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G = C92F7order 378 = 2·33·7

2nd semidirect product of C9 and F7 acting via F7/C7=C6

metacyclic, supersoluble, monomial

Aliases: C92F7, C634C6, D634C3, C72(C9⋊C6), C63⋊C31C2, C21.2(C3×S3), C3.2(C3⋊F7), (C3×C7⋊C3).2S3, SmallGroup(378,19)

Series: Derived Chief Lower central Upper central

C1C63 — C92F7
C1C3C21C63C63⋊C3 — C92F7
C63 — C92F7
C1

Generators and relations for C92F7
 G = < a,b,c | a9=b7=c6=1, ab=ba, cac-1=a5, cbc-1=b5 >

63C2
21C3
21S3
63C6
7C32
14C9
9D7
3C7⋊C3
7D9
21C3×S3
73- 1+2
3D21
9F7
2C7⋊C9
7C9⋊C6
3C3⋊F7

Character table of C92F7

 class 123A3B3C6A6B79A9B9C21A21B63A63B63C63D63E63F
 size 16322121636366424266666666
ρ11111111111111111111    trivial
ρ21-1111-1-1111111111111    linear of order 2
ρ3111ζ3ζ32ζ3ζ3211ζ32ζ311111111    linear of order 3
ρ41-11ζ3ζ32ζ65ζ611ζ32ζ311111111    linear of order 6
ρ51-11ζ32ζ3ζ6ζ6511ζ3ζ3211111111    linear of order 6
ρ6111ζ32ζ3ζ32ζ311ζ3ζ3211111111    linear of order 3
ρ720222002-1-1-122-1-1-1-1-1-1    orthogonal lifted from S3
ρ8202-1--3-1+-3002-1ζ65ζ622-1-1-1-1-1-1    complex lifted from C3×S3
ρ9202-1+-3-1--3002-1ζ6ζ6522-1-1-1-1-1-1    complex lifted from C3×S3
ρ106060000-1600-1-1-1-1-1-1-1-1    orthogonal lifted from F7
ρ1160-300006000-3-3000000    orthogonal lifted from C9⋊C6
ρ126060000-1-300-1-11-21/21+21/21+21/21-21/21+21/21-21/2    orthogonal lifted from C3⋊F7
ρ136060000-1-300-1-11+21/21-21/21-21/21+21/21-21/21+21/2    orthogonal lifted from C3⋊F7
ρ1460-30000-10001+21/21-21/295ζ7695ζ7594ζ7694ζ75-2ζ94ζ7494ζ7394ζ79492ζ7692ζ739ζ749ζ795ζ7495ζ7-2ζ94ζ7594ζ7494ζ7394ζ7294ζ79492ζ7492ζ729ζ759ζ73ζ98ζ7298ζ797ζ7697ζ7395ζ7495ζ79ζ769ζ74-2ζ9ζ739ζ729ζ7998ζ7598ζ7397ζ7497ζ72-2ζ95ζ7595ζ7495ζ7395ζ7295ζ79594ζ7494ζ798ζ7298ζ7-2ζ97ζ7697ζ7597ζ7497ζ7297ζ79794ζ7694ζ7592ζ7492ζ7298ζ7698ζ74-2ζ98ζ7398ζ7298ζ79894ζ7494ζ792ζ7692ζ739ζ729ζ7    orthogonal faithful
ρ1560-30000-10001+21/21-21/298ζ7698ζ74-2ζ98ζ7398ζ7298ζ79894ζ7494ζ792ζ7692ζ739ζ729ζ798ζ7298ζ7-2ζ97ζ7697ζ7597ζ7497ζ7297ζ79794ζ7694ζ7592ζ7492ζ7295ζ7495ζ7-2ζ94ζ7594ζ7494ζ7394ζ7294ζ79492ζ7492ζ729ζ759ζ7395ζ7695ζ7594ζ7694ζ75-2ζ94ζ7494ζ7394ζ79492ζ7692ζ739ζ749ζ7ζ98ζ7298ζ797ζ7697ζ7395ζ7495ζ79ζ769ζ74-2ζ9ζ739ζ729ζ7998ζ7598ζ7397ζ7497ζ72-2ζ95ζ7595ζ7495ζ7395ζ7295ζ79594ζ7494ζ7    orthogonal faithful
ρ1660-30000-10001-21/21+21/295ζ7495ζ7-2ζ94ζ7594ζ7494ζ7394ζ7294ζ79492ζ7492ζ729ζ759ζ7398ζ7698ζ74-2ζ98ζ7398ζ7298ζ79894ζ7494ζ792ζ7692ζ739ζ729ζ795ζ7695ζ7594ζ7694ζ75-2ζ94ζ7494ζ7394ζ79492ζ7692ζ739ζ749ζ7ζ98ζ7298ζ797ζ7697ζ7395ζ7495ζ79ζ769ζ74-2ζ9ζ739ζ729ζ7998ζ7598ζ7397ζ7497ζ72-2ζ95ζ7595ζ7495ζ7395ζ7295ζ79594ζ7494ζ798ζ7298ζ7-2ζ97ζ7697ζ7597ζ7497ζ7297ζ79794ζ7694ζ7592ζ7492ζ72    orthogonal faithful
ρ1760-30000-10001-21/21+21/298ζ7298ζ7-2ζ97ζ7697ζ7597ζ7497ζ7297ζ79794ζ7694ζ7592ζ7492ζ7298ζ7598ζ7397ζ7497ζ72-2ζ95ζ7595ζ7495ζ7395ζ7295ζ79594ζ7494ζ798ζ7698ζ74-2ζ98ζ7398ζ7298ζ79894ζ7494ζ792ζ7692ζ739ζ729ζ795ζ7495ζ7-2ζ94ζ7594ζ7494ζ7394ζ7294ζ79492ζ7492ζ729ζ759ζ7395ζ7695ζ7594ζ7694ζ75-2ζ94ζ7494ζ7394ζ79492ζ7692ζ739ζ749ζ7ζ98ζ7298ζ797ζ7697ζ7395ζ7495ζ79ζ769ζ74-2ζ9ζ739ζ729ζ79    orthogonal faithful
ρ1860-30000-10001-21/21+21/2ζ98ζ7298ζ797ζ7697ζ7395ζ7495ζ79ζ769ζ74-2ζ9ζ739ζ729ζ7995ζ7695ζ7594ζ7694ζ75-2ζ94ζ7494ζ7394ζ79492ζ7692ζ739ζ749ζ798ζ7598ζ7397ζ7497ζ72-2ζ95ζ7595ζ7495ζ7395ζ7295ζ79594ζ7494ζ798ζ7298ζ7-2ζ97ζ7697ζ7597ζ7497ζ7297ζ79794ζ7694ζ7592ζ7492ζ7298ζ7698ζ74-2ζ98ζ7398ζ7298ζ79894ζ7494ζ792ζ7692ζ739ζ729ζ795ζ7495ζ7-2ζ94ζ7594ζ7494ζ7394ζ7294ζ79492ζ7492ζ729ζ759ζ73    orthogonal faithful
ρ1960-30000-10001+21/21-21/298ζ7598ζ7397ζ7497ζ72-2ζ95ζ7595ζ7495ζ7395ζ7295ζ79594ζ7494ζ7ζ98ζ7298ζ797ζ7697ζ7395ζ7495ζ79ζ769ζ74-2ζ9ζ739ζ729ζ7998ζ7298ζ7-2ζ97ζ7697ζ7597ζ7497ζ7297ζ79794ζ7694ζ7592ζ7492ζ7298ζ7698ζ74-2ζ98ζ7398ζ7298ζ79894ζ7494ζ792ζ7692ζ739ζ729ζ795ζ7495ζ7-2ζ94ζ7594ζ7494ζ7394ζ7294ζ79492ζ7492ζ729ζ759ζ7395ζ7695ζ7594ζ7694ζ75-2ζ94ζ7494ζ7394ζ79492ζ7692ζ739ζ749ζ7    orthogonal faithful

