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## G = (C3×C24).C4order 288 = 25·32

### 4th non-split extension by C3×C24 of C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — (C3×C24).C4
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C4×C3⋊S3 — C32⋊M4(2) — (C3×C24).C4
 Lower central C32 — C3×C6 — C3×C12 — (C3×C24).C4
 Upper central C1 — C2 — C4 — C8

Generators and relations for (C3×C24).C4
G = < a,b,c | a3=b24=1, c4=b12, ab=ba, cac-1=a-1b8, cbc-1=a-1b19 >

Character table of (C3×C24).C4

 class 1 2A 2B 3A 3B 4A 4B 4C 6A 6B 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D 24A 24B 24C 24D 24E 24F 24G 24H size 1 1 18 4 4 2 9 9 4 4 2 2 18 18 36 36 36 36 4 4 4 4 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 -1 1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ5 1 1 -1 1 1 1 -1 -1 1 1 1 1 -1 -1 i i -i -i 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ6 1 1 -1 1 1 1 -1 -1 1 1 -1 -1 1 1 i -i -i i 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ7 1 1 -1 1 1 1 -1 -1 1 1 1 1 -1 -1 -i -i i i 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ8 1 1 -1 1 1 1 -1 -1 1 1 -1 -1 1 1 -i i i -i 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ9 2 2 -2 2 2 -2 2 2 2 2 0 0 0 0 0 0 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ11 2 -2 0 2 2 0 2i -2i -2 -2 √-2 -√-2 √2 -√2 0 0 0 0 0 0 0 0 -√-2 √-2 √-2 √-2 √-2 -√-2 -√-2 -√-2 complex lifted from C8.C4 ρ12 2 -2 0 2 2 0 -2i 2i -2 -2 -√-2 √-2 √2 -√2 0 0 0 0 0 0 0 0 √-2 -√-2 -√-2 -√-2 -√-2 √-2 √-2 √-2 complex lifted from C8.C4 ρ13 2 -2 0 2 2 0 -2i 2i -2 -2 √-2 -√-2 -√2 √2 0 0 0 0 0 0 0 0 -√-2 √-2 √-2 √-2 √-2 -√-2 -√-2 -√-2 complex lifted from C8.C4 ρ14 2 -2 0 2 2 0 2i -2i -2 -2 -√-2 √-2 -√2 √2 0 0 0 0 0 0 0 0 √-2 -√-2 -√-2 -√-2 -√-2 √-2 √-2 √-2 complex lifted from C8.C4 ρ15 4 4 0 1 -2 4 0 0 -2 1 4 4 0 0 0 0 0 0 -2 -2 1 1 -2 -2 -2 1 1 -2 1 1 orthogonal lifted from C32⋊C4 ρ16 4 4 0 -2 1 4 0 0 1 -2 4 4 0 0 0 0 0 0 1 1 -2 -2 1 1 1 -2 -2 1 -2 -2 orthogonal lifted from C32⋊C4 ρ17 4 4 0 -2 1 4 0 0 1 -2 -4 -4 0 0 0 0 0 0 1 1 -2 -2 -1 -1 -1 2 2 -1 2 2 orthogonal lifted from C2×C32⋊C4 ρ18 4 4 0 1 -2 4 0 0 -2 1 -4 -4 0 0 0 0 0 0 -2 -2 1 1 2 2 2 -1 -1 2 -1 -1 orthogonal lifted from C2×C32⋊C4 ρ19 4 4 0 -2 1 -4 0 0 1 -2 0 0 0 0 0 0 0 0 -1 -1 2 2 3i 3i -3i 0 0 -3i 0 0 complex lifted from C4⋊(C32⋊C4) ρ20 4 4 0 1 -2 -4 0 0 -2 1 0 0 0 0 0 0 0 0 2 2 -1 -1 0 0 0 3i -3i 0 3i -3i complex lifted from C4⋊(C32⋊C4) ρ21 4 4 0 1 -2 -4 0 0 -2 1 0 0 0 0 0 0 0 0 2 2 -1 -1 0 0 0 -3i 3i 0 -3i 3i complex lifted from C4⋊(C32⋊C4) ρ22 4 4 0 -2 1 -4 0 0 1 -2 0 0 0 0 0 0 0 0 -1 -1 2 2 -3i -3i 3i 0 0 3i 0 0 complex lifted from C4⋊(C32⋊C4) ρ23 4 -4 0 -2 1 0 0 0 -1 2 2√-2 -2√-2 0 0 0 0 0 0 -3i 3i 0 0 -2ζ83+ζ8 -2ζ87+ζ85 ζ87-2ζ85 -√-2 -√-2 ζ83-2ζ8 √-2 √-2 complex faithful ρ24 4 -4 0 1 -2 0 0 0 2 -1 2√-2 -2√-2 0 0 0 0 0 0 0 0 -3i 3i √-2 -√-2 -√-2 -2ζ87+ζ85 ζ87-2ζ85 √-2 -2ζ83+ζ8 ζ83-2ζ8 complex faithful ρ25 4 -4 0 1 -2 0 0 0 2 -1 2√-2 -2√-2 0 0 0 0 0 0 0 0 3i -3i √-2 -√-2 -√-2 ζ87-2ζ85 -2ζ87+ζ85 √-2 ζ83-2ζ8 -2ζ83+ζ8 complex faithful ρ26 4 -4 0 -2 1 0 0 0 -1 2 -2√-2 2√-2 0 0 0 0 0 0 3i -3i 0 0 ζ87-2ζ85 ζ83-2ζ8 -2ζ83+ζ8 √-2 √-2 -2ζ87+ζ85 -√-2 -√-2 complex faithful ρ27 4 -4 0 -2 1 0 0 0 -1 2 2√-2 -2√-2 0 0 0 0 0 0 3i -3i 0 0 ζ83-2ζ8 ζ87-2ζ85 -2ζ87+ζ85 -√-2 -√-2 -2ζ83+ζ8 √-2 √-2 complex faithful ρ28 4 -4 0 1 -2 0 0 0 2 -1 -2√-2 2√-2 0 0 0 0 0 0 0 0 -3i 3i -√-2 √-2 √-2 -2ζ83+ζ8 ζ83-2ζ8 -√-2 -2ζ87+ζ85 ζ87-2ζ85 complex faithful ρ29 4 -4 0 1 -2 0 0 0 2 -1 -2√-2 2√-2 0 0 0 0 0 0 0 0 3i -3i -√-2 √-2 √-2 ζ83-2ζ8 -2ζ83+ζ8 -√-2 ζ87-2ζ85 -2ζ87+ζ85 complex faithful ρ30 4 -4 0 -2 1 0 0 0 -1 2 -2√-2 2√-2 0 0 0 0 0 0 -3i 3i 0 0 -2ζ87+ζ85 -2ζ83+ζ8 ζ83-2ζ8 √-2 √-2 ζ87-2ζ85 -√-2 -√-2 complex faithful

