Copied to
clipboard

G = C636C6order 378 = 2·33·7

6th semidirect product of C63 and C6 acting faithfully

metacyclic, supersoluble, monomial

Aliases: C636C6, C74(C9⋊C6), D92(C7⋊C3), (C7×D9)⋊2C3, C63⋊C32C2, C21.7(C3×S3), C92(C2×C7⋊C3), C3.2(S3×C7⋊C3), (C3×C7⋊C3).4S3, SmallGroup(378,14)

Series: Derived Chief Lower central Upper central

C1C63 — C636C6
C1C3C21C63C63⋊C3 — C636C6
C63 — C636C6
C1

Generators and relations for C636C6
 G = < a,b | a63=b6=1, bab-1=a23 >

9C2
21C3
3S3
63C6
7C32
14C9
9C14
3C7⋊C3
21C3×S3
73- 1+2
3S3×C7
9C2×C7⋊C3
2C7⋊C9
7C9⋊C6
3S3×C7⋊C3

Character table of C636C6

 class 123A3B3C6A6B7A7B9A9B9C14A14B21A21B63A63B63C63D63E63F
 size 192212163633364242272766666666
ρ11111111111111111111111    trivial
ρ21-1111-1-111111-1-111111111    linear of order 2
ρ3111ζ3ζ32ζ3ζ32111ζ32ζ31111111111    linear of order 3
ρ41-11ζ32ζ3ζ6ζ65111ζ3ζ32-1-111111111    linear of order 6
ρ51-11ζ3ζ32ζ65ζ6111ζ32ζ3-1-111111111    linear of order 6
ρ6111ζ32ζ3ζ32ζ3111ζ3ζ321111111111    linear of order 3
ρ7202220022-1-1-10022-1-1-1-1-1-1    orthogonal lifted from S3
ρ8202-1+-3-1--30022-1ζ6ζ650022-1-1-1-1-1-1    complex lifted from C3×S3
ρ9202-1--3-1+-30022-1ζ65ζ60022-1-1-1-1-1-1    complex lifted from C3×S3
ρ103330000-1+-7/2-1--7/2300-1+-7/2-1--7/2-1--7/2-1+-7/2-1--7/2-1--7/2-1+-7/2-1+-7/2-1+-7/2-1--7/2    complex lifted from C7⋊C3
ρ113330000-1--7/2-1+-7/2300-1--7/2-1+-7/2-1+-7/2-1--7/2-1+-7/2-1+-7/2-1--7/2-1--7/2-1--7/2-1+-7/2    complex lifted from C7⋊C3
ρ123-330000-1--7/2-1+-7/23001+-7/21--7/2-1+-7/2-1--7/2-1+-7/2-1+-7/2-1--7/2-1--7/2-1--7/2-1+-7/2    complex lifted from C2×C7⋊C3
ρ133-330000-1+-7/2-1--7/23001--7/21+-7/2-1--7/2-1+-7/2-1--7/2-1--7/2-1+-7/2-1+-7/2-1+-7/2-1--7/2    complex lifted from C2×C7⋊C3
ρ1460-300006600000-3-3000000    orthogonal lifted from C9⋊C6
ρ156060000-1+-7-1--7-30000-1--7-1+-71+-7/21+-7/21--7/21--7/21--7/21+-7/2    complex lifted from S3×C7⋊C3
ρ166060000-1--7-1+-7-30000-1+-7-1--71--7/21--7/21+-7/21+-7/21+-7/21--7/2    complex lifted from S3×C7⋊C3
ρ1760-30000-1--7-1+-7000001--7/21+-7/2ζ95ζ7295ζ794ζ7494ζ7292ζ7492ζ79ζ749ζ7ζ98ζ7498ζ794ζ7294ζ792ζ7292ζ79ζ749ζ7297ζ7597ζ7395ζ7695ζ7592ζ7692ζ739ζ769ζ75ζ95ζ7695ζ7394ζ7694ζ7592ζ7592ζ739ζ759ζ7395ζ7595ζ7394ζ7694ζ7392ζ7692ζ759ζ769ζ7598ζ7498ζ7297ζ7297ζ794ζ7494ζ7292ζ7492ζ7    complex faithful
ρ1860-30000-1+-7-1--7000001+-7/21--7/2ζ95ζ7695ζ7394ζ7694ζ7592ζ7592ζ739ζ759ζ7395ζ7595ζ7394ζ7694ζ7392ζ7692ζ759ζ769ζ75ζ98ζ7498ζ794ζ7294ζ792ζ7292ζ79ζ749ζ7298ζ7498ζ7297ζ7297ζ794ζ7494ζ7292ζ7492ζ7ζ95ζ7295ζ794ζ7494ζ7292ζ7492ζ79ζ749ζ797ζ7597ζ7395ζ7695ζ7592ζ7692ζ739ζ769ζ75    complex faithful
ρ1960-30000-1+-7-1--7000001+-7/21--7/295ζ7595ζ7394ζ7694ζ7392ζ7692ζ759ζ769ζ7597ζ7597ζ7395ζ7695ζ7592ζ7692ζ739ζ769ζ7598ζ7498ζ7297ζ7297ζ794ζ7494ζ7292ζ7492ζ7ζ95ζ7295ζ794ζ7494ζ7292ζ7492ζ79ζ749ζ7ζ98ζ7498ζ794ζ7294ζ792ζ7292ζ79ζ749ζ72ζ95ζ7695ζ7394ζ7694ζ7592ζ7592ζ739ζ759ζ73    complex faithful
ρ2060-30000-1--7-1+-7000001--7/21+-7/298ζ7498ζ7297ζ7297ζ794ζ7494ζ7292ζ7492ζ7ζ95ζ7295ζ794ζ7494ζ7292ζ7492ζ79ζ749ζ795ζ7595ζ7394ζ7694ζ7392ζ7692ζ759ζ769ζ7597ζ7597ζ7395ζ7695ζ7592ζ7692ζ739ζ769ζ75ζ95ζ7695ζ7394ζ7694ζ7592ζ7592ζ739ζ759ζ73ζ98ζ7498ζ794ζ7294ζ792ζ7292ζ79ζ749ζ72    complex faithful
ρ2160-30000-1+-7-1--7000001+-7/21--7/297ζ7597ζ7395ζ7695ζ7592ζ7692ζ739ζ769ζ75ζ95ζ7695ζ7394ζ7694ζ7592ζ7592ζ739ζ759ζ73ζ95ζ7295ζ794ζ7494ζ7292ζ7492ζ79ζ749ζ7ζ98ζ7498ζ794ζ7294ζ792ζ7292ζ79ζ749ζ7298ζ7498ζ7297ζ7297ζ794ζ7494ζ7292ζ7492ζ795ζ7595ζ7394ζ7694ζ7392ζ7692ζ759ζ769ζ75    complex faithful
ρ2260-30000-1--7-1+-7000001--7/21+-7/2ζ98ζ7498ζ794ζ7294ζ792ζ7292ζ79ζ749ζ7298ζ7498ζ7297ζ7297ζ794ζ7494ζ7292ζ7492ζ7ζ95ζ7695ζ7394ζ7694ζ7592ζ7592ζ739ζ759ζ7395ζ7595ζ7394ζ7694ζ7392ζ7692ζ759ζ769ζ7597ζ7597ζ7395ζ7695ζ7592ζ7692ζ739ζ769ζ75ζ95ζ7295ζ794ζ7494ζ7292ζ7492ζ79ζ749ζ7    complex faithful

