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G = C3×C4⋊C4order 48 = 24·3

Direct product of C3 and C4⋊C4

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

Aliases: C3×C4⋊C4, C4⋊C12, C123C4, C6.3Q8, C6.13D4, C2.(C3×Q8), (C2×C4).1C6, C2.2(C3×D4), C6.11(C2×C4), C2.2(C2×C12), (C2×C12).7C2, C22.3(C2×C6), (C2×C6).14C22, SmallGroup(48,22)

Series: Derived Chief Lower central Upper central

C1C2 — C3×C4⋊C4
C1C2C22C2×C6C2×C12 — C3×C4⋊C4
C1C2 — C3×C4⋊C4
C1C2×C6 — C3×C4⋊C4

Generators and relations for C3×C4⋊C4
 G = < a,b,c | a3=b4=c4=1, ab=ba, ac=ca, cbc-1=b-1 >

2C4
2C4
2C12
2C12

Character table of C3×C4⋊C4

 class 12A2B2C3A3B4A4B4C4D4E4F6A6B6C6D6E6F12A12B12C12D12E12F12G12H12I12J12K12L
 size 111111222222111111222222222222
ρ1111111111111111111111111111111    trivial
ρ21111111-1-1-11-11111111-1-1-1-1-1-1-1111-1    linear of order 2
ρ3111111-11-1-1-11111111-11-1-1-1-111-1-1-11    linear of order 2
ρ4111111-1-111-1-1111111-1-11111-1-1-1-1-1-1    linear of order 2
ρ51111ζ3ζ321-1-1-11-1ζ3ζ3ζ3ζ32ζ32ζ32ζ32ζ65ζ65ζ65ζ6ζ6ζ6ζ6ζ3ζ3ζ32ζ65    linear of order 6
ρ61111ζ3ζ32-11-1-1-11ζ3ζ3ζ3ζ32ζ32ζ32ζ6ζ3ζ65ζ65ζ6ζ6ζ32ζ32ζ65ζ65ζ6ζ3    linear of order 6
ρ71111ζ32ζ3-11-1-1-11ζ32ζ32ζ32ζ3ζ3ζ3ζ65ζ32ζ6ζ6ζ65ζ65ζ3ζ3ζ6ζ6ζ65ζ32    linear of order 6
ρ81111ζ32ζ3111111ζ32ζ32ζ32ζ3ζ3ζ3ζ3ζ32ζ32ζ32ζ3ζ3ζ3ζ3ζ32ζ32ζ3ζ32    linear of order 3
ρ91111ζ32ζ31-1-1-11-1ζ32ζ32ζ32ζ3ζ3ζ3ζ3ζ6ζ6ζ6ζ65ζ65ζ65ζ65ζ32ζ32ζ3ζ6    linear of order 6
ρ101111ζ3ζ32111111ζ3ζ3ζ3ζ32ζ32ζ32ζ32ζ3ζ3ζ3ζ32ζ32ζ32ζ32ζ3ζ3ζ32ζ3    linear of order 3
ρ111111ζ32ζ3-1-111-1-1ζ32ζ32ζ32ζ3ζ3ζ3ζ65ζ6ζ32ζ32ζ3ζ3ζ65ζ65ζ6ζ6ζ65ζ6    linear of order 6
ρ121111ζ3ζ32-1-111-1-1ζ3ζ3ζ3ζ32ζ32ζ32ζ6ζ65ζ3ζ3ζ32ζ32ζ6ζ6ζ65ζ65ζ6ζ65    linear of order 6
ρ131-11-111-i-1i-ii11-1-1-11-1-i-1i-ii-i1-1i-ii1    linear of order 4
ρ141-11-111i1i-i-i-11-1-1-11-1i1i-ii-i-11-ii-i-1    linear of order 4
ρ151-11-111-i1-iii-11-1-1-11-1-i1-ii-ii-11i-ii-1    linear of order 4
ρ161-11-111i-1-ii-i11-1-1-11-1i-1-ii-ii1-1-ii-i1    linear of order 4
ρ171-11-1ζ62ζ32ζ2-1ζ2ζ2ζ21ζ62ζ65ζ65ζ6ζ32ζ6ζ43ζ32ζ65ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32ζ32ζ6ζ4ζ3ζ43ζ3ζ4ζ32ζ62    linear of order 12
ρ181-11-1ζ32ζ62ζ2-1ζ2ζ2ζ21ζ32ζ6ζ6ζ65ζ62ζ65ζ4ζ3ζ6ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3ζ62ζ65ζ43ζ32ζ4ζ32ζ43ζ3ζ32    linear of order 12
ρ191-11-1ζ62ζ32ζ21ζ2ζ2ζ2-1ζ62ζ65ζ65ζ6ζ32ζ6ζ4ζ32ζ62ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32ζ6ζ32ζ43ζ3ζ4ζ3ζ43ζ32ζ65    linear of order 12
ρ201-11-1ζ32ζ62ζ21ζ2ζ2ζ2-1ζ32ζ6ζ6ζ65ζ62ζ65ζ4ζ3ζ32ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3ζ65ζ62ζ43ζ32ζ4ζ32ζ43ζ3ζ6    linear of order 12
ρ211-11-1ζ62ζ32ζ21ζ2ζ2ζ2-1ζ62ζ65ζ65ζ6ζ32ζ6ζ43ζ32ζ62ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32ζ6ζ32ζ4ζ3ζ43ζ3ζ4ζ32ζ65    linear of order 12
ρ221-11-1ζ62ζ32ζ2-1ζ2ζ2ζ21ζ62ζ65ζ65ζ6ζ32ζ6ζ4ζ32ζ65ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32ζ32ζ6ζ43ζ3ζ4ζ3ζ43ζ32ζ62    linear of order 12
ρ231-11-1ζ32ζ62ζ2-1ζ2ζ2ζ21ζ32ζ6ζ6ζ65ζ62ζ65ζ43ζ3ζ6ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3ζ62ζ65ζ4ζ32ζ43ζ32ζ4ζ3ζ32    linear of order 12
ρ241-11-1ζ32ζ62ζ21ζ2ζ2ζ2-1ζ32ζ6ζ6ζ65ζ62ζ65ζ43ζ3ζ32ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3ζ65ζ62ζ4ζ32ζ43ζ32ζ4ζ3ζ6    linear of order 12
ρ252-2-2222000000-2-222-2-2000000000000    orthogonal lifted from D4
ρ2622-2-222000000-22-2-2-22000000000000    symplectic lifted from Q8, Schur index 2
ρ272-2-22-1--3-1+-30000001+-31+-3-1--3-1+-31--31--3000000000000    complex lifted from C3×D4
ρ2822-2-2-1--3-1+-30000001+-3-1--31+-31--31--3-1+-3000000000000    complex lifted from C3×Q8
ρ2922-2-2-1+-3-1--30000001--3-1+-31--31+-31+-3-1--3000000000000    complex lifted from C3×Q8
ρ302-2-22-1+-3-1--30000001--31--3-1+-3-1--31+-31+-3000000000000    complex lifted from C3×D4

