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## G = C4⋊C4.237D6order 192 = 26·3

### 15th non-split extension by C4⋊C4 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C4⋊C4.237D6
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×Dic6 — C22×Dic6 — C4⋊C4.237D6
 Lower central C3 — C6 — C12 — C4⋊C4.237D6
 Upper central C1 — C22 — C22×C4 — C42⋊C2

Generators and relations for C4⋊C4.237D6
G = < a,b,c,d | a4=b4=c6=1, d2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=ab-1, dcd-1=c-1 >

Subgroups: 360 in 150 conjugacy classes, 63 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×2], C6, C6 [×2], C6 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×8], Q8 [×10], C23, Dic3 [×4], C12 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×2], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], M4(2) [×2], C22×C4, C22×C4, C2×Q8 [×9], C3⋊C8 [×2], Dic6 [×4], Dic6 [×6], C2×Dic3 [×6], C2×C12 [×2], C2×C12 [×4], C2×C12 [×2], C22×C6, Q8⋊C4 [×4], C42⋊C2, C2×M4(2), C22×Q8, C2×C3⋊C8 [×2], C4.Dic3 [×2], C4×C12, C3×C22⋊C4, C3×C4⋊C4 [×2], C2×Dic6 [×6], C2×Dic6 [×3], C22×Dic3, C22×C12, C23.38D4, C6.SD16 [×4], C2×C4.Dic3, C3×C42⋊C2, C22×Dic6, C4⋊C4.237D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C22×S3, C2×C22⋊C4, C8.C22 [×2], D6⋊C4 [×4], S3×C2×C4, C2×D12, C2×C3⋊D4, C23.38D4, C2×D6⋊C4, Q8.14D6 [×2], C4⋊C4.237D6

Smallest permutation representation of C4⋊C4.237D6
On 96 points
Generators in S96
```(1 19 4 22)(2 20 5 23)(3 21 6 24)(7 16 10 13)(8 17 11 14)(9 18 12 15)(25 36 39 32)(26 34 37 33)(27 35 38 31)(28 46 44 42)(29 47 45 40)(30 48 43 41)(49 96 52 93)(50 91 53 94)(51 92 54 95)(55 87 58 90)(56 88 59 85)(57 89 60 86)(61 69 64 72)(62 70 65 67)(63 71 66 68)(73 82 76 79)(74 83 77 80)(75 84 78 81)
(1 91 16 70)(2 95 17 68)(3 93 18 72)(4 94 13 67)(5 92 14 71)(6 96 15 69)(7 65 22 53)(8 63 23 51)(9 61 24 49)(10 62 19 50)(11 66 20 54)(12 64 21 52)(25 58 40 81)(26 56 41 79)(27 60 42 83)(28 74 35 89)(29 78 36 87)(30 76 34 85)(31 86 44 77)(32 90 45 75)(33 88 43 73)(37 59 48 82)(38 57 46 80)(39 55 47 84)
(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 33 4 34)(2 32 5 36)(3 31 6 35)(7 41 10 48)(8 40 11 47)(9 42 12 46)(13 30 16 43)(14 29 17 45)(15 28 18 44)(19 37 22 26)(20 39 23 25)(21 38 24 27)(49 74 52 77)(50 73 53 76)(51 78 54 75)(55 68 58 71)(56 67 59 70)(57 72 60 69)(61 89 64 86)(62 88 65 85)(63 87 66 90)(79 94 82 91)(80 93 83 96)(81 92 84 95)```

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A 6B 6C 6D 6E 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12N order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 6 6 8 8 8 8 12 12 12 12 12 ··· 12 size 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 12 12 12 12 2 2 2 4 4 12 12 12 12 2 2 2 2 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C4 S3 D4 D4 D6 D6 C4×S3 D12 C3⋊D4 C3⋊D4 C8.C22 Q8.14D6 kernel C4⋊C4.237D6 C6.SD16 C2×C4.Dic3 C3×C42⋊C2 C22×Dic6 C2×Dic6 C42⋊C2 C2×C12 C22×C6 C4⋊C4 C22×C4 C2×C4 C2×C4 C2×C4 C23 C6 C2 # reps 1 4 1 1 1 8 1 3 1 2 1 4 4 2 2 2 4

Matrix representation of C4⋊C4.237D6 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 7 14 0 0 0 0 59 66 0 0 0 0 57 12 7 59 0 0 45 57 14 66
,
 27 0 0 0 0 0 0 27 0 0 0 0 0 0 11 50 48 48 0 0 23 34 50 48 0 0 8 9 39 23 0 0 72 8 50 62
,
 0 1 0 0 0 0 72 1 0 0 0 0 0 0 72 72 0 0 0 0 1 0 0 0 0 0 28 12 0 1 0 0 16 28 72 1
,
 3 28 0 0 0 0 31 70 0 0 0 0 0 0 30 43 0 0 0 0 13 43 0 0 0 0 67 26 46 27 0 0 26 32 0 27

`G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,7,59,57,45,0,0,14,66,12,57,0,0,0,0,7,14,0,0,0,0,59,66],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,11,23,8,72,0,0,50,34,9,8,0,0,48,50,39,50,0,0,48,48,23,62],[0,72,0,0,0,0,1,1,0,0,0,0,0,0,72,1,28,16,0,0,72,0,12,28,0,0,0,0,0,72,0,0,0,0,1,1],[3,31,0,0,0,0,28,70,0,0,0,0,0,0,30,13,67,26,0,0,43,43,26,32,0,0,0,0,46,0,0,0,0,0,27,27] >;`

C4⋊C4.237D6 in GAP, Magma, Sage, TeX

`C_4\rtimes C_4._{237}D_6`
`% in TeX`

`G:=Group("C4:C4.237D6");`
`// GroupNames label`

`G:=SmallGroup(192,563);`
`// by ID`

`G=gap.SmallGroup(192,563);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,422,387,58,1684,438,102,6278]);`
`// Polycyclic`

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

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