Smallest permutation representation of C636C6
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)
(2 12 59 9 26 24)(3 23 54 17 51 47)(4 34 49 25 13 7)(5 45 44 33 38 30)(6 56 39 41 63 53)(8 15 29 57 50 36)(10 37 19)(11 48 14 18 62 42)(16 40 52 58 61 31)(20 21 32 27 35 60)(22 43)(28 46 55)

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

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

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

Matrix representation of C636C6 in GL6(𝔽127)

10076106205795
5124107863289
57951161865121
328910998671
65121241001829
671275198116
,
100000
1261260000
000010
0000126126
104012601260
23231111

G:=sub<GL(6,GF(127))| [100,51,57,32,65,6,76,24,95,89,121,71,106,107,116,109,24,27,20,86,18,98,100,51,57,32,65,6,18,98,95,89,121,71,29,116],[1,126,0,0,104,23,0,126,0,0,0,23,0,0,0,0,126,1,0,0,0,0,0,1,0,0,1,126,126,1,0,0,0,126,0,1] >;

C636C6 in GAP, Magma, Sage, TeX

C_{63}\rtimes_6C_6
% in TeX

G:=Group("C63:6C6");
// GroupNames label

G:=SmallGroup(378,14);
// by ID

G=gap.SmallGroup(378,14);
# by ID

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

G:=Group<a,b|a^63=b^6=1,b*a*b^-1=a^23>;
// generators/relations

Export

Subgroup lattice of C636C6 in TeX
Character table of C636C6 in TeX

׿
×
𝔽