Smallest permutation representation of C3×C4⋊C4
Regular action on 48 points
Generators in S48
(1 28 23)(2 25 24)(3 26 21)(4 27 22)(5 9 18)(6 10 19)(7 11 20)(8 12 17)(13 45 34)(14 46 35)(15 47 36)(16 48 33)(29 40 44)(30 37 41)(31 38 42)(32 39 43)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)
(1 38 8 34)(2 37 5 33)(3 40 6 36)(4 39 7 35)(9 16 25 41)(10 15 26 44)(11 14 27 43)(12 13 28 42)(17 45 23 31)(18 48 24 30)(19 47 21 29)(20 46 22 32)

G:=sub<Sym(48)| (1,28,23)(2,25,24)(3,26,21)(4,27,22)(5,9,18)(6,10,19)(7,11,20)(8,12,17)(13,45,34)(14,46,35)(15,47,36)(16,48,33)(29,40,44)(30,37,41)(31,38,42)(32,39,43), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,38,8,34)(2,37,5,33)(3,40,6,36)(4,39,7,35)(9,16,25,41)(10,15,26,44)(11,14,27,43)(12,13,28,42)(17,45,23,31)(18,48,24,30)(19,47,21,29)(20,46,22,32)>;

G:=Group( (1,28,23)(2,25,24)(3,26,21)(4,27,22)(5,9,18)(6,10,19)(7,11,20)(8,12,17)(13,45,34)(14,46,35)(15,47,36)(16,48,33)(29,40,44)(30,37,41)(31,38,42)(32,39,43), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,38,8,34)(2,37,5,33)(3,40,6,36)(4,39,7,35)(9,16,25,41)(10,15,26,44)(11,14,27,43)(12,13,28,42)(17,45,23,31)(18,48,24,30)(19,47,21,29)(20,46,22,32) );

G=PermutationGroup([(1,28,23),(2,25,24),(3,26,21),(4,27,22),(5,9,18),(6,10,19),(7,11,20),(8,12,17),(13,45,34),(14,46,35),(15,47,36),(16,48,33),(29,40,44),(30,37,41),(31,38,42),(32,39,43)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48)], [(1,38,8,34),(2,37,5,33),(3,40,6,36),(4,39,7,35),(9,16,25,41),(10,15,26,44),(11,14,27,43),(12,13,28,42),(17,45,23,31),(18,48,24,30),(19,47,21,29),(20,46,22,32)])

Matrix representation of C3×C4⋊C4 in GL3(𝔽13) generated by

100
090
009
,
1200
0012
010
,
500
005
050
G:=sub<GL(3,GF(13))| [1,0,0,0,9,0,0,0,9],[12,0,0,0,0,1,0,12,0],[5,0,0,0,0,5,0,5,0] >;

C3×C4⋊C4 in GAP, Magma, Sage, TeX

C_3\times C_4\rtimes C_4
% in TeX

G:=Group("C3xC4:C4");
// GroupNames label

G:=SmallGroup(48,22);
// by ID

G=gap.SmallGroup(48,22);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-2,120,141,66]);
// Polycyclic

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

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