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## G = C6.1182+ 1+4order 192 = 26·3

### 27th non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C4○D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C6.1182+ 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — C2×C3⋊D4 — C4×C3⋊D4 — C6.1182+ 1+4
 Lower central C3 — C2×C6 — C6.1182+ 1+4
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for C6.1182+ 1+4
G = < a,b,c,d,e | a6=b4=c2=1, d2=b2, e2=a3, ab=ba, cac=a-1, ad=da, ae=ea, cbc=a3b-1, bd=db, ebe-1=a3b, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 528 in 228 conjugacy classes, 107 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×Q8, C22×S3, C22×C6, C2×C4⋊C4, C4×D4, C4×Q8, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, C4×Dic3, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C22×C12, C6×Q8, D43Q8, Dic3.D4, Dic34D4, C12⋊Q8, Dic3.Q8, S3×C4⋊C4, C4.D12, C2×C4⋊Dic3, C4×C3⋊D4, Q8×Dic3, D63Q8, C3×C22⋊Q8, C6.1182+ 1+4
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C4○D4, C24, C22×S3, C22×Q8, C2×C4○D4, 2+ 1+4, D42S3, S3×Q8, S3×C23, D43Q8, C2×D42S3, C2×S3×Q8, D4○D12, C6.1182+ 1+4

Smallest permutation representation of C6.1182+ 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 86 15 79)(2 87 16 80)(3 88 17 81)(4 89 18 82)(5 90 13 83)(6 85 14 84)(7 20 92 27)(8 21 93 28)(9 22 94 29)(10 23 95 30)(11 24 96 25)(12 19 91 26)(31 62 38 55)(32 63 39 56)(33 64 40 57)(34 65 41 58)(35 66 42 59)(36 61 37 60)(43 74 50 67)(44 75 51 68)(45 76 52 69)(46 77 53 70)(47 78 54 71)(48 73 49 72)
(2 6)(3 5)(7 93)(8 92)(9 91)(10 96)(11 95)(12 94)(13 17)(14 16)(20 24)(21 23)(25 27)(28 30)(32 36)(33 35)(37 39)(40 42)(44 48)(45 47)(49 51)(52 54)(55 65)(56 64)(57 63)(58 62)(59 61)(60 66)(67 77)(68 76)(69 75)(70 74)(71 73)(72 78)(79 89)(80 88)(81 87)(82 86)(83 85)(84 90)
(1 38 15 31)(2 39 16 32)(3 40 17 33)(4 41 18 34)(5 42 13 35)(6 37 14 36)(7 68 92 75)(8 69 93 76)(9 70 94 77)(10 71 95 78)(11 72 96 73)(12 67 91 74)(19 43 26 50)(20 44 27 51)(21 45 28 52)(22 46 29 53)(23 47 30 54)(24 48 25 49)(55 79 62 86)(56 80 63 87)(57 81 64 88)(58 82 65 89)(59 83 66 90)(60 84 61 85)
(1 46 4 43)(2 47 5 44)(3 48 6 45)(7 66 10 63)(8 61 11 64)(9 62 12 65)(13 51 16 54)(14 52 17 49)(15 53 18 50)(19 38 22 41)(20 39 23 42)(21 40 24 37)(25 36 28 33)(26 31 29 34)(27 32 30 35)(55 91 58 94)(56 92 59 95)(57 93 60 96)(67 82 70 79)(68 83 71 80)(69 84 72 81)(73 88 76 85)(74 89 77 86)(75 90 78 87)```

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

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 1 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 C2 S3 Q8 D6 D6 D6 D6 C4○D4 2+ 1+4 D4⋊2S3 S3×Q8 D4○D12 kernel C6.1182+ 1+4 Dic3.D4 Dic3⋊4D4 C12⋊Q8 Dic3.Q8 S3×C4⋊C4 C4.D12 C2×C4⋊Dic3 C4×C3⋊D4 Q8×Dic3 D6⋊3Q8 C3×C22⋊Q8 C22⋊Q8 C3⋊D4 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C12 C6 C4 C22 C2 # reps 1 2 2 1 2 1 2 1 1 1 1 1 1 4 2 3 1 1 4 1 2 2 2

Matrix representation of C6.1182+ 1+4 in GL6(𝔽13)

 12 12 0 0 0 0 1 0 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 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 8 5 0 0 0 0 3 5 0 0 0 0 0 0 1 12 0 0 0 0 2 12
,
 1 0 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 2 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 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 1 12 0 0 0 0 2 12
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 5 0 0 0 0 0 10 8 0 0 0 0 0 0 5 0 0 0 0 0 10 8

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

C6.1182+ 1+4 in GAP, Magma, Sage, TeX

`C_6._{118}2_+^{1+4}`
`% in TeX`

`G:=Group("C6.118ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1194);`
`// by ID`

`G=gap.SmallGroup(192,1194);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,100,570,185,192,6278]);`
`// Polycyclic`

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

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