×C2×C4 [×2], C2×D12, C4○D12 [×4], C22×Dic3 [×2], C2×C3⋊D4 [×2], C22×C12, C23.38C23, C122Q8 [×2], C427S3 [×2], C23.21D6 [×4], C4.D12 [×4], C3×C42⋊C2, C22×Dic6, C2×C4○D12, C42.92D6
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, D12 [×4], C22×S3 [×7], C22×D4, 2- 1+4 [×2], C2×D12 [×6], S3×C23, C23.38C23, C22×D12, Q8○D12 [×2], C42.92D6

Smallest permutation representation of C42.92D6
On 96 points
Generators in S96
```(1 59 7 53)(2 60 8 54)(3 49 9 55)(4 50 10 56)(5 51 11 57)(6 52 12 58)(13 87 19 93)(14 88 20 94)(15 89 21 95)(16 90 22 96)(17 91 23 85)(18 92 24 86)(25 75 31 81)(26 76 32 82)(27 77 33 83)(28 78 34 84)(29 79 35 73)(30 80 36 74)(37 62 43 68)(38 63 44 69)(39 64 45 70)(40 65 46 71)(41 66 47 72)(42 67 48 61)
(1 90 65 27)(2 85 66 34)(3 92 67 29)(4 87 68 36)(5 94 69 31)(6 89 70 26)(7 96 71 33)(8 91 72 28)(9 86 61 35)(10 93 62 30)(11 88 63 25)(12 95 64 32)(13 43 80 56)(14 38 81 51)(15 45 82 58)(16 40 83 53)(17 47 84 60)(18 42 73 55)(19 37 74 50)(20 44 75 57)(21 39 76 52)(22 46 77 59)(23 41 78 54)(24 48 79 49)
(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)
(1 58 7 52)(2 51 8 57)(3 56 9 50)(4 49 10 55)(5 54 11 60)(6 59 12 53)(13 35 19 29)(14 28 20 34)(15 33 21 27)(16 26 22 32)(17 31 23 25)(18 36 24 30)(37 67 43 61)(38 72 44 66)(39 65 45 71)(40 70 46 64)(41 63 47 69)(42 68 48 62)(73 87 79 93)(74 92 80 86)(75 85 81 91)(76 90 82 96)(77 95 83 89)(78 88 84 94)```

`G:=sub<Sym(96)| (1,59,7,53)(2,60,8,54)(3,49,9,55)(4,50,10,56)(5,51,11,57)(6,52,12,58)(13,87,19,93)(14,88,20,94)(15,89,21,95)(16,90,22,96)(17,91,23,85)(18,92,24,86)(25,75,31,81)(26,76,32,82)(27,77,33,83)(28,78,34,84)(29,79,35,73)(30,80,36,74)(37,62,43,68)(38,63,44,69)(39,64,45,70)(40,65,46,71)(41,66,47,72)(42,67,48,61), (1,90,65,27)(2,85,66,34)(3,92,67,29)(4,87,68,36)(5,94,69,31)(6,89,70,26)(7,96,71,33)(8,91,72,28)(9,86,61,35)(10,93,62,30)(11,88,63,25)(12,95,64,32)(13,43,80,56)(14,38,81,51)(15,45,82,58)(16,40,83,53)(17,47,84,60)(18,42,73,55)(19,37,74,50)(20,44,75,57)(21,39,76,52)(22,46,77,59)(23,41,78,54)(24,48,79,49), (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), (1,58,7,52)(2,51,8,57)(3,56,9,50)(4,49,10,55)(5,54,11,60)(6,59,12,53)(13,35,19,29)(14,28,20,34)(15,33,21,27)(16,26,22,32)(17,31,23,25)(18,36,24,30)(37,67,43,61)(38,72,44,66)(39,65,45,71)(40,70,46,64)(41,63,47,69)(42,68,48,62)(73,87,79,93)(74,92,80,86)(75,85,81,91)(76,90,82,96)(77,95,83,89)(78,88,84,94)>;`

`G:=Group( (1,59,7,53)(2,60,8,54)(3,49,9,55)(4,50,10,56)(5,51,11,57)(6,52,12,58)(13,87,19,93)(14,88,20,94)(15,89,21,95)(16,90,22,96)(17,91,23,85)(18,92,24,86)(25,75,31,81)(26,76,32,82)(27,77,33,83)(28,78,34,84)(29,79,35,73)(30,80,36,74)(37,62,43,68)(38,63,44,69)(39,64,45,70)(40,65,46,71)(41,66,47,72)(42,67,48,61), (1,90,65,27)(2,85,66,34)(3,92,67,29)(4,87,68,36)(5,94,69,31)(6,89,70,26)(7,96,71,33)(8,91,72,28)(9,86,61,35)(10,93,62,30)(11,88,63,25)(12,95,64,32)(13,43,80,56)(14,38,81,51)(15,45,82,58)(16,40,83,53)(17,47,84,60)(18,42,73,55)(19,37,74,50)(20,44,75,57)(21,39,76,52)(22,46,77,59)(23,41,78,54)(24,48,79,49), (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), (1,58,7,52)(2,51,8,57)(3,56,9,50)(4,49,10,55)(5,54,11,60)(6,59,12,53)(13,35,19,29)(14,28,20,34)(15,33,21,27)(16,26,22,32)(17,31,23,25)(18,36,24,30)(37,67,43,61)(38,72,44,66)(39,65,45,71)(40,70,46,64)(41,63,47,69)(42,68,48,62)(73,87,79,93)(74,92,80,86)(75,85,81,91)(76,90,82,96)(77,95,83,89)(78,88,84,94) );`

