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## G = C6.522+ 1+4order 192 = 26·3

### 52nd 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 — C6.522+ 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — C2×C3⋊D4 — C4×C3⋊D4 — C6.522+ 1+4
 Lower central C3 — C2×C6 — C6.522+ 1+4
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for C6.522+ 1+4
G = < a,b,c,d,e | a6=b4=e2=1, c2=a3, d2=b2, ab=ba, ac=ca, dad-1=eae=a-1, cbc-1=b-1, dbd-1=a3b, be=eb, cd=dc, ce=ec, ede=a3b2d >

Subgroups: 528 in 228 conjugacy classes, 105 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, 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, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C22×C12, C6×Q8, D43Q8, Dic3.D4, Dic34D4, Dic6⋊C4, Dic3.Q8, S3×C4⋊C4, D6⋊Q8, C2×Dic3⋊C4, C4×C3⋊D4, Dic3⋊Q8, D63Q8, C3×C22⋊Q8, C6.522+ 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, S3×Q8, S3×C23, D43Q8, D46D6, C2×S3×Q8, S3×C4○D4, C6.522+ 1+4

Smallest permutation representation of C6.522+ 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 53 14 45)(2 54 15 46)(3 49 16 47)(4 50 17 48)(5 51 18 43)(6 52 13 44)(7 55 91 63)(8 56 92 64)(9 57 93 65)(10 58 94 66)(11 59 95 61)(12 60 96 62)(19 39 27 31)(20 40 28 32)(21 41 29 33)(22 42 30 34)(23 37 25 35)(24 38 26 36)(67 82 75 90)(68 83 76 85)(69 84 77 86)(70 79 78 87)(71 80 73 88)(72 81 74 89)
(1 33 4 36)(2 34 5 31)(3 35 6 32)(7 75 10 78)(8 76 11 73)(9 77 12 74)(13 40 16 37)(14 41 17 38)(15 42 18 39)(19 46 22 43)(20 47 23 44)(21 48 24 45)(25 52 28 49)(26 53 29 50)(27 54 30 51)(55 82 58 79)(56 83 59 80)(57 84 60 81)(61 88 64 85)(62 89 65 86)(63 90 66 87)(67 94 70 91)(68 95 71 92)(69 96 72 93)
(1 81 14 89)(2 80 15 88)(3 79 16 87)(4 84 17 86)(5 83 18 85)(6 82 13 90)(7 20 91 28)(8 19 92 27)(9 24 93 26)(10 23 94 25)(11 22 95 30)(12 21 96 29)(31 59 39 61)(32 58 40 66)(33 57 41 65)(34 56 42 64)(35 55 37 63)(36 60 38 62)(43 71 51 73)(44 70 52 78)(45 69 53 77)(46 68 54 76)(47 67 49 75)(48 72 50 74)
(2 6)(3 5)(7 92)(8 91)(9 96)(10 95)(11 94)(12 93)(13 15)(16 18)(19 23)(20 22)(25 27)(28 30)(31 35)(32 34)(37 39)(40 42)(43 47)(44 46)(49 51)(52 54)(55 64)(56 63)(57 62)(58 61)(59 66)(60 65)(67 76)(68 75)(69 74)(70 73)(71 78)(72 77)(79 88)(80 87)(81 86)(82 85)(83 90)(84 89)```

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

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

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

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 S3×Q8 D4⋊6D6 S3×C4○D4 kernel C6.522+ 1+4 Dic3.D4 Dic3⋊4D4 Dic6⋊C4 Dic3.Q8 S3×C4⋊C4 D6⋊Q8 C2×Dic3⋊C4 C4×C3⋊D4 Dic3⋊Q8 D6⋊3Q8 C3×C22⋊Q8 C22⋊Q8 C3⋊D4 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 Dic3 C6 C22 C2 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 C6.522+ 1+4 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 12 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 0 8 0 0 0 0 8 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 6 1
,
 8 0 0 0 0 0 0 5 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 8 0 0 0 0 0 0 8
,
 5 0 0 0 0 0 0 8 0 0 0 0 0 0 12 0 0 0 0 0 1 1 0 0 0 0 0 0 1 9 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 7 12

`G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,6,0,0,0,0,0,1],[8,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[5,0,0,0,0,0,0,8,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,9,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,7,0,0,0,0,0,12] >;`

C6.522+ 1+4 in GAP, Magma, Sage, TeX

`C_6._{52}2_+^{1+4}`
`% in TeX`

`G:=Group("C6.52ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1195);`
`// by ID`

`G=gap.SmallGroup(192,1195);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,100,570,409,80,6278]);`
`// Polycyclic`

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

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