Smallest permutation representation of C92F7
On 63 points
Generators in S63
(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)
(1 32 12 27 62 53 39)(2 33 13 19 63 54 40)(3 34 14 20 55 46 41)(4 35 15 21 56 47 42)(5 36 16 22 57 48 43)(6 28 17 23 58 49 44)(7 29 18 24 59 50 45)(8 30 10 25 60 51 37)(9 31 11 26 61 52 38)
(2 3 5 9 8 6)(4 7)(10 44 63 46 36 26)(11 37 58 54 34 22)(12 39 62 53 32 27)(13 41 57 52 30 23)(14 43 61 51 28 19)(15 45 56 50 35 24)(16 38 60 49 33 20)(17 40 55 48 31 25)(18 42 59 47 29 21)

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

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

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

Matrix representation of C92F7 in GL6(𝔽127)

12421211949109
1251221081021867
674935871867
781840756078
925555909538
723737928957
,
101010
010101
397738773877
508950885088
1260001260
0126000126
,
3977003877
3888003989
00126000
001100
89501010
8838126126126126

G:=sub<GL(6,GF(127))| [124,125,67,78,92,72,2,122,49,18,55,37,121,108,35,40,55,37,19,102,87,75,90,92,49,18,18,60,95,89,109,67,67,78,38,57],[1,0,39,50,126,0,0,1,77,89,0,126,1,0,38,50,0,0,0,1,77,88,0,0,1,0,38,50,126,0,0,1,77,88,0,126],[39,38,0,0,89,88,77,88,0,0,50,38,0,0,126,1,1,126,0,0,0,1,0,126,38,39,0,0,1,126,77,89,0,0,0,126] >;

C92F7 in GAP, Magma, Sage, TeX

C_9\rtimes_2F_7
% in TeX

G:=Group("C9:2F7");
// GroupNames label

G:=SmallGroup(378,19);
// by ID

G=gap.SmallGroup(378,19);
# by ID

G:=PCGroup([5,-2,-3,-3,-7,-3,3962,997,327,2163,368,6304]);
// Polycyclic

G:=Group<a,b,c|a^9=b^7=c^6=1,a*b=b*a,c*a*c^-1=a^5,c*b*c^-1=b^5>;
// generators/relations

Export

Subgroup lattice of C92F7 in TeX
Character table of C92F7 in TeX

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