direct product, metabelian, soluble, monomial
Aliases: Q8×C32⋊C4, C32⋊8(C4×Q8), (Q8×C32)⋊3C4, C32⋊4Q8⋊3C4, C4.7(C2×C32⋊C4), C3⋊S3.6(C2×Q8), (Q8×C3⋊S3).4C2, (C3×C12).7(C2×C4), (C4×C32⋊C4).2C2, C4⋊(C32⋊C4).3C2, C3⋊S3.12(C4○D4), (C2×C3⋊S3).38C23, (C4×C3⋊S3).40C22, C3⋊Dic3.24(C2×C4), (C3×C6).33(C22×C4), C2.11(C22×C32⋊C4), (C2×C32⋊C4).25C22, SmallGroup(288,938)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C32⋊C4
G = < a,b,c,d,e | a4=c3=d3=e4=1, b2=a2, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ede-1=cd=dc, ece-1=c-1d >
Subgroups: 512 in 108 conjugacy classes, 38 normal (12 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C2×C4, Q8, Q8, C32, Dic3, C12, D6, C42, C4⋊C4, C2×Q8, C3⋊S3, C3×C6, Dic6, C4×S3, C3×Q8, C4×Q8, C3⋊Dic3, C3×C12, C32⋊C4, C32⋊C4, C2×C3⋊S3, S3×Q8, C32⋊4Q8, C4×C3⋊S3, Q8×C32, C2×C32⋊C4, C2×C32⋊C4, C4×C32⋊C4, C4⋊(C32⋊C4), Q8×C3⋊S3, Q8×C32⋊C4
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, C22×C4, C2×Q8, C4○D4, C4×Q8, C32⋊C4, C2×C32⋊C4, C22×C32⋊C4, Q8×C32⋊C4
Character table of Q8×C32⋊C4
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 6A | 6B | 12A | 12B | 12C | 12D | 12E | 12F | |
| size | 1 | 1 | 9 | 9 | 4 | 4 | 2 | 2 | 2 | 9 | 9 | 9 | 9 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | |
| 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 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | linear of order 2 |
| ρ6 | 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 |
| ρ7 | 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 |
| ρ8 | 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 |
| ρ9 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | i | i | -i | -i | -i | -1 | -1 | i | -i | i | -i | i | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ10 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -i | -i | i | i | i | -1 | 1 | i | -i | i | -i | -i | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | linear of order 4 |
| ρ11 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -i | -i | i | i | i | -1 | -1 | -i | i | -i | i | -i | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ12 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | i | i | -i | -i | -i | -1 | 1 | -i | i | -i | i | i | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | linear of order 4 |
| ρ13 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -i | -i | i | i | -i | 1 | 1 | -i | i | i | -i | i | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | linear of order 4 |
| ρ14 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | i | i | -i | -i | i | 1 | -1 | -i | i | i | -i | -i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | linear of order 4 |
| ρ15 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | i | i | -i | -i | i | 1 | 1 | i | -i | -i | i | -i | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | linear of order 4 |
| ρ16 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -i | -i | i | i | -i | 1 | -1 | i | -i | -i | i | i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | linear of order 4 |
| ρ17 | 2 | -2 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ18 | 2 | -2 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ19 | 2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ20 | 2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ21 | 4 | 4 | 0 | 0 | 1 | -2 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | 1 | 2 | -1 | 2 | -1 | -2 | orthogonal lifted from C2×C32⋊C4 |
| ρ22 | 4 | 4 | 0 | 0 | -2 | 1 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | 2 | 1 | -2 | -1 | 2 | -1 | orthogonal lifted from C2×C32⋊C4 |
| ρ23 | 4 | 4 | 0 | 0 | -2 | 1 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | 2 | -1 | 2 | 1 | -2 | -1 | orthogonal lifted from C2×C32⋊C4 |
| ρ24 | 4 | 4 | 0 | 0 | 1 | -2 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | 1 | -2 | 1 | -2 | 1 | -2 | orthogonal lifted from C32⋊C4 |
| ρ25 | 4 | 4 | 0 | 0 | -2 | 1 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | -2 | -1 | 2 | -1 | 2 | 1 | orthogonal lifted from C2×C32⋊C4 |
