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G = C79⋊C6order 474 = 2·3·79

The semidirect product of C79 and C6 acting faithfully

metacyclic, supersoluble, monomial, Z-group

Aliases: C79⋊C6, D79⋊C3, C79⋊C3⋊C2, SmallGroup(474,1)

Series: Derived Chief Lower central Upper central

C1C79 — C79⋊C6
C1C79C79⋊C3 — C79⋊C6
C79 — C79⋊C6
C1

Generators and relations for C79⋊C6
 G = < a,b | a79=b6=1, bab-1=a56 >

79C2
79C3
79C6

Character table of C79⋊C6

 class 123A3B6A6B79A79B79C79D79E79F79G79H79I79J79K79L79M
 size 179797979796666666666666
ρ11111111111111111111    trivial
ρ21-111-1-11111111111111    linear of order 2
ρ311ζ32ζ3ζ32ζ31111111111111    linear of order 3
ρ41-1ζ32ζ3ζ6ζ651111111111111    linear of order 6
ρ51-1ζ3ζ32ζ65ζ61111111111111    linear of order 6
ρ611ζ3ζ32ζ3ζ321111111111111    linear of order 3
ρ7600000ζ796479507944793579297915ζ79737965795979207914796ζ79757966796279177913794ζ796779517940793979287912ζ79707958794979307921799ζ79777948794679337931792ζ79717953794579347926798ζ7978795679557924792379ζ796179607942793779197918ζ79747943794179387936795ζ795779547947793279257922ζ7976797279697910797793ζ796879637952792779167911    orthogonal faithful
ρ8600000ζ79777948794679337931792ζ796479507944793579297915ζ7976797279697910797793ζ79707958794979307921799ζ79757966796279177913794ζ79747943794179387936795ζ79737965795979207914796ζ796179607942793779197918ζ79717953794579347926798ζ796879637952792779167911ζ7978795679557924792379ζ795779547947793279257922ζ796779517940793979287912    orthogonal faithful
ρ9600000ζ79707958794979307921799ζ796779517940793979287912ζ79717953794579347926798ζ7978795679557924792379ζ796179607942793779197918ζ79757966796279177913794ζ796879637952792779167911ζ79777948794679337931792ζ79747943794179387936795ζ7976797279697910797793ζ796479507944793579297915ζ79737965795979207914796ζ795779547947793279257922    orthogonal faithful
ρ10600000ζ795779547947793279257922ζ7976797279697910797793ζ79777948794679337931792ζ79737965795979207914796ζ796479507944793579297915ζ7978795679557924792379ζ79757966796279177913794ζ796779517940793979287912ζ79707958794979307921799ζ796179607942793779197918ζ796879637952792779167911ζ79747943794179387936795ζ79717953794579347926798    orthogonal faithful
ρ11600000ζ796879637952792779167911ζ79747943794179387936795ζ7978795679557924792379ζ7976797279697910797793ζ795779547947793279257922ζ796779517940793979287912ζ79777948794679337931792ζ79737965795979207914796ζ796479507944793579297915ζ79707958794979307921799ζ79717953794579347926798ζ796179607942793779197918ζ79757966796279177913794    orthogonal faithful
ρ12600000ζ7978795679557924792379ζ795779547947793279257922ζ79747943794179387936795ζ796479507944793579297915ζ79777948794679337931792ζ796179607942793779197918ζ7976797279697910797793ζ79707958794979307921799ζ79757966796279177913794ζ79717953794579347926798ζ796779517940793979287912ζ796879637952792779167911ζ79737965795979207914796    orthogonal faithful
ρ13600000ζ79737965795979207914796ζ79717953794579347926798ζ79707958794979307921799ζ796879637952792779167911ζ796779517940793979287912ζ796479507944793579297915ζ796179607942793779197918ζ795779547947793279257922ζ7978795679557924792379ζ79777948794679337931792ζ7976797279697910797793ζ79757966796279177913794ζ79747943794179387936795    orthogonal faithful
ρ14600000ζ79747943794179387936795ζ79777948794679337931792ζ795779547947793279257922ζ79757966796279177913794ζ7976797279697910797793ζ796879637952792779167911ζ796479507944793579297915ζ79717953794579347926798ζ79737965795979207914796ζ796779517940793979287912ζ796179607942793779197918ζ7978795679557924792379ζ79707958794979307921799    orthogonal faithful
ρ15600000ζ79717953794579347926798ζ796179607942793779197918ζ796779517940793979287912ζ79747943794179387936795ζ796879637952792779167911ζ79737965795979207914796ζ7978795679557924792379ζ7976797279697910797793ζ795779547947793279257922ζ796479507944793579297915ζ79757966796279177913794ζ79707958794979307921799ζ79777948794679337931792    orthogonal faithful
ρ16600000ζ796779517940793979287912ζ796879637952792779167911ζ796179607942793779197918ζ795779547947793279257922ζ7978795679557924792379ζ79707958794979307921799ζ79747943794179387936795ζ796479507944793579297915ζ79777948794679337931792ζ79757966796279177913794ζ79737965795979207914796ζ79717953794579347926798ζ7976797279697910797793    orthogonal faithful
ρ17600000ζ7976797279697910797793ζ79757966796279177913794ζ796479507944793579297915ζ79717953794579347926798ζ79737965795979207914796ζ795779547947793279257922ζ79707958794979307921799ζ796879637952792779167911ζ796779517940793979287912ζ7978795679557924792379ζ79747943794179387936795ζ79777948794679337931792ζ796179607942793779197918    orthogonal faithful
ρ18600000ζ796179607942793779197918ζ7978795679557924792379ζ796879637952792779167911ζ79777948794679337931792ζ79747943794179387936795ζ79717953794579347926798ζ795779547947793279257922ζ79757966796279177913794ζ7976797279697910797793ζ79737965795979207914796ζ79707958794979307921799ζ796779517940793979287912ζ796479507944793579297915    orthogonal faithful
ρ19600000ζ79757966796279177913794ζ79707958794979307921799ζ79737965795979207914796ζ796179607942793779197918ζ79717953794579347926798ζ7976797279697910797793ζ796779517940793979287912ζ79747943794179387936795ζ796879637952792779167911ζ795779547947793279257922ζ79777948794679337931792ζ796479507944793579297915ζ7978795679557924792379    orthogonal faithful

