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G = C245Dic3order 192 = 26·3

1st semidirect product of C24 and Dic3 acting via Dic3/C3=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C245Dic3, C33C2≀C4, (C23×C6)⋊3C4, (C2×D4).7D6, (C2×C12).3D4, C22≀C2.2S3, C12.D42C2, C22⋊C41Dic3, (C6×D4).5C22, (C22×C6).14D4, C6.19(C23⋊C4), C23.7D62C2, C23.5(C3⋊D4), C23.6(C2×Dic3), C2.4(C23.7D6), C22.12(C6.D4), (C3×C22⋊C4)⋊1C4, (C2×C4).5(C3⋊D4), (C3×C22≀C2).1C2, (C22×C6).13(C2×C4), (C2×C6).94(C22⋊C4), SmallGroup(192,95)

Series: Derived Chief Lower central Upper central

C1C22×C6 — C245Dic3
C1C3C6C2×C6C22×C6C6×D4C23.7D6 — C245Dic3
C3C6C2×C6C22×C6 — C245Dic3
C1C2C22C2×D4C22≀C2

Generators and relations for C245Dic3
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e6=1, f2=e3, ab=ba, eae-1=ac=ca, ad=da, faf-1=abcd, bc=cb, ebe-1=bd=db, fbf-1=bcd, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=e-1 >

Subgroups: 304 in 94 conjugacy classes, 23 normal (all characteristic)
C1, C2, C2 [×5], C3, C4 [×3], C22, C22 [×12], C6, C6 [×5], C8, C2×C4, C2×C4 [×2], D4 [×3], C23 [×2], C23 [×4], Dic3, C12 [×2], C2×C6, C2×C6 [×12], C22⋊C4, C22⋊C4 [×2], M4(2), C2×D4, C2×D4, C24, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×D4 [×3], C22×C6 [×2], C22×C6 [×4], C23⋊C4, C4.D4, C22≀C2, C4.Dic3, C6.D4, C3×C22⋊C4, C3×C22⋊C4, C6×D4, C6×D4, C23×C6, C2≀C4, C12.D4, C23.7D6, C3×C22≀C2, C245Dic3
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], Dic3 [×2], D6, C22⋊C4, C2×Dic3, C3⋊D4 [×2], C23⋊C4, C6.D4, C2≀C4, C23.7D6, C245Dic3

Character table of C245Dic3

 class 12A2B2C2D2E2F34A4B4C4D6A6B6C6D6E6F6G6H6I6J8A8B12A12B12C
 size 1124444248242422244444482424888
ρ1111111111111111111111111111    trivial
ρ21111111111-1-11111111111-1-1111    linear of order 2
ρ311111-1-111-111111-11-1-11-11-1-1-1-11    linear of order 2
ρ411111-1-111-1-1-1111-11-1-11-1111-1-11    linear of order 2
ρ5111-11-1-11-11-ii111-11-1-11-1-1i-i11-1    linear of order 4
ρ6111-11-1-11-11i-i111-11-1-11-1-1-ii11-1    linear of order 4
ρ7111-11111-1-1-ii111111111-1-ii-1-1-1    linear of order 4
ρ8111-11111-1-1i-i111111111-1i-i-1-1-1    linear of order 4
ρ92222-2002-20002220-200-2020000-2    orthogonal lifted from D4
ρ1022222-2-2-12-200-1-1-11-111-11-10011-1    orthogonal lifted from D6
ρ11222-2-200220002220-200-20-200002    orthogonal lifted from D4
ρ122222222-12200-1-1-1-1-1-1-1-1-1-100-1-1-1    orthogonal lifted from S3
ρ13222-2222-1-2-200-1-1-1-1-1-1-1-1-1100111    symplectic lifted from Dic3, Schur index 2
ρ14222-22-2-2-1-2200-1-1-11-111-11100-1-11    symplectic lifted from Dic3, Schur index 2
ρ152222-200-1-2000-1-1-1-31-3--31--3-100-3--31    complex lifted from C3⋊D4
ρ16222-2-200-12000-1-1-1-31-3--31--3100--3-3-1    complex lifted from C3⋊D4
ρ17222-2-200-12000-1-1-1--31--3-31-3100-3--3-1    complex lifted from C3⋊D4
ρ182222-200-1-2000-1-1-1--31--3-31-3-100--3-31    complex lifted from C3⋊D4
ρ194-4000-22400000-4020-220-2000000    orthogonal lifted from C2≀C4
ρ204-40002-2400000-40-202-202000000    orthogonal lifted from C2≀C4
ρ2144-4000040000-44-4000000000000    orthogonal lifted from C23⋊C4
ρ2244-40000-200002-220-2-3002-30000000    complex lifted from C23.7D6
ρ234-4000-22-20000-2-322-3-1--301+-3-1+-301--3000000    complex faithful
ρ244-40002-2-200002-32-2-31--30-1+-31+-30-1--3000000    complex faithful
ρ2544-40000-200002-2202-300-2-30000000    complex lifted from C23.7D6
ρ264-40002-2-20000-2-322-31+-30-1--31--30-1+-3000000    complex faithful
ρ274-4000-22-200002-32-2-3-1+-301--3-1--301+-3000000    complex faithful

