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G = C8.(C32⋊C4)  order 288 = 25·32

1st non-split extension by C8 of C32⋊C4 acting via C32⋊C4/C3⋊S3=C2

metabelian, soluble, monomial

Aliases: (C3×C24).5C4, C8.1(C32⋊C4), C3⋊Dic3.36D4, C324C8.8C4, C324(C8.C4), C32⋊M4(2).3C2, (C2×C3⋊S3).6Q8, (C8×C3⋊S3).13C2, C4.11(C2×C32⋊C4), (C3×C6).14(C4⋊C4), (C3×C12).15(C2×C4), C2.7(C4⋊(C32⋊C4)), (C4×C3⋊S3).83C22, SmallGroup(288,419)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C8.(C32⋊C4)
C1C32C3×C6C3⋊Dic3C4×C3⋊S3C32⋊M4(2) — C8.(C32⋊C4)
C32C3×C6C3×C12 — C8.(C32⋊C4)
C1C2C4C8

Generators and relations for C8.(C32⋊C4)
 G = < a,b,c,d | a8=b3=c3=1, d4=a4, ab=ba, ac=ca, dad-1=a-1, dcd-1=bc=cb, dbd-1=b-1c >

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 C8.(C32⋊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-2-2-22-2--2000000002-2-2-2-2222    complex lifted from C8.C4
ρ122-20220-2i2i-2-22-2-2--200000000-22222-2-2-2    complex lifted from C8.C4
ρ132-202202i-2i-2-22-2--2-200000000-22222-2-2-2    complex lifted from C8.C4
ρ142-20220-2i2i-2-2-22--2-2000000002-2-2-2-2222    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)
ρ234-401-20002-122-2200000000-3i3i2-2-2ζ83+2ζ887852ζ87+2ζ85838    complex faithful
ρ244-40-21000-12-22220000003i-3i008785838ζ87+2ζ8522ζ83+2ζ8-2-2    complex faithful
ρ254-401-20002-1-2222000000003i-3i-222838ζ87+2ζ85-28785ζ83+2ζ8    complex faithful
ρ264-40-21000-12-2222000000-3i3i00ζ83+2ζ8ζ87+2ζ85838228785-2-2    complex faithful
ρ274-40-21000-1222-220000003i-3i008388785ζ83+2ζ8-2-2ζ87+2ζ8522    complex faithful
ρ284-401-20002-1-222200000000-3i3i-222ζ87+2ζ85838-2ζ83+2ζ88785    complex faithful
ρ294-40-21000-1222-22000000-3i3i00ζ87+2ζ85ζ83+2ζ88785-2-283822    complex faithful
ρ304-401-20002-122-22000000003i-3i2-2-28785ζ83+2ζ82838ζ87+2ζ85    complex faithful

Smallest permutation representation of C8.(C32⋊C4)
On 48 points
Generators in S48
(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 25)(2 39 26)(3 40 27)(4 33 28)(5 34 29)(6 35 30)(7 36 31)(8 37 32)(9 46 20)(10 47 21)(11 48 22)(12 41 23)(13 42 24)(14 43 17)(15 44 18)(16 45 19)
(9 20 46)(10 21 47)(11 22 48)(12 23 41)(13 24 42)(14 17 43)(15 18 44)(16 19 45)
(1 11 3 9 5 15 7 13)(2 10 4 16 6 14 8 12)(17 32 41 39 21 28 45 35)(18 31 42 38 22 27 46 34)(19 30 43 37 23 26 47 33)(20 29 44 36 24 25 48 40)

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

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

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

Matrix representation of C8.(C32⋊C4) in GL4(𝔽73) generated by

63044
06344
00510
00051
,
7272720
1001
00072
00172
,
1010
0110
00721
00720
,
2703334
2703333
19464646
462700
G:=sub<GL(4,GF(73))| [63,0,0,0,0,63,0,0,4,4,51,0,4,4,0,51],[72,1,0,0,72,0,0,0,72,0,0,1,0,1,72,72],[1,0,0,0,0,1,0,0,1,1,72,72,0,0,1,0],[27,27,19,46,0,0,46,27,33,33,46,0,34,33,46,0] >;

C8.(C32⋊C4) in GAP, Magma, Sage, TeX

C_8.(C_3^2\rtimes C_4)
% in TeX

G:=Group("C8.(C3^2:C4)");
// GroupNames label

G:=SmallGroup(288,419);
// by ID

G=gap.SmallGroup(288,419);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,176,100,675,80,9413,691,12550,2372]);
// Polycyclic

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

Export

Subgroup lattice of C8.(C32⋊C4) in TeX
Character table of C8.(C32⋊C4) in TeX

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