G = C8⋊(C32⋊C4) order 288 = 25·32
2nd semidirect product of C8 and C32⋊C4 acting via C32⋊C4/C3⋊S3=C2
Aliases: (C3×C24)⋊3C4, C8⋊2(C32⋊C4), C3⋊S3.5SD16, C3⋊Dic3.6Q8, C32⋊4C8⋊11C4, C32⋊4(C4.Q8), C4.8(C2×C32⋊C4), (C8×C3⋊S3).10C2, (C2×C3⋊S3).38D4, (C3×C6).11(C4⋊C4), (C3×C12).12(C2×C4), C4⋊(C32⋊C4).6C2, C2.4(C4⋊(C32⋊C4)), (C4×C3⋊S3).80C22, SmallGroup(288,416)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊(C32⋊C4)
G = < a,b,c,d | a8=b3=c3=d4=1, ab=ba, ac=ca, dad-1=a3, dcd-1=bc=cb, dbd-1=b-1c >
Subgroups: 328 in 58 conjugacy classes, 18 normal (14 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C8, C2×C4, C32, Dic3, C12, D6, C4⋊C4, C2×C8, C3⋊S3, C3×C6, C3⋊C8, C24, C4×S3, C4.Q8, C3⋊Dic3, C3×C12, C32⋊C4, C2×C3⋊S3, S3×C8, C32⋊4C8, C3×C24, C4×C3⋊S3, C2×C32⋊C4, C8×C3⋊S3, C4⋊(C32⋊C4), C8⋊(C32⋊C4)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C4⋊C4, SD16, C4.Q8, C32⋊C4, C2×C32⋊C4, C4⋊(C32⋊C4), C8⋊(C32⋊C4)
Character table of C8⋊(C32⋊C4)
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | |
| size | 1 | 1 | 9 | 9 | 4 | 4 | 2 | 18 | 36 | 36 | 36 | 36 | 4 | 4 | 2 | 2 | 18 | 18 | 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 | i | -i | -i | i | 1 | 1 | -1 | -1 | 1 | 1 | 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 | -i | -i | i | i | 1 | 1 | 1 | 1 | -1 | -1 | 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 | i | i | -i | -i | 1 | 1 | 1 | 1 | -1 | -1 | 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 | -i | i | i | -i | 1 | 1 | -1 | -1 | 1 | 1 | 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 | 0 | 0 | 0 | 0 | 2 | 2 | 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 | 0 | 0 | 0 | 0 | 2 | 2 | 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 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -√-2 | √-2 | √-2 | -√-2 | 0 | 0 | 0 | 0 | √-2 | √-2 | -√-2 | -√-2 | -√-2 | -√-2 | √-2 | √-2 | complex lifted from SD16 |
| ρ12 | 2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | √-2 | -√-2 | √-2 | -√-2 | 0 | 0 | 0 | 0 | -√-2 | -√-2 | √-2 | √-2 | √-2 | √-2 | -√-2 | -√-2 | complex lifted from SD16 |
| ρ13 | 2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -√-2 | √-2 | -√-2 | √-2 | 0 | 0 | 0 | 0 | √-2 | √-2 | -√-2 | -√-2 | -√-2 | -√-2 | √-2 | √-2 | complex lifted from SD16 |
| ρ14 | 2 | -2 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | √-2 | -√-2 | -√-2 | √-2 | 0 | 0 | 0 | 0 | -√-2 | -√-2 | √-2 | √-2 | √-2 | √-2 | -√-2 | -√-2 | complex lifted from SD16 |
| ρ15 | 4 | 4 | 0 | 0 | -2 | 1 | 4 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | 4 | 4 | 0 | 0 | 1 | -2 | 1 | -2 | -2 | 1 | -2 | -2 | 1 | 1 | 1 | -2 | orthogonal lifted from C32⋊C4 |
| ρ16 | 4 | 4 | 0 | 0 | -2 | 1 | 4 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | -4 | -4 | 0 | 0 | 1 | -2 | 1 | -2 | 2 | -1 | 2 | 2 | -1 | -1 | -1 | 2 | orthogonal lifted from C2×C32⋊C4 |
