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## G = Q8×C32⋊C4order 288 = 25·32

### Direct product of Q8 and C32⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — Q8×C32⋊C4
 Chief series C1 — C32 — C3⋊S3 — C2×C3⋊S3 — C2×C32⋊C4 — C4×C32⋊C4 — Q8×C32⋊C4
 Lower central C32 — C3×C6 — Q8×C32⋊C4
 Upper central C1 — C2 — Q8

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 [×2], C3 [×2], C4 [×3], C4 [×8], C22, S3 [×4], C6 [×2], C2×C4 [×7], Q8, Q8 [×3], C32, Dic3 [×6], C12 [×6], D6 [×2], C42 [×3], C4⋊C4 [×3], C2×Q8, C3⋊S3 [×2], C3×C6, Dic6 [×6], C4×S3 [×6], C3×Q8 [×2], C4×Q8, C3⋊Dic3 [×3], C3×C12 [×3], C32⋊C4 [×2], C32⋊C4 [×3], C2×C3⋊S3, S3×Q8 [×2], C324Q8 [×3], C4×C3⋊S3 [×3], Q8×C32, C2×C32⋊C4, C2×C32⋊C4 [×3], C4×C32⋊C4 [×3], C4⋊(C32⋊C4) [×3], Q8×C3⋊S3, Q8×C32⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], Q8 [×2], C23, C22×C4, C2×Q8, C4○D4, C4×Q8, C32⋊C4, C2×C32⋊C4 [×3], 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

Smallest permutation representation of Q8×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 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

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