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G = C56⋊C6order 336 = 24·3·7

2nd semidirect product of C56 and C6 acting faithfully

metacyclic, supersoluble, monomial

Aliases: C82F7, C562C6, D28.2C6, Dic144C6, C56⋊C2⋊C3, C7⋊C31SD16, C71(C3×SD16), C4⋊F7.2C2, C4.F74C2, C4.8(C2×F7), C28.8(C2×C6), C14.1(C3×D4), C2.3(C4⋊F7), (C8×C7⋊C3)⋊2C2, (C2×C7⋊C3).1D4, (C4×C7⋊C3).8C22, SmallGroup(336,9)

Series: Derived Chief Lower central Upper central

C1C28 — C56⋊C6
C1C7C14C28C4×C7⋊C3C4⋊F7 — C56⋊C6
C7C14C28 — C56⋊C6
C1C2C4C8

Generators and relations for C56⋊C6
 G = < a,b | a56=b6=1, bab-1=a3 >

28C2
7C3
14C22
14C4
7C6
28C6
4D7
7Q8
7D4
7C12
14C12
14C2×C6
2D14
2Dic7
4F7
7SD16
7C3×Q8
7C3×D4
7C24
2C2×F7
2C7⋊C12
7C3×SD16

Character table of C56⋊C6

 class 12A2B3A3B4A4B6A6B6C6D78A8B12A12B12C12D1424A24B24C24D28A28B56A56B56C56D
 size 11287722877282862214142828614141414666666
ρ111111111111111111111111111111    trivial
ρ211-1111111-1-11-1-111111-1-1-1-111-1-1-1-1    linear of order 2
ρ311-1111-111-1-111111-1-111111111111    linear of order 2
ρ4111111-111111-1-111-1-11-1-1-1-111-1-1-1-1    linear of order 2
ρ5111ζ32ζ311ζ3ζ32ζ3ζ32111ζ3ζ32ζ3ζ321ζ3ζ32ζ3ζ32111111    linear of order 3
ρ611-1ζ32ζ311ζ3ζ32ζ65ζ61-1-1ζ3ζ32ζ3ζ321ζ65ζ6ζ65ζ611-1-1-1-1    linear of order 6
ρ711-1ζ3ζ321-1ζ32ζ3ζ6ζ65111ζ32ζ3ζ6ζ651ζ32ζ3ζ32ζ3111111    linear of order 6
ρ8111ζ3ζ321-1ζ32ζ3ζ32ζ31-1-1ζ32ζ3ζ6ζ651ζ6ζ65ζ6ζ6511-1-1-1-1    linear of order 6
ρ9111ζ32ζ31-1ζ3ζ32ζ3ζ321-1-1ζ3ζ32ζ65ζ61ζ65ζ6ζ65ζ611-1-1-1-1    linear of order 6
ρ10111ζ3ζ3211ζ32ζ3ζ32ζ3111ζ32ζ3ζ32ζ31ζ32ζ3ζ32ζ3111111    linear of order 3
ρ1111-1ζ3ζ3211ζ32ζ3ζ6ζ651-1-1ζ32ζ3ζ32ζ31ζ6ζ65ζ6ζ6511-1-1-1-1    linear of order 6
ρ1211-1ζ32ζ31-1ζ3ζ32ζ65ζ6111ζ3ζ32ζ65ζ61ζ3ζ32ζ3ζ32111111    linear of order 6
ρ1322022-202200200-2-20020000-2-20000    orthogonal lifted from D4
ρ14220-1+-3-1--3-20-1--3-1+-3002001+-31--30020000-2-20000    complex lifted from C3×D4
ρ15220-1--3-1+-3-20-1+-3-1--3002001--31+-30020000-2-20000    complex lifted from C3×D4
ρ162-202200-2-2002-2--20000-2--2-2-2--200-2--2--2-2    complex lifted from SD16
ρ172-202200-2-2002--2-20000-2-2--2--2-200--2-2-2--2    complex lifted from SD16
ρ182-20-1+-3-1--3001+-31--3002--2-20000-2ζ83ζ328ζ32ζ87ζ385ζ3ζ87ζ3285ζ32ζ83ζ38ζ300--2-2-2--2    complex lifted from C3×SD16
ρ192-20-1--3-1+-3001--31+-3002--2-20000-2ζ83ζ38ζ3ζ87ζ3285ζ32ζ87ζ385ζ3ζ83ζ328ζ3200--2-2-2--2    complex lifted from C3×SD16
ρ202-20-1+-3-1--3001+-31--3002-2--20000-2ζ87ζ3285ζ32ζ83ζ38ζ3ζ83ζ328ζ32ζ87ζ385ζ300-2--2--2-2    complex lifted from C3×SD16
ρ212-20-1--3-1+-3001--31+-3002-2--20000-2ζ87ζ385ζ3ζ83ζ328ζ32ζ83ζ38ζ3ζ87ζ3285ζ3200-2--2--2-2    complex lifted from C3×SD16
ρ2266000600000-1660000-10000-1-1-1-1-1-1    orthogonal lifted from F7
ρ2366000600000-1-6-60000-10000-1-11111    orthogonal lifted from C2×F7
ρ2466000-600000-1000000-1000011-7-777    orthogonal lifted from C4⋊F7
ρ2566000-600000-1000000-100001177-7-7    orthogonal lifted from C4⋊F7
ρ266-6000000000-1-3-23-2000010000-77ζ83ζ7483ζ7283ζ7838ζ748ζ728ζ7ζ87ζ7487ζ7287ζ78785ζ7485ζ7285ζ7ζ87ζ7687ζ7587ζ738785ζ7685ζ7585ζ7383ζ7483ζ7283ζ78ζ748ζ728ζ78    complex faithful
ρ276-6000000000-13-2-3-20000100007-7ζ87ζ7687ζ7587ζ738785ζ7685ζ7585ζ7383ζ7483ζ7283ζ78ζ748ζ728ζ78ζ83ζ7483ζ7283ζ7838ζ748ζ728ζ7ζ87ζ7487ζ7287ζ78785ζ7485ζ7285ζ7    complex faithful
ρ286-6000000000-13-2-3-2000010000-77ζ87ζ7487ζ7287ζ78785ζ7485ζ7285ζ7ζ83ζ7483ζ7283ζ7838ζ748ζ728ζ783ζ7483ζ7283ζ78ζ748ζ728ζ78ζ87ζ7687ζ7587ζ738785ζ7685ζ7585ζ73    complex faithful
ρ296-6000000000-1-3-23-20000100007-783ζ7483ζ7283ζ78ζ748ζ728ζ78ζ87ζ7687ζ7587ζ738785ζ7685ζ7585ζ73ζ87ζ7487ζ7287ζ78785ζ7485ζ7285ζ7ζ83ζ7483ζ7283ζ7838ζ748ζ728ζ7    complex faithful

