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G = C3×C4○D4order 48 = 24·3

Direct product of C3 and C4○D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C3×C4○D4, D4C12, Q8C12, D42C6, Q83C6, C6.13C23, C12.21C22, C4(C3×D4), C4(C3×Q8), (C2×C4)⋊3C6, C12(C3×D4), C12(C3×Q8), (C2×C12)⋊7C2, (C3×D4)⋊5C2, C4.5(C2×C6), (C3×Q8)⋊5C2, C22.(C2×C6), C2.3(C22×C6), (C2×C6).2C22, SmallGroup(48,47)

Series: Derived Chief Lower central Upper central

C1C2 — C3×C4○D4
C1C2C6C2×C6C3×D4 — C3×C4○D4
C1C2 — C3×C4○D4
C1C12 — C3×C4○D4

Generators and relations for C3×C4○D4
 G = < a,b,c,d | a3=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

2C2
2C2
2C2
2C6
2C6
2C6

Character table of C3×C4○D4

 class 12A2B2C2D3A3B4A4B4C4D4E6A6B6C6D6E6F6G6H12A12B12C12D12E12F12G12H12I12J
 size 112221111222112222221111222222
ρ1111111111111111111111111111111    trivial
ρ211-11111-1-1-11-111-1111-11-1-1-1-1-111-1-1-1    linear of order 2
ρ31111-111-1-1-1-111111-1-111-1-1-1-1-1-1-111-1    linear of order 2
ρ411-11-111111-1-111-11-1-1-1111111-1-1-1-11    linear of order 2
ρ5111-1111-1-11-1-1111-1111-1-1-1-1-11-1-1-1-11    linear of order 2
ρ611-1-111111-1-1111-1-111-1-11111-1-1-111-1    linear of order 2
ρ711-1-1-111-1-111111-1-1-1-1-1-1-1-1-1-1111111    linear of order 2
ρ8111-1-11111-11-1111-1-1-11-11111-111-1-1-1    linear of order 2
ρ911-11-1ζ3ζ32111-1-1ζ3ζ32ζ6ζ3ζ65ζ6ζ65ζ32ζ3ζ3ζ32ζ32ζ32ζ65ζ6ζ6ζ65ζ3    linear of order 6
ρ1011111ζ32ζ311111ζ32ζ3ζ3ζ32ζ32ζ3ζ32ζ3ζ32ζ32ζ3ζ3ζ3ζ32ζ3ζ3ζ32ζ32    linear of order 3
ρ1111-111ζ3ζ32-1-1-11-1ζ3ζ32ζ6ζ3ζ3ζ32ζ65ζ32ζ65ζ65ζ6ζ6ζ6ζ3ζ32ζ6ζ65ζ65    linear of order 6
ρ1211-1-11ζ32ζ311-1-11ζ32ζ3ζ65ζ6ζ32ζ3ζ6ζ65ζ32ζ32ζ3ζ3ζ65ζ6ζ65ζ3ζ32ζ6    linear of order 6
ρ13111-11ζ32ζ3-1-11-1-1ζ32ζ3ζ3ζ6ζ32ζ3ζ32ζ65ζ6ζ6ζ65ζ65ζ3ζ6ζ65ζ65ζ6ζ32    linear of order 6
ρ14111-1-1ζ3ζ3211-11-1ζ3ζ32ζ32ζ65ζ65ζ6ζ3ζ6ζ3ζ3ζ32ζ32ζ6ζ3ζ32ζ6ζ65ζ65    linear of order 6
ρ1511-111ζ32ζ3-1-1-11-1ζ32ζ3ζ65ζ32ζ32ζ3ζ6ζ3ζ6ζ6ζ65ζ65ζ65ζ32ζ3ζ65ζ6ζ6    linear of order 6
ρ16111-11ζ3ζ32-1-11-1-1ζ3ζ32ζ32ζ65ζ3ζ32ζ3ζ6ζ65ζ65ζ6ζ6ζ32ζ65ζ6ζ6ζ65ζ3    linear of order 6
ρ1711111ζ3ζ3211111ζ3ζ32ζ32ζ3ζ3ζ32ζ3ζ32ζ3ζ3ζ32ζ32ζ32ζ3ζ32ζ32ζ3ζ3    linear of order 3
ρ18111-1-1ζ32ζ311-11-1ζ32ζ3ζ3ζ6ζ6ζ65ζ32ζ65ζ32ζ32ζ3ζ3ζ65ζ32ζ3ζ65ζ6ζ6    linear of order 6
