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## G = C6.1482+ 1+4order 192 = 26·3

### 57th 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 — C6.1482+ 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — C2×C3⋊D4 — D6⋊3D4 — C6.1482+ 1+4
 Lower central C3 — C2×C6 — C6.1482+ 1+4
 Upper central C1 — C22 — C2×C4○D4

Generators and relations for C6.1482+ 1+4
G = < a,b,c,d,e | a6=b4=c2=e2=1, d2=b2, ab=ba, ac=ca, ad=da, eae=a-1, cbc=b-1, bd=db, be=eb, cd=dc, ece=a3c, ede=a3b2d >

Subgroups: 824 in 312 conjugacy classes, 113 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×C6, C4×D4, C4×Q8, C4⋊D4, C41D4, C2×C4○D4, C2×C4○D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, S3×C2×C4, C2×D12, Q83S3, C2×C3⋊D4, C22×C12, C6×D4, C6×Q8, C3×C4○D4, Q86D4, C4×C3⋊D4, C127D4, D63D4, C123D4, Q8×Dic3, C2×Q83S3, C6×C4○D4, C6.1482+ 1+4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C3⋊D4, C22×S3, C22×D4, C2×C4○D4, 2+ 1+4, C2×C3⋊D4, S3×C23, Q86D4, S3×C4○D4, D4○D12, C22×C3⋊D4, C6.1482+ 1+4

Smallest permutation representation of C6.1482+ 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 37 7 43)(2 38 8 44)(3 39 9 45)(4 40 10 46)(5 41 11 47)(6 42 12 48)(13 31 19 25)(14 32 20 26)(15 33 21 27)(16 34 22 28)(17 35 23 29)(18 36 24 30)(49 91 55 85)(50 92 56 86)(51 93 57 87)(52 94 58 88)(53 95 59 89)(54 96 60 90)(61 73 67 79)(62 74 68 80)(63 75 69 81)(64 76 70 82)(65 77 71 83)(66 78 72 84)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)
(1 19 7 13)(2 20 8 14)(3 21 9 15)(4 22 10 16)(5 23 11 17)(6 24 12 18)(25 43 31 37)(26 44 32 38)(27 45 33 39)(28 46 34 40)(29 47 35 41)(30 48 36 42)(49 67 55 61)(50 68 56 62)(51 69 57 63)(52 70 58 64)(53 71 59 65)(54 72 60 66)(73 91 79 85)(74 92 80 86)(75 93 81 87)(76 94 82 88)(77 95 83 89)(78 96 84 90)
(1 64)(2 63)(3 62)(4 61)(5 66)(6 65)(7 70)(8 69)(9 68)(10 67)(11 72)(12 71)(13 49)(14 54)(15 53)(16 52)(17 51)(18 50)(19 55)(20 60)(21 59)(22 58)(23 57)(24 56)(25 85)(26 90)(27 89)(28 88)(29 87)(30 86)(31 91)(32 96)(33 95)(34 94)(35 93)(36 92)(37 76)(38 75)(39 74)(40 73)(41 78)(42 77)(43 82)(44 81)(45 80)(46 79)(47 84)(48 83)```

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

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

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

45 conjugacy classes

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

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 C4○D4 C3⋊D4 2+ 1+4 S3×C4○D4 D4○D12 kernel C6.1482+ 1+4 C4×C3⋊D4 C12⋊7D4 D6⋊3D4 C12⋊3D4 Q8×Dic3 C2×Q8⋊3S3 C6×C4○D4 C2×C4○D4 C3×Q8 C22×C4 C2×D4 C2×Q8 Dic3 Q8 C6 C2 C2 # reps 1 3 3 3 3 1 1 1 1 4 3 3 1 4 8 1 2 2

Matrix representation of C6.1482+ 1+4 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 1 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 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 5 0 0 0 0 5 0
,
 0 1 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 0 8 0 0 0 0 5 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 5 0 0 0 0 0 0 5
,
 12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0

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

C6.1482+ 1+4 in GAP, Magma, Sage, TeX

`C_6._{148}2_+^{1+4}`
`% in TeX`

`G:=Group("C6.148ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1393);`
`// by ID`

`G=gap.SmallGroup(192,1393);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,758,219,1571,80,6278]);`
`// Polycyclic`

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

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