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G = C62.3D4  order 288 = 25·32

3rd non-split extension by C62 of D4 acting faithfully

non-abelian, soluble, monomial

Aliases: C62.3D4, (C3×C6).4D8, D6⋊S3⋊4C4, C22.8S3≀C2, C3⋊Dic3.4D4, (C3×C6).4SD16, C2.1(C32⋊D8), C32⋊3(D4⋊C4), C62.C22⋊1C2, C2.1(C32⋊2SD16), C2.9(S32⋊C4), (C2×C32⋊2C8)⋊1C2, C3⋊Dic3.9(C2×C4), (C2×D6⋊S3).1C2, (C3×C6).9(C22⋊C4), (C2×C3⋊Dic3).1C22, SmallGroup(288,387)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3⋊Dic3 — C62.3D4
Chief series
C1 — C32 — C3×C6 — C3⋊Dic3 — C2×C3⋊Dic3 — C2×D6⋊S3 — C62.3D4
Lower central
C32 — C3×C6 — C3⋊Dic3 — C62.3D4
Upper central
C1 — C22

Generators and relations for C62.3D4
 G = < a,b,c,d | a6=b6=d2=1, c4=b3, ab=ba, cac-1=a3b4, dad=a-1, cbc-1=a2b3, bd=db, dcd=a3b3c3 >

Subgroups: 456 in 90 conjugacy classes, 19 normal (17 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C8, C2×C4, D4, C23, C32, Dic3, C12, D6, C2×C6, C4⋊C4, C2×C8, C2×D4, C3×S3, C3×C6, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, D4⋊C4, C3×Dic3, C3⋊Dic3, S3×C6, C62, Dic3⋊C4, C2×C3⋊D4, C32⋊2C8, D6⋊S3, D6⋊S3, C6×Dic3, C2×C3⋊Dic3, S3×C2×C6, C62.C22, C2×C32⋊2C8, C2×D6⋊S3, C62.3D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, D8, SD16, D4⋊C4, S3≀C2, S32⋊C4, C32⋊D8, C32⋊2SD16, C62.3D4

Character table of C62.3D4

 class 12A2B2C2D2E3A3B4A4B4C4D6A6B6C6D6E6F6G6H6I6J8A8B8C8D12A12B12C12D
 size 111112124412121818444444121212121818181812121212
ρ1111111111111111111111111111111    trivial
ρ21111-1-111-1-111111111-1-1-1-11111-1-1-1-1    linear of order 2
ρ31111-1-1111111111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ411111111-1-1111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ511-1-1-1111i-i-11-1-11-11-111-1-1-ii-ii-i-iii    linear of order 4
ρ611-1-11-111-ii-11-1-11-11-1-1-111-ii-iiii-i-i    linear of order 4
ρ711-1-11-111i-i-11-1-11-11-1-1-111i-ii-i-i-iii    linear of order 4
ρ811-1-1-1111-ii-11-1-11-11-111-1-1i-ii-iii-i-i    linear of order 4
ρ922-2-20022002-2-2-22-22-2000000000000    orthogonal lifted from D4
ρ102222002200-2-2222222000000000000    orthogonal lifted from D4
ρ112-22-200220000-22-2-2-220000-√2-√2√2√20000    orthogonal lifted from D8
ρ122-22-200220000-22-2-2-220000√2√2-√2-√20000    orthogonal lifted from D8
ρ132-2-22002200002-2-22-2-20000√-2-√-2-√-2√-20000    complex lifted from SD16
ρ142-2-22002200002-2-22-2-20000-√-2√-2√-2-√-20000    complex lifted from SD16
ρ15444422-210000-2-2-2111-1-1-1-100000000    orthogonal lifted from S3≀C2
ρ164444001-22200111-2-2-200000000-1-1-1-1    orthogonal lifted from S3≀C2
ρ174444001-2-2-200111-2-2-2000000001111    orthogonal lifted from S3≀C2
ρ184444-2-2-210000-2-2-2111111100000000    orthogonal lifted from S3≀C2
ρ1944-4-42-2-21000022-2-11-111-1-100000000    orthogonal lifted from S32⋊C4
ρ2044-4-4-22-21000022-2-11-1-1-11100000000    orthogonal lifted from S32⋊C4
ρ214-4-44001-200001-1-1-22200000000-√3√3-√3√3    symplectic lifted from C32⋊2SD16, Schur index 2
ρ224-4-44001-200001-1-1-22200000000√3-√3√3-√3    symplectic lifted from C32⋊2SD16, Schur index 2
ρ234-44-400-2100002-22-1-11√-3-√-3-√-3√-300000000    complex lifted from C32⋊D8
ρ244-4-4400-210000-2221-1-1-√-3√-3-√-3√-300000000    complex lifted from C32⋊2SD16
ρ254-44-4001-20000-11-122-200000000-√-3√-3√-3-√-3    complex lifted from C32⋊D8
ρ264-4-4400-210000-2221-1-1√-3-√-3√-3-√-300000000    complex lifted from C32⋊2SD16
ρ2744-4-4001-2-2i2i00-1-112-2200000000-i-iii    complex lifted from S32⋊C4
ρ2844-4-4001-22i-2i00-1-112-2200000000ii-i-i    complex lifted from S32⋊C4
ρ294-44-4001-20000-11-122-200000000√-3-√-3-√-3√-3    complex lifted from C32⋊D8
ρ304-44-400-2100002-22-1-11-√-3√-3√-3-√-300000000    complex lifted from C32⋊D8

