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G = C425Dic3order 192 = 26·3

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C425Dic3, (C4×C12)⋊3C4, (C6×D4)⋊2C4, (C2×D4)⋊2Dic3, (C2×D4).10D6, C41D4.3S3, C32(C42⋊C4), (C22×C6).17D4, C6.25(C23⋊C4), C23.7D68C2, C23.8(C3⋊D4), (C6×D4).173C22, C2.10(C23.7D6), C22.16(C6.D4), (C2×C12).10(C2×C4), (C3×C41D4).7C2, (C2×C4).3(C2×Dic3), (C2×C6).103(C22⋊C4), SmallGroup(192,104)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C425Dic3
C1C3C6C2×C6C22×C6C6×D4C23.7D6 — C425Dic3
C3C6C2×C6C2×C12 — C425Dic3
C1C2C22C2×D4C41D4

Generators and relations for C425Dic3
 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 [×4], C3, C4 [×5], C22, C22 [×7], C6, C6 [×4], C2×C4, C2×C4 [×3], D4 [×6], C23 [×2], C23, Dic3 [×2], C12 [×3], C2×C6, C2×C6 [×7], C42, C22⋊C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×Dic3 [×2], C2×C12, C2×C12, C3×D4 [×6], C22×C6 [×2], C22×C6, C23⋊C4 [×2], C41D4, C6.D4 [×2], C4×C12, C6×D4 [×2], C6×D4 [×2], C42⋊C4, C23.7D6 [×2], C3×C41D4, C425Dic3
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, C42⋊C4, C23.7D6, C425Dic3

Character table of C425Dic3

 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 C425Dic3
On 24 points - transitive group 24T354
Generators in S24
(1 17 9 14)(2 15 7 18)(3 13 8 16)(4 12)(5 10)(6 11)(19 22)(20 23)(21 24)
(1 14 9 17)(2 18 7 15)(3 16 8 13)(4 24 12 21)(5 22 10 19)(6 20 11 23)
(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 11)(2 10)(3 12)(4 8)(5 7)(6 9)(13 21 16 24)(14 20 17 23)(15 19 18 22)

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

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

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

G:=TransitiveGroup(24,354);

Matrix representation of C425Dic3 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] >;

C425Dic3 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 C425Dic3 in TeX

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