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G = Q8×C32⋊C4order 288 = 25·32

Direct product of Q8 and C32⋊C4

direct product, metabelian, soluble, monomial

Aliases: Q8×C32⋊C4, C328(C4×Q8), (Q8×C32)⋊3C4, C324Q83C4, 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

C1C3×C6 — Q8×C32⋊C4
C1C32C3⋊S3C2×C3⋊S3C2×C32⋊C4C4×C32⋊C4 — Q8×C32⋊C4
C32C3×C6 — Q8×C32⋊C4
C1C2Q8

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 12A2B2C3A3B4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P6A6B12A12B12C12D12E12F
 size 119944222999918181818181818181844888888
ρ1111111111111111111111111111111    trivial
ρ2111111-1-11-1-1-1-1-11-11111-1-111-111-1-1-1    linear of order 2
ρ3111111111-1-1-1-1-111-1-1-1-1-1111111111    linear of order 2
ρ4111111-1-11111111-1-1-1-1-11-111-111-1-1-1    linear of order 2
ρ5111111-11-11111-1-11-1-111-1-1111-1-1-1-11    linear of order 2
ρ61111111-1-1-1-1-1-11-1-1-1-1111111-1-1-111-1    linear of order 2
ρ7111111-11-1-1-1-1-11-1111-1-11-1111-1-1-1-11    linear of order 2
ρ81111111-1-11111-1-1-111-1-1-1111-1-1-111-1    linear of order 2
ρ911-1-111111ii-i-i-i-1-1i-ii-ii-111111111    linear of order 4
ρ1011-1-111-1-11-i-iiii-11i-ii-i-i111-111-1-1-1    linear of order 4
ρ1111-1-111111-i-iiii-1-1-ii-ii-i-111111111    linear of order 4
ρ1211-1-111-1-11ii-i-i-i-11-ii-iii111-111-1-1-1    linear of order 4
ρ1311-1-1111-1-1-i-iii-i11-iii-ii-111-1-1-111-1    linear of order 4
ρ1411-1-111-11-1ii-i-ii1-1-iii-i-i1111-1-1-1-11    linear of order 4
ρ1511-1-1111-1-1ii-i-ii11i-i-ii-i-111-1-1-111-1    linear of order 4
ρ1611-1-111-11-1-i-iii-i1-1i-i-iii1111-1-1-1-11    linear of order 4
ρ172-22-222000-22-22000000000-2-2000000    symplectic lifted from Q8, Schur index 2
ρ182-22-2220002-22-2000000000-2-2000000    symplectic lifted from Q8, Schur index 2
ρ192-2-2222000-2i2i2i-2i000000000-2-2000000    complex lifted from C4○D4
ρ202-2-22220002i-2i-2i2i000000000-2-2000000    complex lifted from C4○D4
ρ2144001-2-44-40000000000000-2112-12-1-2    orthogonal lifted from C2×C32⋊C4
ρ224400-21-4-4400000000000001-221-2-12-1    orthogonal lifted from C2×C32⋊C4
ρ234400-214-4-400000000000001-22-121-2-1    orthogonal lifted from C2×C32⋊C4
ρ2444001-24440000000000000-211-21-21-2    orthogonal lifted from C32⋊C4
ρ254400-21-44-400000000000001-2-2-12-121    orthogonal lifted from C2×C32⋊C4
ρ2644001-2-4-440000000000000-21-1-212-12    orthogonal lifted from C2×C32⋊C4
ρ2744001-24-4-40000000000000-21-12-1-212    orthogonal lifted from C2×C32⋊C4
ρ284400-2144400000000000001-2-21-21-21    orthogonal lifted from C32⋊C4
ρ298-800-420000000000000000-24000000    symplectic faithful, Schur index 2
ρ308-8002-400000000000000004-2000000    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)

0120000
100000
001000
000100
000010
000001
,
390000
9100000
0012000
0001200
0000120
0000012
,
100000
010000
0001200
0011200
0000012
0000112
,
100000
010000
001000
000100
0000121
0000120
,
1200000
0120000
000010
000001
000100
001000

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

Character table of Q8×C32⋊C4 in TeX

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