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## G = C6.102+ 1+4order 192 = 26·3

### 10th 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 — C6.102+ 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — S3×C4⋊C4 — C6.102+ 1+4
 Lower central C3 — C2×C6 — C6.102+ 1+4
 Upper central C1 — C22 — C2×C4⋊C4

Generators and relations for C6.102+ 1+4
G = < a,b,c,d,e | a6=b4=c2=1, d2=b2, e2=a3, bab-1=a-1, ac=ca, ad=da, ae=ea, cbc=a3b-1, dbd-1=a3b, be=eb, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 520 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, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C22×S3, C22×C6, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4×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, C6.D4, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C2×Dic6, S3×C2×C4, S3×C2×C4, C2×C3⋊D4, C22×C12, C22×C12, D43Q8, Dic6⋊C4, C12⋊Q8, Dic3.Q8, S3×C4⋊C4, D6⋊Q8, C4.D12, C12.48D4, C12.48D4, C4×C3⋊D4, C4×C3⋊D4, C6×C4⋊C4, C6.102+ 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, C4○D12, S3×Q8, S3×C23, D43Q8, C2×C4○D12, D46D6, C2×S3×Q8, C6.102+ 1+4

Smallest permutation representation of C6.102+ 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 12 17 93)(2 11 18 92)(3 10 13 91)(4 9 14 96)(5 8 15 95)(6 7 16 94)(19 83 30 88)(20 82 25 87)(21 81 26 86)(22 80 27 85)(23 79 28 90)(24 84 29 89)(31 76 42 71)(32 75 37 70)(33 74 38 69)(34 73 39 68)(35 78 40 67)(36 77 41 72)(43 56 54 61)(44 55 49 66)(45 60 50 65)(46 59 51 64)(47 58 52 63)(48 57 53 62)
(1 24)(2 19)(3 20)(4 21)(5 22)(6 23)(7 87)(8 88)(9 89)(10 90)(11 85)(12 86)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(37 49)(38 50)(39 51)(40 52)(41 53)(42 54)(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 91)(80 92)(81 93)(82 94)(83 95)(84 96)
(1 77 17 72)(2 78 18 67)(3 73 13 68)(4 74 14 69)(5 75 15 70)(6 76 16 71)(7 39 94 34)(8 40 95 35)(9 41 96 36)(10 42 91 31)(11 37 92 32)(12 38 93 33)(19 66 30 55)(20 61 25 56)(21 62 26 57)(22 63 27 58)(23 64 28 59)(24 65 29 60)(43 90 54 79)(44 85 49 80)(45 86 50 81)(46 87 51 82)(47 88 52 83)(48 89 53 84)
(1 86 4 89)(2 87 5 90)(3 88 6 85)(7 22 10 19)(8 23 11 20)(9 24 12 21)(13 83 16 80)(14 84 17 81)(15 79 18 82)(25 95 28 92)(26 96 29 93)(27 91 30 94)(31 66 34 63)(32 61 35 64)(33 62 36 65)(37 56 40 59)(38 57 41 60)(39 58 42 55)(43 78 46 75)(44 73 47 76)(45 74 48 77)(49 68 52 71)(50 69 53 72)(51 70 54 67)```

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

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

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

45 conjugacy classes

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

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + - + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 Q8 D6 D6 C4○D4 C4○D12 2+ 1+4 S3×Q8 D4⋊6D6 kernel C6.102+ 1+4 Dic6⋊C4 C12⋊Q8 Dic3.Q8 S3×C4⋊C4 D6⋊Q8 C4.D12 C12.48D4 C4×C3⋊D4 C6×C4⋊C4 C2×C4⋊C4 C3⋊D4 C4⋊C4 C22×C4 C12 C4 C6 C22 C2 # reps 1 1 1 2 1 2 1 3 3 1 1 4 4 3 4 8 1 2 2

Matrix representation of C6.102+ 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
,
 4 11 0 0 0 0 1 9 0 0 0 0 0 0 12 0 0 0 0 0 1 1 0 0 0 0 0 0 1 2 0 0 0 0 12 12
,
 7 3 0 0 0 0 10 6 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 6 10 0 0 0 0 3 7 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 8 3 0 0 0 0 0 5
,
 5 0 0 0 0 0 0 5 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 11 0 0 0 0 1 1

`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],[4,1,0,0,0,0,11,9,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,2,12],[7,10,0,0,0,0,3,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,3,0,0,0,0,10,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,3,5],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,11,1] >;`

C6.102+ 1+4 in GAP, Magma, Sage, TeX

`C_6._{10}2_+^{1+4}`
`% in TeX`

`G:=Group("C6.10ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1070);`
`// by ID`

`G=gap.SmallGroup(192,1070);`
`# by ID`

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

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

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