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G = (C3×C24).C4order 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, C324C8.7C4, C323(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

C1C3×C12 — (C3×C24).C4
C1C32C3×C6C3⋊Dic3C4×C3⋊S3C32⋊M4(2) — (C3×C24).C4
C32C3×C6C3×C12 — (C3×C24).C4
C1C2C4C8

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 >

18C2
2C3
2C3
9C22
9C4
2C6
2C6
6S3
6S3
6S3
6S3
9C2×C4
9C8
18C8
18C8
2C12
2C12
6Dic3
6D6
6D6
6Dic3
2C3⋊S3
9M4(2)
9M4(2)
9C2×C8
2C24
2C24
6C3⋊C8
6C4×S3
6C4×S3
6C3⋊C8
9C8.C4
6S3×C8
6S3×C8
2C322C8
2C322C8

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--22-200000000--2-2-2-2-2--2--2--2    complex lifted from C8.C4
ρ122-20220-2i2i-2-2--2-22-200000000-2--2--2--2--2-2-2-2    complex lifted from C8.C4
ρ132-20220-2i2i-2-2-2--2-2200000000--2-2-2-2-2--2--2--2    complex lifted from C8.C4
ρ142-202202i-2i-2-2--2-2-2200000000-2--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)
ρ234-40-21000-122-2-2-2000000-3i3i00-2ζ838-2ζ8785ζ87-2ζ85--2--2ζ83-2ζ8-2-2    complex faithful
ρ244-401-20002-12-2-2-200000000-3i3i-2--2--2-2ζ8785ζ87-2ζ85-2-2ζ838ζ83-2ζ8    complex faithful
ρ254-401-20002-12-2-2-2000000003i-3i-2--2--2ζ87-2ζ85-2ζ8785-2ζ83-2ζ8-2ζ838    complex faithful
ρ264-40-21000-12-2-22-20000003i-3i00ζ87-2ζ85ζ83-2ζ8-2ζ838-2-2-2ζ8785--2--2    complex faithful
ρ274-40-21000-122-2-2-20000003i-3i00ζ83-2ζ8ζ87-2ζ85-2ζ8785--2--2-2ζ838-2-2    complex faithful
ρ284-401-20002-1-2-22-200000000-3i3i--2-2-2-2ζ838ζ83-2ζ8--2-2ζ8785ζ87-2ζ85    complex faithful
ρ294-401-20002-1-2-22-2000000003i-3i--2-2-2ζ83-2ζ8-2ζ838--2ζ87-2ζ85-2ζ8785    complex faithful
ρ304-40-21000-12-2-22-2000000-3i3i00-2ζ8785-2ζ838ζ83-2ζ8-2-2ζ87-2ζ85--2--2    complex faithful

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

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

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

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

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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