ρ1911-11-1ζ32ζ3111-1-1ζ32ζ3ζ65ζ32ζ6ζ65ζ6ζ3ζ32ζ32ζ3ζ3ζ3ζ6ζ65ζ65ζ6ζ32    linear of order 6
ρ201111-1ζ3ζ32-1-1-1-11ζ3ζ32ζ32ζ3ζ65ζ6ζ3ζ32ζ65ζ65ζ6ζ6ζ6ζ65ζ6ζ32ζ3ζ65    linear of order 6
ρ2111-1-11ζ3ζ3211-1-11ζ3ζ32ζ6ζ65ζ3ζ32ζ65ζ6ζ3ζ3ζ32ζ32ζ6ζ65ζ6ζ32ζ3ζ65    linear of order 6
ρ221111-1ζ32ζ3-1-1-1-11ζ32ζ3ζ3ζ32ζ6ζ65ζ32ζ3ζ6ζ6ζ65ζ65ζ65ζ6ζ65ζ3ζ32ζ6    linear of order 6
ρ2311-1-1-1ζ3ζ32-1-1111ζ3ζ32ζ6ζ65ζ65ζ6ζ65ζ6ζ65ζ65ζ6ζ6ζ32ζ3ζ32ζ32ζ3ζ3    linear of order 6
ρ2411-1-1-1ζ32ζ3-1-1111ζ32ζ3ζ65ζ6ζ6ζ65ζ6ζ65ζ6ζ6ζ65ζ65ζ3ζ32ζ3ζ3ζ32ζ32    linear of order 6
ρ252-200022-2i2i000-2-20000002i-2i2i-2i000000    complex lifted from C4○D4
ρ262-2000222i-2i000-2-2000000-2i2i-2i2i000000    complex lifted from C4○D4
ρ272-2000-1+-3-1--3-2i2i0001--31+-30000004ζ343ζ34ζ3243ζ32000000    complex faithful
ρ282-2000-1--3-1+-3-2i2i0001+-31--30000004ζ3243ζ324ζ343ζ3000000    complex faithful
ρ292-2000-1+-3-1--32i-2i0001--31+-300000043ζ34ζ343ζ324ζ32000000    complex faithful
ρ302-2000-1--3-1+-32i-2i0001+-31--300000043ζ324ζ3243ζ34ζ3000000    complex faithful

Permutation representations of C3×C4○D4
On 24 points - transitive group 24T17
Generators in S24
(1 13 9)(2 14 10)(3 15 11)(4 16 12)(5 18 21)(6 19 22)(7 20 23)(8 17 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 21 3 23)(2 22 4 24)(5 15 7 13)(6 16 8 14)(9 18 11 20)(10 19 12 17)
(1 23)(2 24)(3 21)(4 22)(5 15)(6 16)(7 13)(8 14)(9 20)(10 17)(11 18)(12 19)

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

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

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

G:=TransitiveGroup(24,17);

Matrix representation of C3×C4○D4 in GL2(𝔽13) generated by

30
03
,
50
05
,
80
05
,
05
80
G:=sub<GL(2,GF(13))| [3,0,0,3],[5,0,0,5],[8,0,0,5],[0,8,5,0] >;

C3×C4○D4 in GAP, Magma, Sage, TeX

C_3\times C_4\circ D_4
% in TeX

G:=Group("C3xC4oD4");
// GroupNames label

G:=SmallGroup(48,47);
// by ID

G=gap.SmallGroup(48,47);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-2,261,102]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^4=d^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c>;
// generators/relations

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