| ρ17 | 4 | 4 | 0 | 0 | 1 | -2 | 4 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | 4 | 4 | 0 | 0 | -2 | 1 | -2 | 1 | 1 | -2 | 1 | 1 | -2 | -2 | -2 | 1 | orthogonal lifted from C32⋊C4 |
| ρ18 | 4 | 4 | 0 | 0 | 1 | -2 | 4 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | -4 | -4 | 0 | 0 | -2 | 1 | -2 | 1 | -1 | 2 | -1 | -1 | 2 | 2 | 2 | -1 | orthogonal lifted from C2×C32⋊C4 |
| ρ19 | 4 | 4 | 0 | 0 | 1 | -2 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 2 | -1 | 2 | -1 | -3i | 0 | 3i | -3i | 0 | 0 | 0 | 3i | complex lifted from C4⋊(C32⋊C4) |
| ρ20 | 4 | 4 | 0 | 0 | -2 | 1 | -4 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | 2 | 0 | 3i | 0 | 0 | 3i | -3i | -3i | 0 | complex lifted from C4⋊(C32⋊C4) |
| ρ21 | 4 | 4 | 0 | 0 | -2 | 1 | -4 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | 2 | 0 | -3i | 0 | 0 | -3i | 3i | 3i | 0 | complex lifted from C4⋊(C32⋊C4) |
| ρ22 | 4 | 4 | 0 | 0 | 1 | -2 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 2 | -1 | 2 | -1 | 3i | 0 | -3i | 3i | 0 | 0 | 0 | -3i | complex lifted from C4⋊(C32⋊C4) |
| ρ23 | 4 | -4 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | -2√-2 | 2√-2 | 0 | 0 | 0 | -3i | 0 | 3i | 2ζ83-ζ8 | -√-2 | -ζ87+2ζ85 | 2ζ87-ζ85 | √-2 | √-2 | -√-2 | -ζ83+2ζ8 | complex faithful |
| ρ24 | 4 | -4 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | 2√-2 | -2√-2 | 0 | 0 | 0 | -3i | 0 | 3i | 2ζ87-ζ85 | √-2 | -ζ83+2ζ8 | 2ζ83-ζ8 | -√-2 | -√-2 | √-2 | -ζ87+2ζ85 | complex faithful |
| ρ25 | 4 | -4 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | 2√-2 | -2√-2 | 0 | 0 | 0 | 3i | 0 | -3i | -ζ87+2ζ85 | √-2 | 2ζ83-ζ8 | -ζ83+2ζ8 | -√-2 | -√-2 | √-2 | 2ζ87-ζ85 | complex faithful |
| ρ26 | 4 | -4 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2 | -2√-2 | 2√-2 | 0 | 0 | -3i | 0 | 3i | 0 | -√-2 | -ζ83+2ζ8 | √-2 | √-2 | -ζ87+2ζ85 | 2ζ87-ζ85 | 2ζ83-ζ8 | -√-2 | complex faithful |
| ρ27 | 4 | -4 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | -2√-2 | 2√-2 | 0 | 0 | 0 | 3i | 0 | -3i | -ζ83+2ζ8 | -√-2 | 2ζ87-ζ85 | -ζ87+2ζ85 | √-2 | √-2 | -√-2 | 2ζ83-ζ8 | complex faithful |
| ρ28 | 4 | -4 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2 | -2√-2 | 2√-2 | 0 | 0 | 3i | 0 | -3i | 0 | -√-2 | 2ζ83-ζ8 | √-2 | √-2 | 2ζ87-ζ85 | -ζ87+2ζ85 | -ζ83+2ζ8 | -√-2 | complex faithful |
| ρ29 | 4 | -4 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2 | 2√-2 | -2√-2 | 0 | 0 | 3i | 0 | -3i | 0 | √-2 | 2ζ87-ζ85 | -√-2 | -√-2 | 2ζ83-ζ8 | -ζ83+2ζ8 | -ζ87+2ζ85 | √-2 | complex faithful |
| ρ30 | 4 | -4 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2 | 2√-2 | -2√-2 | 0 | 0 | -3i | 0 | 3i | 0 | √-2 | -ζ87+2ζ85 | -√-2 | -√-2 | -ζ83+2ζ8 | 2ζ83-ζ8 | 2ζ87-ζ85 | √-2 | complex faithful |
►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)
