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## G = C6.752- 1+4order 192 = 26·3

### 30th 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 — C6.752- 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — C23.16D6 — C6.752- 1+4
 Lower central C3 — C2×C6 — C6.752- 1+4
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for C6.752- 1+4
G = < a,b,c,d,e | a6=b4=e2=1, c2=a3, d2=a3b2, bab-1=a-1, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=a3c, ce=ec, ede=b2d >

Subgroups: 448 in 206 conjugacy classes, 103 normal (31 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, C2×C4, Q8, C23, Dic3, Dic3, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×Q8, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×Q8, C22×C6, C2×C4⋊C4, C42⋊C2, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C2×Dic6, C22×Dic3, C22×C12, C6×Q8, C23.41C23, C23.16D6, Dic3.D4, C12⋊Q8, Dic3.Q8, C2×Dic3⋊C4, C12.48D4, Dic3⋊Q8, C3×C22⋊Q8, C6.752- 1+4
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C24, C22×S3, C22×Q8, 2+ 1+4, 2- 1+4, S3×Q8, S3×C23, C23.41C23, D46D6, C2×S3×Q8, Q8○D12, C6.752- 1+4

Smallest permutation representation of C6.752- 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 64 17 60)(2 63 18 59)(3 62 13 58)(4 61 14 57)(5 66 15 56)(6 65 16 55)(7 54 96 48)(8 53 91 47)(9 52 92 46)(10 51 93 45)(11 50 94 44)(12 49 95 43)(19 75 30 71)(20 74 25 70)(21 73 26 69)(22 78 27 68)(23 77 28 67)(24 76 29 72)(31 87 42 83)(32 86 37 82)(33 85 38 81)(34 90 39 80)(35 89 40 79)(36 88 41 84)
(1 21 4 24)(2 22 5 19)(3 23 6 20)(7 81 10 84)(8 82 11 79)(9 83 12 80)(13 28 16 25)(14 29 17 26)(15 30 18 27)(31 43 34 46)(32 44 35 47)(33 45 36 48)(37 50 40 53)(38 51 41 54)(39 52 42 49)(55 74 58 77)(56 75 59 78)(57 76 60 73)(61 72 64 69)(62 67 65 70)(63 68 66 71)(85 93 88 96)(86 94 89 91)(87 95 90 92)
(1 38 14 36)(2 39 15 31)(3 40 16 32)(4 41 17 33)(5 42 18 34)(6 37 13 35)(7 76 93 69)(8 77 94 70)(9 78 95 71)(10 73 96 72)(11 74 91 67)(12 75 92 68)(19 52 27 43)(20 53 28 44)(21 54 29 45)(22 49 30 46)(23 50 25 47)(24 51 26 48)(55 86 62 79)(56 87 63 80)(57 88 64 81)(58 89 65 82)(59 90 66 83)(60 85 61 84)
(7 96)(8 91)(9 92)(10 93)(11 94)(12 95)(31 42)(32 37)(33 38)(34 39)(35 40)(36 41)(43 49)(44 50)(45 51)(46 52)(47 53)(48 54)(79 89)(80 90)(81 85)(82 86)(83 87)(84 88)```

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + - + + + + + - - - image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 Q8 D6 D6 D6 D6 2+ 1+4 2- 1+4 S3×Q8 D4⋊6D6 Q8○D12 kernel C6.752- 1+4 C23.16D6 Dic3.D4 C12⋊Q8 Dic3.Q8 C2×Dic3⋊C4 C12.48D4 Dic3⋊Q8 C3×C22⋊Q8 C22⋊Q8 C2×Dic3 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C6 C6 C22 C2 C2 # reps 1 2 2 2 4 1 1 2 1 1 4 2 3 1 1 1 1 2 2 2

Matrix representation of C6.752- 1+4 in GL8(𝔽13)

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

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

C6.752- 1+4 in GAP, Magma, Sage, TeX

`C_6._{75}2_-^{1+4}`
`% in TeX`

`G:=Group("C6.75ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1182);`
`// by ID`

`G=gap.SmallGroup(192,1182);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,387,100,1123,185,80,6278]);`
`// Polycyclic`

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

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