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## G = C62.84C23order 288 = 25·32

### 79th non-split extension by C62 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.84C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×D6⋊S3 — C62.84C23
 Lower central C32 — C62 — C62.84C23
 Upper central C1 — C22 — C2×C4

Generators and relations for C62.84C23
G = < a,b,c,d,e | a6=b6=c2=d2=1, e2=b3, ab=ba, ac=ca, dad=a-1, ae=ea, cbc=b-1, bd=db, be=eb, dcd=a3c, ece-1=b3c, ede-1=b3d >

Subgroups: 970 in 231 conjugacy classes, 56 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×12], S3 [×4], C6 [×6], C6 [×7], C2×C4, C2×C4 [×2], D4 [×12], C23 [×4], C32, Dic3 [×12], C12 [×4], C12 [×2], D6 [×12], C2×C6 [×2], C2×C6 [×13], C42, C2×D4 [×6], C3×S3 [×4], C3×C6, C3×C6 [×2], D12 [×4], C2×Dic3 [×6], C3⋊D4 [×16], C2×C12 [×2], C2×C12, C3×D4 [×4], C22×S3 [×4], C22×C6 [×4], C41D4, C3⋊Dic3 [×4], C3×C12 [×2], S3×C6 [×12], C62, C4×Dic3 [×3], C2×D12 [×2], C2×C3⋊D4 [×8], C6×D4 [×2], D6⋊S3 [×8], C3×D12 [×4], C2×C3⋊Dic3 [×2], C6×C12, S3×C2×C6 [×4], C123D4 [×2], C4×C3⋊Dic3, C2×D6⋊S3 [×4], C6×D12 [×2], C62.84C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×6], C23, D6 [×6], C2×D4 [×3], C3⋊D4 [×4], C22×S3 [×2], C41D4, S32, S3×D4 [×4], C2×C3⋊D4 [×2], D6⋊S3 [×2], C2×S32, C123D4 [×2], D6⋊D6 [×2], C2×D6⋊S3, C62.84C23

Smallest permutation representation of C62.84C23
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 18 5 16 3 14)(2 13 6 17 4 15)(7 95 11 93 9 91)(8 96 12 94 10 92)(19 30 23 28 21 26)(20 25 24 29 22 27)(31 40 33 42 35 38)(32 41 34 37 36 39)(43 53 45 49 47 51)(44 54 46 50 48 52)(55 63 57 65 59 61)(56 64 58 66 60 62)(67 74 69 76 71 78)(68 75 70 77 72 73)(79 86 83 90 81 88)(80 87 84 85 82 89)
(1 60)(2 55)(3 56)(4 57)(5 58)(6 59)(7 51)(8 52)(9 53)(10 54)(11 49)(12 50)(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 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 75)(8 74)(9 73)(10 78)(11 77)(12 76)(13 38)(14 37)(15 42)(16 41)(17 40)(18 39)(19 47)(20 46)(21 45)(22 44)(23 43)(24 48)(25 50)(26 49)(27 54)(28 53)(29 52)(30 51)(55 80)(56 79)(57 84)(58 83)(59 82)(60 81)(61 89)(62 88)(63 87)(64 86)(65 85)(66 90)(67 92)(68 91)(69 96)(70 95)(71 94)(72 93)
(1 27 16 24)(2 28 17 19)(3 29 18 20)(4 30 13 21)(5 25 14 22)(6 26 15 23)(7 86 93 81)(8 87 94 82)(9 88 95 83)(10 89 96 84)(11 90 91 79)(12 85 92 80)(31 43 42 49)(32 44 37 50)(33 45 38 51)(34 46 39 52)(35 47 40 53)(36 48 41 54)(55 67 65 76)(56 68 66 77)(57 69 61 78)(58 70 62 73)(59 71 63 74)(60 72 64 75)```

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G 6H 6I 6J ··· 6Q 12A ··· 12H order 1 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 6 ··· 6 6 6 6 6 ··· 6 12 ··· 12 size 1 1 1 1 12 12 12 12 2 2 4 2 2 18 18 18 18 2 ··· 2 4 4 4 12 ··· 12 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + - + image C1 C2 C2 C2 S3 D4 D4 D6 D6 C3⋊D4 S32 S3×D4 D6⋊S3 C2×S32 D6⋊D6 kernel C62.84C23 C4×C3⋊Dic3 C2×D6⋊S3 C6×D12 C2×D12 C3⋊Dic3 C3×C12 C2×C12 C22×S3 C12 C2×C4 C6 C4 C22 C2 # reps 1 1 4 2 2 4 2 2 4 8 1 4 2 1 4

Matrix representation of C62.84C23 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 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
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 12 12
,
 7 11 0 0 0 0 11 6 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 6 2 0 0 0 0 2 7 0 0 0 0 0 0 0 12 0 0 0 0 12 0 0 0 0 0 0 0 2 4 0 0 0 0 9 11
,
 0 12 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 1 0 0 0 0 0 0 1

`G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[7,11,0,0,0,0,11,6,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[6,2,0,0,0,0,2,7,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,2,9,0,0,0,0,4,11],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;`

C62.84C23 in GAP, Magma, Sage, TeX

`C_6^2._{84}C_2^3`
`% in TeX`

`G:=Group("C6^2.84C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,562);`
`// by ID`

`G=gap.SmallGroup(288,562);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,422,219,100,1356,9414]);`
`// Polycyclic`

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

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