`G=PermutationGroup([(1,59,7,53),(2,60,8,54),(3,49,9,55),(4,50,10,56),(5,51,11,57),(6,52,12,58),(13,87,19,93),(14,88,20,94),(15,89,21,95),(16,90,22,96),(17,91,23,85),(18,92,24,86),(25,75,31,81),(26,76,32,82),(27,77,33,83),(28,78,34,84),(29,79,35,73),(30,80,36,74),(37,62,43,68),(38,63,44,69),(39,64,45,70),(40,65,46,71),(41,66,47,72),(42,67,48,61)], [(1,90,65,27),(2,85,66,34),(3,92,67,29),(4,87,68,36),(5,94,69,31),(6,89,70,26),(7,96,71,33),(8,91,72,28),(9,86,61,35),(10,93,62,30),(11,88,63,25),(12,95,64,32),(13,43,80,56),(14,38,81,51),(15,45,82,58),(16,40,83,53),(17,47,84,60),(18,42,73,55),(19,37,74,50),(20,44,75,57),(21,39,76,52),(22,46,77,59),(23,41,78,54),(24,48,79,49)], [(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)], [(1,58,7,52),(2,51,8,57),(3,56,9,50),(4,49,10,55),(5,54,11,60),(6,59,12,53),(13,35,19,29),(14,28,20,34),(15,33,21,27),(16,26,22,32),(17,31,23,25),(18,36,24,30),(37,67,43,61),(38,72,44,66),(39,65,45,71),(40,70,46,64),(41,63,47,69),(42,68,48,62),(73,87,79,93),(74,92,80,86),(75,85,81,91),(76,90,82,96),(77,95,83,89),(78,88,84,94)])`

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4N 6A 6B 6C 6D 6E 12A 12B 12C 12D 12E ··· 12N order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 ··· 4 6 6 6 6 6 12 12 12 12 12 ··· 12 size 1 1 1 1 2 2 12 12 2 2 2 2 2 4 4 4 4 12 ··· 12 2 2 2 4 4 2 2 2 2 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 D6 D12 2- 1+4 Q8○D12 kernel C42.92D6 C12⋊2Q8 C42⋊7S3 C23.21D6 C4.D12 C3×C42⋊C2 C22×Dic6 C2×C4○D12 C42⋊C2 C2×C12 C42 C22⋊C4 C4⋊C4 C22×C4 C2×C4 C6 C2 # reps 1 2 2 4 4 1 1 1 1 4 2 2 2 1 8 2 4

Matrix representation of C42.92D6 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 10 7 0 0 0 0 6 3 0 0 0 0 0 0 10 7 0 0 0 0 6 3
,
 1 8 0 0 0 0 3 12 0 0 0 0 0 0 2 4 2 0 0 0 9 11 0 2 0 0 12 0 11 9 0 0 0 12 4 2
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 8 11 8 8 0 0 2 10 5 0 0 0 6 7 5 2 0 0 6 12 11 3
,
 12 0 0 0 0 0 10 1 0 0 0 0 0 0 3 11 11 2 0 0 8 10 4 2 0 0 12 1 11 3 0 0 2 1 5 2

`G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,10,6,0,0,0,0,7,3,0,0,0,0,0,0,10,6,0,0,0,0,7,3],[1,3,0,0,0,0,8,12,0,0,0,0,0,0,2,9,12,0,0,0,4,11,0,12,0,0,2,0,11,4,0,0,0,2,9,2],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,2,6,6,0,0,11,10,7,12,0,0,8,5,5,11,0,0,8,0,2,3],[12,10,0,0,0,0,0,1,0,0,0,0,0,0,3,8,12,2,0,0,11,10,1,1,0,0,11,4,11,5,0,0,2,2,3,2] >;`

C42.92D6 in GAP, Magma, Sage, TeX

`C_4^2._{92}D_6`
`% in TeX`

`G:=Group("C4^2.92D6");`
`// GroupNames label`

`G:=SmallGroup(192,1085);`
`// by ID`

`G=gap.SmallGroup(192,1085);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,675,570,297,192,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=a^2*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;`
`// generators/relations`

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