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## G = C32⋊82+ 1+4order 288 = 25·32

### 2nd semidirect product of C32 and 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C32⋊82+ 1+4
 Chief series C1 — C3 — C32 — C3×C6 — C2×C3⋊S3 — C22×C3⋊S3 — D4×C3⋊S3 — C32⋊82+ 1+4
 Lower central C32 — C3×C6 — C32⋊82+ 1+4
 Upper central C1 — C2 — C2×D4

Generators and relations for C3282+ 1+4
G = < a,b,c,d,e,f | a3=b3=c4=d2=f2=1, e2=c2, ab=ba, ac=ca, ad=da, eae-1=a-1, af=fa, bc=cb, bd=db, ebe-1=b-1, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef=c2e >

Subgroups: 1860 in 498 conjugacy classes, 153 normal (11 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×D4, C2×D4, C4○D4, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, 2+ 1+4, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, C4○D12, S3×D4, D42S3, C2×C3⋊D4, C6×D4, C324Q8, C4×C3⋊S3, C12⋊S3, C2×C3⋊Dic3, C327D4, C6×C12, D4×C32, C22×C3⋊S3, C2×C62, D46D6, C12.59D6, D4×C3⋊S3, C12.D6, C2×C327D4, D4×C3×C6, C3282+ 1+4
Quotients: C1, C2, C22, S3, C23, D6, C24, C3⋊S3, C22×S3, 2+ 1+4, C2×C3⋊S3, S3×C23, C22×C3⋊S3, D46D6, C23×C3⋊S3, C3282+ 1+4

