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G = C32⋊C4⋊Q8order 288 = 25·32

1st semidirect product of C32⋊C4 and Q8 acting via Q8/C4=C2

non-abelian, soluble, monomial, rational

Aliases: C4.5S3≀C2, C32⋊C41Q8, C321(C4⋊Q8), (C3×C12).10D4, C3⋊Dic3.28D4, Dic3.D6.5C2, C6.D6.3C22, C2.6(C2×S3≀C2), (C3×C6).3(C2×D4), C3⋊S3.2(C2×Q8), (C4×C32⋊C4).1C2, C3⋊S3.Q8.1C2, (C2×C3⋊S3).3C23, (C4×C3⋊S3).29C22, (C2×C32⋊C4).13C22, SmallGroup(288,870)

Series: Derived Chief Lower central Upper central

C1C32C2×C3⋊S3 — C32⋊C4⋊Q8
C1C32C3⋊S3C2×C3⋊S3C6.D6C3⋊S3.Q8 — C32⋊C4⋊Q8
C32C2×C3⋊S3 — C32⋊C4⋊Q8
C1C2C4

Generators and relations for C32⋊C4⋊Q8
 G = < a,b,c,d,e | a3=b3=c4=d4=1, e2=d2, cbc-1=ab=ba, cac-1=a-1b, ad=da, ae=ea, bd=db, ebe-1=a-1b-1, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 504 in 106 conjugacy classes, 29 normal (11 characteristic)
C1, C2, C2 [×2], C3 [×2], C4, C4 [×9], C22, S3 [×4], C6 [×2], C2×C4 [×7], Q8 [×4], C32, Dic3 [×6], C12 [×6], D6 [×2], C42, C4⋊C4 [×4], C2×Q8 [×2], C3⋊S3 [×2], C3×C6, Dic6 [×6], C4×S3 [×6], C3×Q8 [×2], C4⋊Q8, C3×Dic3 [×4], C3⋊Dic3, C3×C12, C32⋊C4 [×4], C2×C3⋊S3, S3×Q8 [×2], C6.D6 [×4], C322Q8 [×2], C3×Dic6 [×2], C4×C3⋊S3, C2×C32⋊C4 [×2], C3⋊S3.Q8 [×4], C4×C32⋊C4, Dic3.D6 [×2], C32⋊C4⋊Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×4], C23, C2×D4, C2×Q8 [×2], C4⋊Q8, S3≀C2, C2×S3≀C2, C32⋊C4⋊Q8

Character table of C32⋊C4⋊Q8

 class 12A2B2C3A3B4A4B4C4D4E4F4G4H4I4J6A6B12A12B12C12D12E12F
 size 1199442121212121818181818448824242424
ρ1111111111111111111111111    trivial
ρ2111111-1-11-11-111-1-111-1-1-1-111    linear of order 2
ρ3111111111-1-11-1-1-1-11111-11-11    linear of order 2
ρ4111111-1-111-1-1-1-11111-1-11-1-11    linear of order 2
ρ5111111-11-1-11-1-1-11111-1-1-111-1    linear of order 2
ρ61111111-1-1111-1-1-1-111111-11-1    linear of order 2
ρ7111111-11-11-1-111-1-111-1-111-1-1    linear of order 2
ρ81111111-1-1-1-1111111111-1-1-1-1    linear of order 2
ρ922-2-22220000-2000022220000    orthogonal lifted from D4
ρ1022-2-222-200002000022-2-20000    orthogonal lifted from D4
ρ112-22-2220000002-200-2-2000000    symplectic lifted from Q8, Schur index 2
ρ122-22-222000000-2200-2-2000000    symplectic lifted from Q8, Schur index 2
ρ132-2-222200000000-22-2-2000000    symplectic lifted from Q8, Schur index 2
ρ142-2-2222000000002-2-2-2000000    symplectic lifted from Q8, Schur index 2
ρ154400-21-400-22000001-2-1210-10    orthogonal lifted from C2×S3≀C2
ρ164400-2140022000001-21-2-10-10    orthogonal lifted from S3≀C2
ρ1744001-24220000000-21-210-10-1    orthogonal lifted from S3≀C2
ρ1844001-2-4-220000000-212-1010-1    orthogonal lifted from C2×S3≀C2
ρ194400-21-4002-2000001-2-12-1010    orthogonal lifted from C2×S3≀C2
ρ204400-21400-2-2000001-21-21010    orthogonal lifted from S3≀C2
ρ2144001-2-42-20000000-212-10-101    orthogonal lifted from C2×S3≀C2
ρ2244001-24-2-20000000-21-210101    orthogonal lifted from S3≀C2
ρ238-800-420000000000-24000000    symplectic faithful, Schur index 2
ρ248-8002-400000000004-2000000    symplectic faithful, Schur index 2

Smallest permutation representation of C32⋊C4⋊Q8
On 48 points
Generators in S48
(1 24 22)(2 23 21)(3 42 44)(4 41 43)(5 37 39)(6 40 38)(7 45 47)(8 46 48)(9 19 17)(10 18 20)(11 26 28)(12 25 27)(13 29 31)(14 32 30)(15 33 35)(16 34 36)
(2 21 23)(4 43 41)(6 38 40)(7 47 45)(10 20 18)(12 27 25)(14 30 32)(15 35 33)
(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 9 8 5)(2 10 7 6)(3 11 16 13)(4 12 15 14)(17 48 39 22)(18 45 40 23)(19 46 37 24)(20 47 38 21)(25 33 32 41)(26 34 29 42)(27 35 30 43)(28 36 31 44)
(1 15 8 4)(2 16 7 3)(5 14 9 12)(6 13 10 11)(17 27 39 30)(18 26 40 29)(19 25 37 32)(20 28 38 31)(21 36 47 44)(22 35 48 43)(23 34 45 42)(24 33 46 41)

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

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

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

Matrix representation of C32⋊C4⋊Q8 in GL6(𝔽13)

100000
010000
00121200
001000
00001212
000010
,
100000
010000
000100
00121200
000010
000001
,
0120000
100000
000010
00001212
001000
000100
,
0120000
100000
001000
000100
000010
000001
,
930000
340000
000010
000001
001000
000100

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,12,0,0,0,0,0,12,0,0],[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],[9,3,0,0,0,0,3,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

C32⋊C4⋊Q8 in GAP, Magma, Sage, TeX

C_3^2\rtimes C_4\rtimes Q_8
% in TeX

G:=Group("C3^2:C4:Q8");
// GroupNames label

G:=SmallGroup(288,870);
// by ID

G=gap.SmallGroup(288,870);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,112,141,120,422,219,100,2693,2028,362,797,1203]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^4=d^4=1,e^2=d^2,c*b*c^-1=a*b=b*a,c*a*c^-1=a^-1*b,a*d=d*a,a*e=e*a,b*d=d*b,e*b*e^-1=a^-1*b^-1,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations

Export

Character table of C32⋊C4⋊Q8 in TeX

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