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## G = C6.432+ 1+4order 192 = 26·3

### 43rd non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C6.432+ 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — S3×C4⋊C4 — C6.432+ 1+4
 Lower central C3 — C2×C6 — C6.432+ 1+4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for C6.432+ 1+4
G = < a,b,c,d,e | a6=b4=c2=1, d2=a3b2, e2=a3, bab-1=a-1, ac=ca, ad=da, ae=ea, cbc=a3b-1, bd=db, ebe-1=a3b, cd=dc, ece-1=a3c, ede-1=b2d >

Subgroups: 576 in 238 conjugacy classes, 97 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C4×S3, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C2×C4⋊C4, C42⋊C2, C4×D4, C4⋊D4, C4⋊D4, C22.D4, C42.C2, C422C2, C4×Dic3, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C6.D4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C6×D4, C6×D4, C22.47C24, C23.8D6, Dic34D4, C4.Dic6, S3×C4⋊C4, C23.26D6, C4×C3⋊D4, D4×Dic3, C23.23D6, D63D4, D63D4, C3×C4⋊D4, C6.432+ 1+4
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, D42S3, S3×C23, C22.47C24, C2×D42S3, D46D6, S3×C4○D4, C6.432+ 1+4

Smallest permutation representation of C6.432+ 1+4
On 96 points
Generators in S96
```(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 9 14 91)(2 8 15 96)(3 7 16 95)(4 12 17 94)(5 11 18 93)(6 10 13 92)(19 82 29 86)(20 81 30 85)(21 80 25 90)(22 79 26 89)(23 84 27 88)(24 83 28 87)(31 74 41 70)(32 73 42 69)(33 78 37 68)(34 77 38 67)(35 76 39 72)(36 75 40 71)(43 58 53 62)(44 57 54 61)(45 56 49 66)(46 55 50 65)(47 60 51 64)(48 59 52 63)
(1 22)(2 23)(3 24)(4 19)(5 20)(6 21)(7 90)(8 85)(9 86)(10 87)(11 88)(12 89)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)(37 52)(38 53)(39 54)(40 49)(41 50)(42 51)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(61 73)(62 74)(63 75)(64 76)(65 77)(66 78)(79 94)(80 95)(81 96)(82 91)(83 92)(84 93)
(1 77 17 70)(2 78 18 71)(3 73 13 72)(4 74 14 67)(5 75 15 68)(6 76 16 69)(7 42 92 35)(8 37 93 36)(9 38 94 31)(10 39 95 32)(11 40 96 33)(12 41 91 34)(19 62 26 55)(20 63 27 56)(21 64 28 57)(22 65 29 58)(23 66 30 59)(24 61 25 60)(43 89 50 82)(44 90 51 83)(45 85 52 84)(46 86 53 79)(47 87 54 80)(48 88 49 81)
(1 86 4 89)(2 87 5 90)(3 88 6 85)(7 20 10 23)(8 21 11 24)(9 22 12 19)(13 81 16 84)(14 82 17 79)(15 83 18 80)(25 93 28 96)(26 94 29 91)(27 95 30 92)(31 62 34 65)(32 63 35 66)(33 64 36 61)(37 60 40 57)(38 55 41 58)(39 56 42 59)(43 74 46 77)(44 75 47 78)(45 76 48 73)(49 72 52 69)(50 67 53 70)(51 68 54 71)```

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

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

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

39 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 3 4A 4B 4C 4D 4E 4F 4G ··· 4L 4M 4N 4O 4P 6A 6B 6C 6D 6E 6F 6G 12A 12B 12C 12D 12E 12F order 1 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 ··· 4 4 4 4 4 6 6 6 6 6 6 6 12 12 12 12 12 12 size 1 1 1 1 4 4 4 6 6 2 2 2 2 2 4 4 6 ··· 6 12 12 12 12 2 2 2 4 4 8 8 4 4 4 4 8 8

39 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D6 D6 D6 D6 C4○D4 C4○D4 2+ 1+4 D4⋊2S3 D4⋊6D6 S3×C4○D4 kernel C6.432+ 1+4 C23.8D6 Dic3⋊4D4 C4.Dic6 S3×C4⋊C4 C23.26D6 C4×C3⋊D4 D4×Dic3 C23.23D6 D6⋊3D4 C3×C4⋊D4 C4⋊D4 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C12 D6 C6 C4 C2 C2 # reps 1 2 2 1 1 1 1 1 2 3 1 1 2 1 1 3 4 4 1 2 2 2

Matrix representation of C6.432+ 1+4 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 1 12
,
 0 5 0 0 0 0 5 0 0 0 0 0 0 0 11 4 0 0 0 0 9 2 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 0 1 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 1 0 0 0 0 0 0 1
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 5 0 0 0 0 0 0 8 0 0 0 0 0 0 9 2 0 0 0 0 11 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1

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

C6.432+ 1+4 in GAP, Magma, Sage, TeX

`C_6._{43}2_+^{1+4}`
`% in TeX`

`G:=Group("C6.43ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1173);`
`// by ID`

`G=gap.SmallGroup(192,1173);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,100,1571,185,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^6=b^4=c^2=1,d^2=a^3*b^2,e^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=a^3*b^-1,b*d=d*b,e*b*e^-1=a^3*b,c*d=d*c,e*c*e^-1=a^3*c,e*d*e^-1=b^2*d>;`
`// generators/relations`

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