Permutation representations of C245Dic3
On 24 points - transitive group 24T289
Generators in S24
(1 15)(2 16)(3 17)(4 18)(5 13)(6 14)(8 22)(10 24)(12 20)
(2 16)(4 18)(6 14)(7 21)(9 23)(11 19)
(7 21)(8 22)(9 23)(10 24)(11 19)(12 20)
(1 15)(2 16)(3 17)(4 18)(5 13)(6 14)(7 21)(8 22)(9 23)(10 24)(11 19)(12 20)
(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 21 4 24)(2 20 5 23)(3 19 6 22)(7 18 10 15)(8 17 11 14)(9 16 12 13)

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

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

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

G:=TransitiveGroup(24,289);

On 24 points - transitive group 24T356
Generators in S24
(1 17)(2 15)(3 13)(4 7)(5 8)(6 9)(10 18)(11 16)(12 14)(19 22)(20 23)(21 24)
(1 14)(2 18)(3 16)(4 19)(5 23)(6 21)(7 22)(8 20)(9 24)(10 15)(11 13)(12 17)
(1 12)(2 10)(3 11)(13 16)(14 17)(15 18)
(1 12)(2 10)(3 11)(4 7)(5 8)(6 9)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(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 19 16 22)(14 24 17 21)(15 23 18 20)

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

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

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

G:=TransitiveGroup(24,356);

Matrix representation of C245Dic3 in GL4(𝔽7) generated by

1040
0150
0060
0001
,
2604
2641
0060
5210
,
0632
6042
0060
0001
,
6000
0600
0060
0006
,
3541
6446
5554
0002
,
2321
4211
4355
1165
G:=sub<GL(4,GF(7))| [1,0,0,0,0,1,0,0,4,5,6,0,0,0,0,1],[2,2,0,5,6,6,0,2,0,4,6,1,4,1,0,0],[0,6,0,0,6,0,0,0,3,4,6,0,2,2,0,1],[6,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[3,6,5,0,5,4,5,0,4,4,5,0,1,6,4,2],[2,4,4,1,3,2,3,1,2,1,5,6,1,1,5,5] >;

C245Dic3 in GAP, Magma, Sage, TeX

C_2^4\rtimes_5{\rm Dic}_3
% in TeX

G:=Group("C2^4:5Dic3");
// GroupNames label

G:=SmallGroup(192,95);
// by ID

G=gap.SmallGroup(192,95);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,219,675,297,1684,6278]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^6=1,f^2=e^3,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f^-1=a*b*c*d,b*c=c*b,e*b*e^-1=b*d=d*b,f*b*f^-1=b*c*d,f*c*f^-1=c*d=d*c,c*e=e*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations

Export

Character table of C245Dic3 in TeX

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