Smallest permutation representation of C3282+ 1+4
On 72 points
Generators in S72
```(1 32 24)(2 29 21)(3 30 22)(4 31 23)(5 44 66)(6 41 67)(7 42 68)(8 43 65)(9 19 54)(10 20 55)(11 17 56)(12 18 53)(13 50 25)(14 51 26)(15 52 27)(16 49 28)(33 45 57)(34 46 58)(35 47 59)(36 48 60)(37 63 69)(38 64 70)(39 61 71)(40 62 72)
(1 37 19)(2 38 20)(3 39 17)(4 40 18)(5 28 35)(6 25 36)(7 26 33)(8 27 34)(9 24 69)(10 21 70)(11 22 71)(12 23 72)(13 48 41)(14 45 42)(15 46 43)(16 47 44)(29 64 55)(30 61 56)(31 62 53)(32 63 54)(49 59 66)(50 60 67)(51 57 68)(52 58 65)
(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)
(1 44)(2 43)(3 42)(4 41)(5 24)(6 23)(7 22)(8 21)(9 35)(10 34)(11 33)(12 36)(13 40)(14 39)(15 38)(16 37)(17 45)(18 48)(19 47)(20 46)(25 72)(26 71)(27 70)(28 69)(29 65)(30 68)(31 67)(32 66)(49 63)(50 62)(51 61)(52 64)(53 60)(54 59)(55 58)(56 57)
(1 2 3 4)(5 65 7 67)(6 66 8 68)(9 64 11 62)(10 61 12 63)(13 47 15 45)(14 48 16 46)(17 40 19 38)(18 37 20 39)(21 30 23 32)(22 31 24 29)(25 59 27 57)(26 60 28 58)(33 50 35 52)(34 51 36 49)(41 44 43 42)(53 69 55 71)(54 70 56 72)
(1 41)(2 42)(3 43)(4 44)(5 23)(6 24)(7 21)(8 22)(9 36)(10 33)(11 34)(12 35)(13 37)(14 38)(15 39)(16 40)(17 46)(18 47)(19 48)(20 45)(25 69)(26 70)(27 71)(28 72)(29 68)(30 65)(31 66)(32 67)(49 62)(50 63)(51 64)(52 61)(53 59)(54 60)(55 57)(56 58)```

`G:=sub<Sym(72)| (1,32,24)(2,29,21)(3,30,22)(4,31,23)(5,44,66)(6,41,67)(7,42,68)(8,43,65)(9,19,54)(10,20,55)(11,17,56)(12,18,53)(13,50,25)(14,51,26)(15,52,27)(16,49,28)(33,45,57)(34,46,58)(35,47,59)(36,48,60)(37,63,69)(38,64,70)(39,61,71)(40,62,72), (1,37,19)(2,38,20)(3,39,17)(4,40,18)(5,28,35)(6,25,36)(7,26,33)(8,27,34)(9,24,69)(10,21,70)(11,22,71)(12,23,72)(13,48,41)(14,45,42)(15,46,43)(16,47,44)(29,64,55)(30,61,56)(31,62,53)(32,63,54)(49,59,66)(50,60,67)(51,57,68)(52,58,65), (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), (1,44)(2,43)(3,42)(4,41)(5,24)(6,23)(7,22)(8,21)(9,35)(10,34)(11,33)(12,36)(13,40)(14,39)(15,38)(16,37)(17,45)(18,48)(19,47)(20,46)(25,72)(26,71)(27,70)(28,69)(29,65)(30,68)(31,67)(32,66)(49,63)(50,62)(51,61)(52,64)(53,60)(54,59)(55,58)(56,57), (1,2,3,4)(5,65,7,67)(6,66,8,68)(9,64,11,62)(10,61,12,63)(13,47,15,45)(14,48,16,46)(17,40,19,38)(18,37,20,39)(21,30,23,32)(22,31,24,29)(25,59,27,57)(26,60,28,58)(33,50,35,52)(34,51,36,49)(41,44,43,42)(53,69,55,71)(54,70,56,72), (1,41)(2,42)(3,43)(4,44)(5,23)(6,24)(7,21)(8,22)(9,36)(10,33)(11,34)(12,35)(13,37)(14,38)(15,39)(16,40)(17,46)(18,47)(19,48)(20,45)(25,69)(26,70)(27,71)(28,72)(29,68)(30,65)(31,66)(32,67)(49,62)(50,63)(51,64)(52,61)(53,59)(54,60)(55,57)(56,58)>;`

`G:=Group( (1,32,24)(2,29,21)(3,30,22)(4,31,23)(5,44,66)(6,41,67)(7,42,68)(8,43,65)(9,19,54)(10,20,55)(11,17,56)(12,18,53)(13,50,25)(14,51,26)(15,52,27)(16,49,28)(33,45,57)(34,46,58)(35,47,59)(36,48,60)(37,63,69)(38,64,70)(39,61,71)(40,62,72), (1,37,19)(2,38,20)(3,39,17)(4,40,18)(5,28,35)(6,25,36)(7,26,33)(8,27,34)(9,24,69)(10,21,70)(11,22,71)(12,23,72)(13,48,41)(14,45,42)(15,46,43)(16,47,44)(29,64,55)(30,61,56)(31,62,53)(32,63,54)(49,59,66)(50,60,67)(51,57,68)(52,58,65), (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), (1,44)(2,43)(3,42)(4,41)(5,24)(6,23)(7,22)(8,21)(9,35)(10,34)(11,33)(12,36)(13,40)(14,39)(15,38)(16,37)(17,45)(18,48)(19,47)(20,46)(25,72)(26,71)(27,70)(28,69)(29,65)(30,68)(31,67)(32,66)(49,63)(50,62)(51,61)(52,64)(53,60)(54,59)(55,58)(56,57), (1,2,3,4)(5,65,7,67)(6,66,8,68)(9,64,11,62)(10,61,12,63)(13,47,15,45)(14,48,16,46)(17,40,19,38)(18,37,20,39)(21,30,23,32)(22,31,24,29)(25,59,27,57)(26,60,28,58)(33,50,35,52)(34,51,36,49)(41,44,43,42)(53,69,55,71)(54,70,56,72), (1,41)(2,42)(3,43)(4,44)(5,23)(6,24)(7,21)(8,22)(9,36)(10,33)(11,34)(12,35)(13,37)(14,38)(15,39)(16,40)(17,46)(18,47)(19,48)(20,45)(25,69)(26,70)(27,71)(28,72)(29,68)(30,65)(31,66)(32,67)(49,62)(50,63)(51,64)(52,61)(53,59)(54,60)(55,57)(56,58) );`

`G=PermutationGroup([[(1,32,24),(2,29,21),(3,30,22),(4,31,23),(5,44,66),(6,41,67),(7,42,68),(8,43,65),(9,19,54),(10,20,55),(11,17,56),(12,18,53),(13,50,25),(14,51,26),(15,52,27),(16,49,28),(33,45,57),(34,46,58),(35,47,59),(36,48,60),(37,63,69),(38,64,70),(39,61,71),(40,62,72)], [(1,37,19),(2,38,20),(3,39,17),(4,40,18),(5,28,35),(6,25,36),(7,26,33),(8,27,34),(9,24,69),(10,21,70),(11,22,71),(12,23,72),(13,48,41),(14,45,42),(15,46,43),(16,47,44),(29,64,55),(30,61,56),(31,62,53),(32,63,54),(49,59,66),(50,60,67),(51,57,68),(52,58,65)], [(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)], [(1,44),(2,43),(3,42),(4,41),(5,24),(6,23),(7,22),(8,21),(9,35),(10,34),(11,33),(12,36),(13,40),(14,39),(15,38),(16,37),(17,45),(18,48),(19,47),(20,46),(25,72),(26,71),(27,70),(28,69),(29,65),(30,68),(31,67),(32,66),(49,63),(50,62),(51,61),(52,64),(53,60),(54,59),(55,58),(56,57)], [(1,2,3,4),(5,65,7,67),(6,66,8,68),(9,64,11,62),(10,61,12,63),(13,47,15,45),(14,48,16,46),(17,40,19,38),(18,37,20,39),(21,30,23,32),(22,31,24,29),(25,59,27,57),(26,60,28,58),(33,50,35,52),(34,51,36,49),(41,44,43,42),(53,69,55,71),(54,70,56,72)], [(1,41),(2,42),(3,43),(4,44),(5,23),(6,24),(7,21),(8,22),(9,36),(10,33),(11,34),(12,35),(13,37),(14,38),(15,39),(16,40),(17,46),(18,47),(19,48),(20,45),(25,69),(26,70),(27,71),(28,72),(29,68),(30,65),(31,66),(32,67),(49,62),(50,63),(51,64),(52,61),(53,59),(54,60),(55,57),(56,58)]])`

57 conjugacy classes

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

57 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D6 D6 D6 2+ 1+4 D4⋊6D6 kernel C32⋊82+ 1+4 C12.59D6 D4×C3⋊S3 C12.D6 C2×C32⋊7D4 D4×C3×C6 C6×D4 C2×C12 C3×D4 C22×C6 C32 C3 # reps 1 2 4 4 4 1 4 4 16 8 1 8

Matrix representation of C3282+ 1+4 in GL6(𝔽13)

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

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

C3282+ 1+4 in GAP, Magma, Sage, TeX

`C_3^2\rtimes_82_+^{1+4}`
`% in TeX`

`G:=Group("C3^2:8ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(288,1009);`
`// by ID`

`G=gap.SmallGroup(288,1009);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,675,2693,9414]);`
`// Polycyclic`

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

׿
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