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## G = C34order 81 = 34

### Elementary abelian group of type [3,3,3,3]

Aliases: C34, SmallGroup(81,15)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C34
 Chief series C1 — C3 — C32 — C33 — C34
 Lower central C1 — C34
 Upper central C1 — C34
 Jennings C1 — C34

Generators and relations for C34
G = < a,b,c,d | a3=b3=c3=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >

Subgroups: 212, all normal (2 characteristic)
C1, C3, C32, C33, C34
Quotients: C1, C3, C32, C33, C34

Smallest permutation representation of C34
Regular action on 81 points
Generators in S81
```(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)
(1 11 8)(2 12 9)(3 10 7)(4 63 60)(5 61 58)(6 62 59)(13 19 16)(14 20 17)(15 21 18)(22 29 25)(23 30 26)(24 28 27)(31 37 34)(32 38 35)(33 39 36)(40 46 43)(41 47 44)(42 48 45)(49 56 52)(50 57 53)(51 55 54)(64 71 68)(65 72 69)(66 70 67)(73 79 76)(74 80 77)(75 81 78)
(1 59 32)(2 60 33)(3 58 31)(4 39 12)(5 37 10)(6 38 11)(7 61 34)(8 62 35)(9 63 36)(13 67 40)(14 68 41)(15 69 42)(16 70 43)(17 71 44)(18 72 45)(19 66 46)(20 64 47)(21 65 48)(22 73 49)(23 74 50)(24 75 51)(25 76 52)(26 77 53)(27 78 54)(28 81 55)(29 79 56)(30 80 57)
(1 74 14)(2 75 15)(3 73 13)(4 55 65)(5 56 66)(6 57 64)(7 76 16)(8 77 17)(9 78 18)(10 79 19)(11 80 20)(12 81 21)(22 40 31)(23 41 32)(24 42 33)(25 43 34)(26 44 35)(27 45 36)(28 48 39)(29 46 37)(30 47 38)(49 67 58)(50 68 59)(51 69 60)(52 70 61)(53 71 62)(54 72 63)```

`G:=sub<Sym(81)| (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), (1,11,8)(2,12,9)(3,10,7)(4,63,60)(5,61,58)(6,62,59)(13,19,16)(14,20,17)(15,21,18)(22,29,25)(23,30,26)(24,28,27)(31,37,34)(32,38,35)(33,39,36)(40,46,43)(41,47,44)(42,48,45)(49,56,52)(50,57,53)(51,55,54)(64,71,68)(65,72,69)(66,70,67)(73,79,76)(74,80,77)(75,81,78), (1,59,32)(2,60,33)(3,58,31)(4,39,12)(5,37,10)(6,38,11)(7,61,34)(8,62,35)(9,63,36)(13,67,40)(14,68,41)(15,69,42)(16,70,43)(17,71,44)(18,72,45)(19,66,46)(20,64,47)(21,65,48)(22,73,49)(23,74,50)(24,75,51)(25,76,52)(26,77,53)(27,78,54)(28,81,55)(29,79,56)(30,80,57), (1,74,14)(2,75,15)(3,73,13)(4,55,65)(5,56,66)(6,57,64)(7,76,16)(8,77,17)(9,78,18)(10,79,19)(11,80,20)(12,81,21)(22,40,31)(23,41,32)(24,42,33)(25,43,34)(26,44,35)(27,45,36)(28,48,39)(29,46,37)(30,47,38)(49,67,58)(50,68,59)(51,69,60)(52,70,61)(53,71,62)(54,72,63)>;`

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

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

C34 is a maximal subgroup of   C34⋊C2  C33⋊C9  C32⋊He3  C34.C3  C34⋊C5
C34 is a maximal quotient of   3+ 1+4  3- 1+4

81 conjugacy classes

 class 1 3A ··· 3CB order 1 3 ··· 3 size 1 1 ··· 1

81 irreducible representations

 dim 1 1 type + image C1 C3 kernel C34 C33 # reps 1 80

Matrix representation of C34 in GL4(𝔽7) generated by

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

C34 in GAP, Magma, Sage, TeX

`C_3^4`
`% in TeX`

`G:=Group("C3^4");`
`// GroupNames label`

`G:=SmallGroup(81,15);`
`// by ID`

`G=gap.SmallGroup(81,15);`
`# by ID`

`G:=PCGroup([4,-3,3,3,3]);`
`// Polycyclic`

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

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