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G = C42⋊5Dic3  order 192 = 26·3

3rd semidirect product of C42 and Dic3 acting via Dic3/C3=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42⋊5Dic3, (C4×C12)⋊3C4, (C6×D4)⋊2C4, (C2×D4)⋊2Dic3, (C2×D4).10D6, C4⋊1D4.3S3, C3⋊2(C42⋊C4), (C22×C6).17D4, C6.25(C23⋊C4), C23.7D6⋊8C2, C23.8(C3⋊D4), (C6×D4).173C22, C2.10(C23.7D6), C22.16(C6.D4), (C2×C12).10(C2×C4), (C3×C4⋊1D4).7C2, (C2×C4).3(C2×Dic3), (C2×C6).103(C22⋊C4), SmallGroup(192,104)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C12 — C42⋊5Dic3
Chief series
C1 — C3 — C6 — C2×C6 — C22×C6 — C6×D4 — C23.7D6 — C42⋊5Dic3
Lower central
C3 — C6 — C2×C6 — C2×C12 — C42⋊5Dic3
Upper central
C1 — C2 — C22 — C2×D4 — C4⋊1D4

Generators and relations for C42⋊5Dic3
 G = < a,b,c,d | a4=b4=c6=1, d2=c3, ab=ba, cac-1=a-1, dad-1=a-1b, cbc-1=b-1, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 304 in 86 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C2×C6, C2×C6, C42, C22⋊C4, C2×D4, C2×D4, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C23⋊C4, C4⋊1D4, C6.D4, C4×C12, C6×D4, C6×D4, C42⋊C4, C23.7D6, C3×C4⋊1D4, C42⋊5Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, C2×Dic3, C3⋊D4, C23⋊C4, C6.D4, C42⋊C4, C23.7D6, C42⋊5Dic3

Character table of C42⋊5Dic3

 class 12A2B2C2D2E34A4B4C4D4E4F4G6A6B6C6D6E6F6G12A12B12C12D12E12F
 size 1124482444242424242228888444444
ρ1111111111111111111111111111    trivial
ρ21111111111-1-1-1-11111111111111    linear of order 2
ρ311111-111-1-11-11-111111-1-1-1-11-1-11    linear of order 2
ρ411111-111-1-1-11-1111111-1-1-1-11-1-11    linear of order 2
ρ5111-1-1111-1-1ii-i-i111-1-111-1-11-1-11    linear of order 4
ρ6111-1-1111-1-1-i-iii111-1-111-1-11-1-11    linear of order 4
ρ7111-1-1-11111i-i-ii111-1-1-1-1111111    linear of order 4
ρ8111-1-1-11111-iii-i111-1-1-1-1111111    linear of order 4
ρ922222-2-12-2-20000-1-1-1-1-11111-111-1    orthogonal lifted from D6
ρ102222-202-2000000222-220000-200-2    orthogonal lifted from D4
ρ11222-2202-20000002222-20000-200-2    orthogonal lifted from D4
ρ12222222-12220000-1-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ13222-2-2-2-12220000-1-1-11111-1-1-1-1-1-1    symplectic lifted from Dic3, Schur index 2
ρ14222-2-22-12-2-20000-1-1-111-1-111-111-1    symplectic lifted from Dic3, Schur index 2
ρ152222-20-1-2000000-1-1-11-1-√-3√-3√-3-√-31√-3-√-31    complex lifted from C3⋊D4
ρ162222-20-1-2000000-1-1-11-1√-3-√-3-√-3√-31-√-3√-31    complex lifted from C3⋊D4
ρ17222-220-1-2000000-1-1-1-11-√-3√-3-√-3√-31-√-3√-31    complex lifted from C3⋊D4
ρ18222-220-1-2000000-1-1-1-11√-3-√-3√-3-√-31√-3-√-31    complex lifted from C3⋊D4
ρ194-4000040-220000-4000000220-2-20    orthogonal lifted from C42⋊C4
ρ204-40000402-20000-4000000-2-20220    orthogonal lifted from C42⋊C4
ρ2144-4000400000004-4-40000000000    orthogonal lifted from C23⋊C4
ρ224-40000-202-2000022√-3-2√-300001-√-31+√-30-1+√-3-1-√-30    complex faithful
ρ234-40000-202-200002-2√-32√-300001+√-31-√-30-1-√-3-1+√-30    complex faithful
ρ2444-4000-20000000-222000000-2√-3002√-3    complex lifted from C23.7D6
ρ254-40000-20-2200002-2√-32√-30000-1-√-3-1+√-301+√-31-√-30    complex faithful
ρ264-40000-20-22000022√-3-2√-30000-1+√-3-1-√-301-√-31+√-30    complex faithful
ρ2744-4000-20000000-2220000002√-300-2√-3    complex lifted from C23.7D6