Smallest permutation representation of C62.3D4
►On 48 points
Generators in S48
(1 28 18 35 14 41)(2 36)(3 43 16 37 20 30)(4 38)(5 32 22 39 10 45)(6 40)(7 47 12 33 24 26)(8 34)(9 31)(11 25)(13 27)(15 29)(17 48)(19 42)(21 44)(23 46)

(1 5)(2 23 15 6 19 11)(3 7)(4 13 21 8 9 17)(10 14)(12 16)(18 22)(20 24)(25 36 46 29 40 42)(26 30)(27 44 34 31 48 38)(28 32)(33 37)(35 39)(41 45)(43 47)

(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)

(2 34)(3 7)(4 40)(6 38)(8 36)(9 46)(10 22)(11 44)(12 20)(13 42)(14 18)(15 48)(16 24)(17 29)(19 27)(21 25)(23 31)(26 43)(28 41)(30 47)(32 45)(33 37)
 
G:=sub<Sym(48)| (1,28,18,35,14,41)(2,36)(3,43,16,37,20,30)(4,38)(5,32,22,39,10,45)(6,40)(7,47,12,33,24,26)(8,34)(9,31)(11,25)(13,27)(15,29)(17,48)(19,42)(21,44)(23,46), (1,5)(2,23,15,6,19,11)(3,7)(4,13,21,8,9,17)(10,14)(12,16)(18,22)(20,24)(25,36,46,29,40,42)(26,30)(27,44,34,31,48,38)(28,32)(33,37)(35,39)(41,45)(43,47), (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), (2,34)(3,7)(4,40)(6,38)(8,36)(9,46)(10,22)(11,44)(12,20)(13,42)(14,18)(15,48)(16,24)(17,29)(19,27)(21,25)(23,31)(26,43)(28,41)(30,47)(32,45)(33,37)>;
 
G:=Group( (1,28,18,35,14,41)(2,36)(3,43,16,37,20,30)(4,38)(5,32,22,39,10,45)(6,40)(7,47,12,33,24,26)(8,34)(9,31)(11,25)(13,27)(15,29)(17,48)(19,42)(21,44)(23,46), (1,5)(2,23,15,6,19,11)(3,7)(4,13,21,8,9,17)(10,14)(12,16)(18,22)(20,24)(25,36,46,29,40,42)(26,30)(27,44,34,31,48,38)(28,32)(33,37)(35,39)(41,45)(43,47), (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), (2,34)(3,7)(4,40)(6,38)(8,36)(9,46)(10,22)(11,44)(12,20)(13,42)(14,18)(15,48)(16,24)(17,29)(19,27)(21,25)(23,31)(26,43)(28,41)(30,47)(32,45)(33,37) );
 
G=PermutationGroup([[(1,28,18,35,14,41),(2,36),(3,43,16,37,20,30),(4,38),(5,32,22,39,10,45),(6,40),(7,47,12,33,24,26),(8,34),(9,31),(11,25),(13,27),(15,29),(17,48),(19,42),(21,44),(23,46)], [(1,5),(2,23,15,6,19,11),(3,7),(4,13,21,8,9,17),(10,14),(12,16),(18,22),(20,24),(25,36,46,29,40,42),(26,30),(27,44,34,31,48,38),(28,32),(33,37),(35,39),(41,45),(43,47)], [(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)], [(2,34),(3,7),(4,40),(6,38),(8,36),(9,46),(10,22),(11,44),(12,20),(13,42),(14,18),(15,48),(16,24),(17,29),(19,27),(21,25),(23,31),(26,43),(28,41),(30,47),(32,45),(33,37)]])
 

Matrix representation of C62.3D4 ►in GL6(𝔽73)

7200000
0720000
001100
0072000
0000720
0000072
,
7200000
0720000
001000
000100
00007272
000010
,
6760000
67670000
0000720
0000072
0072000
001100
,
100000
0720000
001000
00727200
0000720
0000072

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[67,67,0,0,0,0,6,67,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,72,0,0,0,0,0,0,72,0,0],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;
 

C62.3D4 in GAP, Magma, Sage, TeX 

C_6^2._3D_4
 
% in TeX
 
G:=Group("C6^2.3D4");
 
// GroupNames label
 
G:=SmallGroup(288,387);
 
// by ID
 
G=gap.SmallGroup(288,387);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,85,64,422,219,100,2693,2028,691,797,2372]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^6=d^2=1,c^4=b^3,a*b=b*a,c*a*c^-1=a^3*b^4,d*a*d=a^-1,c*b*c^-1=a^2*b^3,b*d=d*b,d*c*d=a^3*b^3*c^3>;
 
// generators/relations
 

Export

Character table of C62.3D4 in TeX

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