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G = C67⋊C6order 402 = 2·3·67

The semidirect product of C67 and C6 acting faithfully

metacyclic, supersoluble, monomial, Z-group

Aliases: C67⋊C6, D67⋊C3, C67⋊C3⋊C2, SmallGroup(402,1)

Series: Derived Chief Lower central Upper central

C1C67 — C67⋊C6
C1C67C67⋊C3 — C67⋊C6
C67 — C67⋊C6
C1

Generators and relations for C67⋊C6
 G = < a,b | a67=b6=1, bab-1=a30 >

67C2
67C3
67C6

Character table of C67⋊C6

 class 123A3B6A6B67A67B67C67D67E67F67G67H67I67J67K
 size 1676767676766666666666
ρ111111111111111111    trivial
ρ21-111-1-111111111111    linear of order 2
ρ311ζ32ζ3ζ32ζ311111111111    linear of order 3
ρ41-1ζ32ζ3ζ6ζ6511111111111    linear of order 6
ρ51-1ζ3ζ32ζ65ζ611111111111    linear of order 6
ρ611ζ3ζ32ζ3ζ3211111111111    linear of order 3
ρ7600000ζ6766673867376730672967ζ67616746674067276721676ζ675067436741672667246717ζ676567606758679677672ζ67596739673667316728678ζ67636753674967186714674ζ67626756675167166711675ζ675767456735673267226710ζ675267486734673367196715ζ67646747674467236720673ζ675567546742672567136712    orthogonal faithful
ρ8600000ζ675767456735673267226710ζ676567606758679677672ζ67596739673667316728678ζ67646747674467236720673ζ675567546742672567136712ζ67616746674067276721676ζ675067436741672667246717ζ675267486734673367196715ζ67626756675167166711675ζ6766673867376730672967ζ67636753674967186714674    orthogonal faithful
ρ9600000ζ67616746674067276721676ζ67596739673667316728678ζ675767456735673267226710ζ675567546742672567136712ζ675267486734673367196715ζ675067436741672667246717ζ6766673867376730672967ζ676567606758679677672ζ67646747674467236720673ζ67636753674967186714674ζ67626756675167166711675    orthogonal faithful
ρ10600000ζ67596739673667316728678ζ675267486734673367196715ζ676567606758679677672ζ67626756675167166711675ζ67646747674467236720673ζ675767456735673267226710ζ67616746674067276721676ζ675567546742672567136712ζ67636753674967186714674ζ675067436741672667246717ζ6766673867376730672967    orthogonal faithful
ρ11600000ζ67626756675167166711675ζ6766673867376730672967ζ67636753674967186714674ζ675767456735673267226710ζ67616746674067276721676ζ67646747674467236720673ζ675567546742672567136712ζ675067436741672667246717ζ67596739673667316728678ζ675267486734673367196715ζ676567606758679677672    orthogonal faithful
ρ12600000ζ675567546742672567136712ζ67626756675167166711675ζ67646747674467236720673ζ675067436741672667246717ζ6766673867376730672967ζ675267486734673367196715ζ676567606758679677672ζ67636753674967186714674ζ67616746674067276721676ζ67596739673667316728678ζ675767456735673267226710    orthogonal faithful
ρ13600000ζ67636753674967186714674ζ675067436741672667246717ζ6766673867376730672967ζ67596739673667316728678ζ675767456735673267226710ζ67626756675167166711675ζ67646747674467236720673ζ67616746674067276721676ζ676567606758679677672ζ675567546742672567136712ζ675267486734673367196715    orthogonal faithful
ρ14600000ζ675267486734673367196715ζ67646747674467236720673ζ675567546742672567136712ζ6766673867376730672967ζ67636753674967186714674ζ676567606758679677672ζ67596739673667316728678ζ67626756675167166711675ζ675067436741672667246717ζ675767456735673267226710ζ67616746674067276721676    orthogonal faithful
ρ15600000ζ67646747674467236720673ζ67636753674967186714674ζ67626756675167166711675ζ67616746674067276721676ζ675067436741672667246717ζ675567546742672567136712ζ675267486734673367196715ζ6766673867376730672967ζ675767456735673267226710ζ676567606758679677672ζ67596739673667316728678    orthogonal faithful
ρ16600000ζ675067436741672667246717ζ675767456735673267226710ζ67616746674067276721676ζ675267486734673367196715ζ676567606758679677672ζ6766673867376730672967ζ67636753674967186714674ζ67596739673667316728678ζ675567546742672567136712ζ67626756675167166711675ζ67646747674467236720673    orthogonal faithful
ρ17600000ζ676567606758679677672ζ675567546742672567136712ζ675267486734673367196715ζ67636753674967186714674ζ67626756675167166711675ζ67596739673667316728678ζ675767456735673267226710ζ67646747674467236720673ζ6766673867376730672967ζ67616746674067276721676ζ675067436741672667246717    orthogonal faithful

Smallest permutation representation of C67⋊C6
On 67 points: primitive
Generators in S67
(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)
(2 39 38 67 30 31)(3 10 8 66 59 61)(4 48 45 65 21 24)(5 19 15 64 50 54)(6 57 52 63 12 17)(7 28 22 62 41 47)(9 37 29 60 32 40)(11 46 36 58 23 33)(13 55 43 56 14 26)(16 35 20 53 34 49)(18 44 27 51 25 42)

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

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

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

Matrix representation of C67⋊C6 in GL6(𝔽1609)

010000
001000
000100
000010
000001
1608208802234802208
,
100000
1105595828792741587
5771073814319931608
11761410176113605457
124913041477120812831036
2443376496425412

G:=sub<GL(6,GF(1609))| [0,0,0,0,0,1608,1,0,0,0,0,208,0,1,0,0,0,802,0,0,1,0,0,234,0,0,0,1,0,802,0,0,0,0,1,208],[1,1105,577,1176,1249,24,0,595,1073,1410,1304,433,0,828,814,176,1477,764,0,792,31,113,1208,964,0,74,993,605,1283,25,0,1587,1608,457,1036,412] >;

C67⋊C6 in GAP, Magma, Sage, TeX

C_{67}\rtimes C_6
% in TeX

G:=Group("C67:C6");
// GroupNames label

G:=SmallGroup(402,1);
// by ID

G=gap.SmallGroup(402,1);
# by ID

G:=PCGroup([3,-2,-3,-67,3566,1004]);
// Polycyclic

G:=Group<a,b|a^67=b^6=1,b*a*b^-1=a^30>;
// generators/relations

Export

Subgroup lattice of C67⋊C6 in TeX
Character table of C67⋊C6 in TeX

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