Smallest permutation representation of C79⋊C6
On 79 points: primitive
Generators in S79
(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)
(2 25 24 79 56 57)(3 49 47 78 32 34)(4 73 70 77 8 11)(5 18 14 76 63 67)(6 42 37 75 39 44)(7 66 60 74 15 21)(9 35 27 72 46 54)(10 59 50 71 22 31)(12 28 17 69 53 64)(13 52 40 68 29 41)(16 45 30 65 36 51)(19 38 20 62 43 61)(23 55 33 58 26 48)

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

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

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

Matrix representation of C79⋊C6 in GL6(𝔽1423)

010000
001000
000100
000010
000001
142288663140063886
,
100000
8397136402083691075
473595925451002943
7562451322175721166
2966712331146342261
1362331125747301100

G:=sub<GL(6,GF(1423))| [0,0,0,0,0,1422,1,0,0,0,0,886,0,1,0,0,0,63,0,0,1,0,0,1400,0,0,0,1,0,63,0,0,0,0,1,886],[1,839,473,756,296,1362,0,713,595,245,671,33,0,640,92,1322,233,1125,0,208,545,175,1146,747,0,369,1002,72,342,301,0,1075,943,1166,261,100] >;

C79⋊C6 in GAP, Magma, Sage, TeX

C_{79}\rtimes C_6
% in TeX

G:=Group("C79:C6");
// GroupNames label

G:=SmallGroup(474,1);
// by ID

G=gap.SmallGroup(474,1);
# by ID

G:=PCGroup([3,-2,-3,-79,4214,626]);
// Polycyclic

G:=Group<a,b|a^79=b^6=1,b*a*b^-1=a^56>;
// generators/relations

Export

Subgroup lattice of C79⋊C6 in TeX
Character table of C79⋊C6 in TeX

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