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G = C3×C4○D4  order 48 = 24·3

Direct product of C3 and C4○D4

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

Aliases: C3×C4○D4, D4○C12, Q8○C12, D4⋊2C6, Q8⋊3C6, 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

Derived series
C1 — C2 — C3×C4○D4
Chief series
C1 — C2 — C6 — C2×C6 — C3×D4 — C3×C4○D4
Lower central
C1 — C2 — C3×C4○D4
Upper central
C1 — C12 — 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 >

C1
C2
2C2
2C2
2C2
C3
C22
C22
C22
C4
C4
C4
C4
C6
2C6
2C6
2C6
D4
D4
D4
C2×C4
C2×C4
C2×C4
Q8
C2×C6
C2×C6
C12
C2×C6
C12
C12
C12
C4○D4
C3×D4
C3×D4
C3×Q8
C3×D4
C2×C12
C2×C12
C2×C12
C3×C4○D4

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+√-30000002ζ4ζ32ζ43ζ32ζ4ζ322ζ43ζ32000000    complex faithful
ρ282-2000-1-√-3-1+√-3-2i2i0001+√-31-√-30000002ζ4ζ322ζ43ζ322ζ4ζ32ζ43ζ3000000    complex faithful
ρ292-2000-1+√-3-1-√-32i-2i0001-√-31+√-30000002ζ43ζ32ζ4ζ32ζ43ζ322ζ4ζ32000000    complex faithful
ρ302-2000-1-√-3-1+√-32i-2i0001+√-31-√-30000002ζ43ζ322ζ4ζ322ζ43ζ32ζ4ζ3000000    complex faithful

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

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

C3×C4○D4 is a maximal subgroup of
 Q8⋊3Dic3  D4.Dic3  D4⋊D6  Q8.13D6  Q8.14D6  D4○D12  Q8○D12  Q8.C18  2- 1+4⋊3C6  D28⋊6C6  D4⋊2F7  Q8⋊3F7
C3×C4○D4 is a maximal quotient of
 D4×C12  Q8×C12  D28⋊6C6  D4⋊2F7  Q8⋊3F7

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
 

Export

Subgroup lattice of C3×C4○D4 in TeX
Character table of C3×C4○D4 in TeX

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