| ρ26 | 4 | 4 | 0 | 0 | 1 | -2 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | -1 | -2 | 1 | 2 | -1 | 2 | orthogonal lifted from C2×C32⋊C4 |
| ρ27 | 4 | 4 | 0 | 0 | 1 | -2 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | -1 | 2 | -1 | -2 | 1 | 2 | orthogonal lifted from C2×C32⋊C4 |
| ρ28 | 4 | 4 | 0 | 0 | -2 | 1 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | -2 | 1 | -2 | 1 | -2 | 1 | orthogonal lifted from C32⋊C4 |
| ρ29 | 8 | -8 | 0 | 0 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ30 | 8 | -8 | 0 | 0 | 2 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►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 24 3 22)(2 23 4 21)(5 44 7 42)(6 43 8 41)(9 40 11 38)(10 39 12 37)(13 30 15 32)(14 29 16 31)(17 26 19 28)(18 25 20 27)(33 45 35 47)(34 48 36 46)
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 10 45)(6 11 46)(7 12 47)(8 9 48)(21 25 30)(22 26 31)(23 27 32)(24 28 29)(33 42 37)(34 43 38)(35 44 39)(36 41 40)
(5 45 10)(6 46 11)(7 47 12)(8 48 9)(33 37 42)(34 38 43)(35 39 44)(36 40 41)
(1 34 3 36)(2 35 4 33)(5 27 12 30)(6 28 9 31)(7 25 10 32)(8 26 11 29)(13 42 20 39)(14 43 17 40)(15 44 18 37)(16 41 19 38)(21 45 23 47)(22 46 24 48)
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), (1,24,3,22)(2,23,4,21)(5,44,7,42)(6,43,8,41)(9,40,11,38)(10,39,12,37)(13,30,15,32)(14,29,16,31)(17,26,19,28)(18,25,20,27)(33,45,35,47)(34,48,36,46), (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,10,45)(6,11,46)(7,12,47)(8,9,48)(21,25,30)(22,26,31)(23,27,32)(24,28,29)(33,42,37)(34,43,38)(35,44,39)(36,41,40), (5,45,10)(6,46,11)(7,47,12)(8,48,9)(33,37,42)(34,38,43)(35,39,44)(36,40,41), (1,34,3,36)(2,35,4,33)(5,27,12,30)(6,28,9,31)(7,25,10,32)(8,26,11,29)(13,42,20,39)(14,43,17,40)(15,44,18,37)(16,41,19,38)(21,45,23,47)(22,46,24,48)>;
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), (1,24,3,22)(2,23,4,21)(5,44,7,42)(6,43,8,41)(9,40,11,38)(10,39,12,37)(13,30,15,32)(14,29,16,31)(17,26,19,28)(18,25,20,27)(33,45,35,47)(34,48,36,46), (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,10,45)(6,11,46)(7,12,47)(8,9,48)(21,25,30)(22,26,31)(23,27,32)(24,28,29)(33,42,37)(34,43,38)(35,44,39)(36,41,40), (5,45,10)(6,46,11)(7,47,12)(8,48,9)(33,37,42)(34,38,43)(35,39,44)(36,40,41), (1,34,3,36)(2,35,4,33)(5,27,12,30)(6,28,9,31)(7,25,10,32)(8,26,11,29)(13,42,20,39)(14,43,17,40)(15,44,18,37)(16,41,19,38)(21,45,23,47)(22,46,24,48) );
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)], [(1,24,3,22),(2,23,4,21),(5,44,7,42),(6,43,8,41),(9,40,11,38),(10,39,12,37),(13,30,15,32),(14,29,16,31),(17,26,19,28),(18,25,20,27),(33,45,35,47),(34,48,36,46)], [(1,19,14),(2,20,15),(3,17,16),(4,18,13),(5,10,45),(6,11,46),(7,12,47),(8,9,48),(21,25,30),(22,26,31),(23,27,32),(24,28,29),(33,42,37),(34,43,38),(35,44,39),(36,41,40)], [(5,45,10),(6,46,11),(7,47,12),(8,48,9),(33,37,42),(34,38,43),(35,39,44),(36,40,41)], [(1,34,3,36),(2,35,4,33),(5,27,12,30),(6,28,9,31),(7,25,10,32),(8,26,11,29),(13,42,20,39),(14,43,17,40),(15,44,18,37),(16,41,19,38),(21,45,23,47),(22,46,24,48)]])
Matrix representation of Q8×C32⋊C4 ►in GL6(𝔽13)
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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 | 1 |
| 3 | 9 | 0 | 0 | 0 | 0 |
| 9 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 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 | 12 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
G:=sub<GL(6,GF(13))| [0,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,9,0,0,0,0,9,10,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;
Q8×C32⋊C4 in GAP, Magma, Sage, TeX
Q_8\times C_3^2\rtimes C_4
% in TeX
G:=Group("Q8xC3^2:C4"); // GroupNames label
G:=SmallGroup(288,938);
// by ID
G=gap.SmallGroup(288,938);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,120,219,100,9413,362,12550,1203]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=c^3=d^3=e^4=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*d*e^-1=c*d=d*c,e*c*e^-1=c^-1*d>;
// generators/relations
Export