Permutation representations of C42⋊5Dic3
►On 24 points - transitive group 24T354
Generators in S24
(1 5)(2 6)(3 4)(7 20 11 23)(8 24 12 21)(9 22 10 19)(13 16)(14 17)(15 18)

(1 17 5 14)(2 15 6 18)(3 13 4 16)(7 23 11 20)(8 21 12 24)(9 19 10 22)

(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)

(1 9)(2 8)(3 7)(4 11)(5 10)(6 12)(13 23 16 20)(14 22 17 19)(15 21 18 24)
 
G:=sub<Sym(24)| (1,5)(2,6)(3,4)(7,20,11,23)(8,24,12,21)(9,22,10,19)(13,16)(14,17)(15,18), (1,17,5,14)(2,15,6,18)(3,13,4,16)(7,23,11,20)(8,21,12,24)(9,19,10,22), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,9)(2,8)(3,7)(4,11)(5,10)(6,12)(13,23,16,20)(14,22,17,19)(15,21,18,24)>;
 
G:=Group( (1,5)(2,6)(3,4)(7,20,11,23)(8,24,12,21)(9,22,10,19)(13,16)(14,17)(15,18), (1,17,5,14)(2,15,6,18)(3,13,4,16)(7,23,11,20)(8,21,12,24)(9,19,10,22), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,9)(2,8)(3,7)(4,11)(5,10)(6,12)(13,23,16,20)(14,22,17,19)(15,21,18,24) );
 
G=PermutationGroup([[(1,5),(2,6),(3,4),(7,20,11,23),(8,24,12,21),(9,22,10,19),(13,16),(14,17),(15,18)], [(1,17,5,14),(2,15,6,18),(3,13,4,16),(7,23,11,20),(8,21,12,24),(9,19,10,22)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,9),(2,8),(3,7),(4,11),(5,10),(6,12),(13,23,16,20),(14,22,17,19),(15,21,18,24)]])
 
G:=TransitiveGroup(24,354);
 

Matrix representation of C42⋊5Dic3 ►in GL4(𝔽7) generated by

2301
1015
4446
0006
,
6330
1556
3331
5210
,
3011
3622
1115
0004
,
1063
2232
1633
3321
G:=sub<GL(4,GF(7))| [2,1,4,0,3,0,4,0,0,1,4,0,1,5,6,6],[6,1,3,5,3,5,3,2,3,5,3,1,0,6,1,0],[3,3,1,0,0,6,1,0,1,2,1,0,1,2,5,4],[1,2,1,3,0,2,6,3,6,3,3,2,3,2,3,1] >;
 

C42⋊5Dic3 in GAP, Magma, Sage, TeX 

C_4^2\rtimes_5{\rm Dic}_3
 
% in TeX
 
G:=Group("C4^2:5Dic3");
 
// GroupNames label
 
G:=SmallGroup(192,104);
 
// by ID
 
G=gap.SmallGroup(192,104);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,219,1571,570,297,136,1684,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=c^3,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b,c*b*c^-1=b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
 
// generators/relations
 

Export

Character table of C42⋊5Dic3 in TeX

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