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G = (C3×C24).C4  order 288 = 25·32

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

metabelian, soluble, monomial

Aliases: (C3×C24).4C4, C8.2(C32⋊C4), C3⋊Dic3.35D4, C32⋊4C8.7C4, C32⋊3(C8.C4), C32⋊M4(2).2C2, (C2×C3⋊S3).5Q8, (C8×C3⋊S3).12C2, C4.10(C2×C32⋊C4), (C3×C6).13(C4⋊C4), (C3×C12).14(C2×C4), C2.6(C4⋊(C32⋊C4)), (C4×C3⋊S3).82C22, SmallGroup(288,418)

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 >

C1
C2
18C2
2C3
2C3
C4
9C22
9C4
2C6
2C6
6S3
6S3
6S3
6S3
C32
C8
9C2×C4
9C8
18C8
18C8
2C12
2C12
6Dic3
6D6
6D6
6Dic3
C3×C6
2C3⋊S3
9M4(2)
9M4(2)
9C2×C8
2C24
2C24
6C3⋊C8
6C4×S3
6C4×S3
6C3⋊C8
C2×C3⋊S3
C3×C12
C3⋊Dic3
9C8.C4
6S3×C8
6S3×C8
C4×C3⋊S3
C3×C24
C32⋊4C8
2C32⋊2C8
2C32⋊2C8
C32⋊M4(2)
C32⋊M4(2)
C8×C3⋊S3
(C3×C24).C4

Character table of (C3×C24).C4

 class 12A2B3A3B4A4B4C6A6B8A8B8C8D8E8F8G8H12A12B12C12D24A24B24C24D24E24F24G24H
 size 1118442994422181836363636444444444444
ρ1111111111111111111111111111111    trivial
ρ21111111111-1-1-1-11-11-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ311111111111111-1-1-1-1111111111111    linear of order 2
ρ41111111111-1-1-1-1-11-111111-1-1-1-1-1-1-1-1    linear of order 2
ρ511-1111-1-11111-1-1ii-i-i111111111111    linear of order 4
ρ611-1111-1-111-1-111i-i-ii1111-1-1-1-1-1-1-1-1    linear of order 4
ρ711-1111-1-11111-1-1-i-iii111111111111    linear of order 4
ρ811-1111-1-111-1-111-iii-i1111-1-1-1-1-1-1-1-1    linear of order 4
ρ922-222-2222200000000-2-2-2-200000000    orthogonal lifted from D4
ρ1022222-2-2-22200000000-2-2-2-200000000    symplectic lifted from Q8, Schur index 2
ρ112-202202i-2i-2-2√-2-√-2√2-√200000000-√-2√-2√-2√-2√-2-√-2-√-2-√-2    complex lifted from C8.C4
ρ122-20220-2i2i-2-2-√-2√-2√2-√200000000√-2-√-2-√-2-√-2-√-2√-2√-2√-2    complex lifted from C8.C4
ρ132-20220-2i2i-2-2√-2-√-2-√2√200000000-√-2√-2√-2√-2√-2-√-2-√-2-√-2    complex lifted from C8.C4
ρ142-202202i-2i-2-2-√-2√-2-√2√200000000√-2-√-2-√-2-√-2-√-2√-2√-2√-2    complex lifted from C8.C4
ρ154401-2400-2144000000-2-211-2-2-211-211    orthogonal lifted from C32⋊C4
ρ16440-214001-24400000011-2-2111-2-21-2-2    orthogonal lifted from C32⋊C4
ρ17440-214001-2-4-400000011-2-2-1-1-122-122    orthogonal lifted from C2×C32⋊C4
ρ184401-2400-21-4-4000000-2-211222-1-12-1-1    orthogonal lifted from C2×C32⋊C4
ρ19440-21-4001-200000000-1-1223i3i-3i00-3i00    complex lifted from C4⋊(C32⋊C4)
ρ204401-2-400-210000000022-1-10003i-3i03i-3i    complex lifted from C4⋊(C32⋊C4)
ρ214401-2-400-210000000022-1-1000-3i3i0-3i3i    complex lifted from C4⋊(C32⋊C4)
ρ22440-21-4001-200000000-1-122-3i-3i3i003i00    complex lifted from C4⋊(C32⋊C4)
ρ234-40-21000-122√-2-2√-2000000-3i3i00-2ζ83+ζ8-2ζ87+ζ85ζ87-2ζ85-√-2-√-2ζ83-2ζ8√-2√-2    complex faithful
ρ244-401-20002-12√-2-2√-200000000-3i3i√-2-√-2-√-2-2ζ87+ζ85ζ87-2ζ85√-2-2ζ83+ζ8ζ83-2ζ8    complex faithful
ρ254-401-20002-12√-2-2√-2000000003i-3i√-2-√-2-√-2ζ87-2ζ85-2ζ87+ζ85√-2ζ83-2ζ8-2ζ83+ζ8    complex faithful
ρ264-40-21000-12-2√-22√-20000003i-3i00ζ87-2ζ85ζ83-2ζ8-2ζ83+ζ8√-2√-2-2ζ87+ζ85-√-2-√-2    complex faithful
ρ274-40-21000-122√-2-2√-20000003i-3i00ζ83-2ζ8ζ87-2ζ85-2ζ87+ζ85-√-2-√-2-2ζ83+ζ8√-2√-2    complex faithful
ρ284-401-20002-1-2√-22√-200000000-3i3i-√-2√-2√-2-2ζ83+ζ8ζ83-2ζ8-√-2-2ζ87+ζ85ζ87-2ζ85    complex faithful
ρ294-401-20002-1-2√-22√-2000000003i-3i-√-2√-2√-2ζ83-2ζ8-2ζ83+ζ8-√-2ζ87-2ζ85-2ζ87+ζ85    complex faithful
ρ304-40-21000-12-2√-22√-2000000-3i3i00-2ζ87+ζ85-2ζ83+ζ8ζ83-2ζ8√-2√-2ζ87-2ζ85-√-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

72100
72000
00721
00720
,
10000
01000
002251
00220
,
0010
0001
462700
02700
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
 

Export

Subgroup lattice of (C3×C24).C4 in TeX
Character table of (C3×C24).C4 in TeX

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