Smallest permutation representation of (C3×C24).C4
On 48 points
Generators in S48
(1 9 24)(2 10 17)(3 11 18)(4 12 19)(5 13 20)(6 14 21)(7 15 22)(8 16 23)(25 41 33)(26 42 34)(27 43 35)(28 44 36)(29 45 37)(30 46 38)(31 47 39)(32 48 40)
(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 26 22 36 5 38 18 48)(2 29 23 39 6 41 19 27)(3 32 24 42 7 44 20 30)(4 35 17 45 8 47 21 33)(9 34 15 28 13 46 11 40)(10 37 16 31 14 25 12 43)

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

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

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

Matrix representation of (C3×C24).C4 in GL4(𝔽73) generated by

 72 1 0 0 72 0 0 0 0 0 72 1 0 0 72 0
,
 10 0 0 0 0 10 0 0 0 0 22 51 0 0 22 0
,
 0 0 1 0 0 0 0 1 46 27 0 0 0 27 0 0
G:=sub<GL(4,GF(73))| [72,72,0,0,1,0,0,0,0,0,72,72,0,0,1,0],[10,0,0,0,0,10,0,0,0,0,22,22,0,0,51,0],[0,0,46,0,0,0,27,27,1,0,0,0,0,1,0,0] >;

(C3×C24).C4 in GAP, Magma, Sage, TeX

(C_3\times C_{24}).C_4
% in TeX

G:=Group("(C3xC24).C4");
// GroupNames label

G:=SmallGroup(288,418);
// by ID

G=gap.SmallGroup(288,418);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,64,100,675,80,9413,691,12550,2372]);
// Polycyclic

G:=Group<a,b,c|a^3=b^24=1,c^4=b^12,a*b=b*a,c*a*c^-1=a^-1*b^8,c*b*c^-1=a^-1*b^19>;
// generators/relations

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