(9 19 46)(10 20 47)(11 21 48)(12 22 41)(13 23 42)(14 24 43)(15 17 44)(16 18 45)
(1 36 29)(2 37 30)(3 38 31)(4 39 32)(5 40 25)(6 33 26)(7 34 27)(8 35 28)(9 19 46)(10 20 47)(11 21 48)(12 22 41)(13 23 42)(14 24 43)(15 17 44)(16 18 45)
(1 9)(2 12)(3 15)(4 10)(5 13)(6 16)(7 11)(8 14)(17 31 44 38)(18 26 45 33)(19 29 46 36)(20 32 47 39)(21 27 48 34)(22 30 41 37)(23 25 42 40)(24 28 43 35)
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), (9,19,46)(10,20,47)(11,21,48)(12,22,41)(13,23,42)(14,24,43)(15,17,44)(16,18,45), (1,36,29)(2,37,30)(3,38,31)(4,39,32)(5,40,25)(6,33,26)(7,34,27)(8,35,28)(9,19,46)(10,20,47)(11,21,48)(12,22,41)(13,23,42)(14,24,43)(15,17,44)(16,18,45), (1,9)(2,12)(3,15)(4,10)(5,13)(6,16)(7,11)(8,14)(17,31,44,38)(18,26,45,33)(19,29,46,36)(20,32,47,39)(21,27,48,34)(22,30,41,37)(23,25,42,40)(24,28,43,35)>;
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), (9,19,46)(10,20,47)(11,21,48)(12,22,41)(13,23,42)(14,24,43)(15,17,44)(16,18,45), (1,36,29)(2,37,30)(3,38,31)(4,39,32)(5,40,25)(6,33,26)(7,34,27)(8,35,28)(9,19,46)(10,20,47)(11,21,48)(12,22,41)(13,23,42)(14,24,43)(15,17,44)(16,18,45), (1,9)(2,12)(3,15)(4,10)(5,13)(6,16)(7,11)(8,14)(17,31,44,38)(18,26,45,33)(19,29,46,36)(20,32,47,39)(21,27,48,34)(22,30,41,37)(23,25,42,40)(24,28,43,35) );
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)], [(9,19,46),(10,20,47),(11,21,48),(12,22,41),(13,23,42),(14,24,43),(15,17,44),(16,18,45)], [(1,36,29),(2,37,30),(3,38,31),(4,39,32),(5,40,25),(6,33,26),(7,34,27),(8,35,28),(9,19,46),(10,20,47),(11,21,48),(12,22,41),(13,23,42),(14,24,43),(15,17,44),(16,18,45)], [(1,9),(2,12),(3,15),(4,10),(5,13),(6,16),(7,11),(8,14),(17,31,44,38),(18,26,45,33),(19,29,46,36),(20,32,47,39),(21,27,48,34),(22,30,41,37),(23,25,42,40),(24,28,43,35)]])
Matrix representation of C8⋊(C32⋊C4) ►in GL6(𝔽73)
| 67 | 6 | 0 | 0 | 0 | 0 |
| 67 | 67 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 0 | 0 | 46 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 8 | 64 | 0 | 0 | 0 | 0 |
| 64 | 65 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
G:=sub<GL(6,GF(73))| [67,67,0,0,0,0,6,67,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,46,0,0,0,0,0,0,46],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[8,64,0,0,0,0,64,65,0,0,0,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,1,0,0,0,0,0,0,1,0,0] >;
C8⋊(C32⋊C4) in GAP, Magma, Sage, TeX
C_8\rtimes (C_3^2\rtimes C_4)
% in TeX
G:=Group("C8:(C3^2:C4)"); // GroupNames label
G:=SmallGroup(288,416);
// by ID
G=gap.SmallGroup(288,416);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,64,675,80,9413,691,12550,2372]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^3=c^3=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^3,d*c*d^-1=b*c=c*b,d*b*d^-1=b^-1*c>;
// generators/relations
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