Smallest permutation representation of C56⋊C6
On 56 points
Generators in S56
(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)
(2 20 26 28 10 4)(3 39 51 55 19 7)(5 21 45 53 37 13)(6 40 14 24 46 16)(8 22)(9 41 33 49 17 25)(11 23 27 47 35 31)(12 42 52 18 44 34)(15 43)(30 48 54 56 38 32)(36 50)

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

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

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

Matrix representation of C56⋊C6 in GL6(𝔽3)

012002
121201
022200
012212
010102
001101
,
100202
011200
010010
000200
020200
010102

G:=sub<GL(6,GF(3))| [0,1,0,0,0,0,1,2,2,1,1,0,2,1,2,2,0,1,0,2,2,2,1,1,0,0,0,1,0,0,2,1,0,2,2,1],[1,0,0,0,0,0,0,1,1,0,2,1,0,1,0,0,0,0,2,2,0,2,2,1,0,0,1,0,0,0,2,0,0,0,0,2] >;

C56⋊C6 in GAP, Magma, Sage, TeX

C_{56}\rtimes C_6
% in TeX

G:=Group("C56:C6");
// GroupNames label

G:=SmallGroup(336,9);
// by ID

G=gap.SmallGroup(336,9);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-7,169,79,867,69,10373,1745]);
// Polycyclic

G:=Group<a,b|a^56=b^6=1,b*a*b^-1=a^3>;
// generators/relations

Export

Subgroup lattice of C56⋊C6 in TeX
Character table of C56⋊C6 in TeX

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