C2xQ8:A4

direct product, non-abelian, soluble

Aliases: C₂×Q₈⋊A₄, C₂⁴.₁₁A₄, C₂³⋊₃SL₂(𝔽₃), Q₈⋊₁(C₂×A₄), (C₂×Q₈)⋊₃A₄, (Q₈×C₂³)⋊₃C₃, (C₂²×Q₈)⋊₈C₆, C₂³.₂₂(C₂×A₄), C₂²⋊(C₂×SL₂(𝔽₃)), C₂².₄(C₂²⋊A₄), C₂.₂(C₂×C₂²⋊A₄), SmallGroup(192,1506)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂ — C₂²×Q₈ — C₂×Q₈⋊A₄

Chief series

C₁ — C₂ — Q₈ — C₂²×Q₈ — Q₈⋊A₄ — C₂×Q₈⋊A₄

Lower central

C₂²×Q₈ — C₂×Q₈⋊A₄

Upper central

C₁ — C₂²

Subgroups: 559 in 175 conjugacy classes, 22 normal (10 characteristic)
C₁, C₂, C₂^[×2], C₂^[×4], C₃, C₄^[×8], C₂²^[×2], C₂²^[×11], C₆^[×3], C₂×C₄^[×28], Q₈^[×4], Q₈^[×20], C₂³, C₂³^[×2], C₂³^[×4], A₄, C₂×C₆, C₂²×C₄^[×14], C₂×Q₈^[×4], C₂×Q₈^[×36], C₂⁴, SL₂(𝔽₃)^[×4], C₂×A₄^[×3], C₂³×C₄, C₂²×Q₈, C₂²×Q₈^[×9], C₂×SL₂(𝔽₃)^[×4], C₂²×A₄, Q₈×C₂³, Q₈⋊A₄, C₂×Q₈⋊A₄

Quotients:
C₁, C₂, C₃, C₆, A₄^[×5], SL₂(𝔽₃)^[×2], C₂×A₄^[×5], C₂×SL₂(𝔽₃), C₂²⋊A₄, Q₈⋊A₄^[×2], C₂×C₂²⋊A₄, C₂×Q₈⋊A₄

Generators and relations
G = < a,b,c,d,e,f | a²=b⁴=d²=e²=f³=1, c²=b², ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc^-1=b^-1, bd=db, be=eb, fbf^-1=c, cd=dc, ce=ec, fcf^-1=bc, fdf^-1=de=ed, fef^-1=d >

Smallest permutation representation
►On 48 points

Generators in S₄₈

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

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

(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)

(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)

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

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

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

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

Matrix representation ►G ⊆ GL₇(𝔽₁₃)

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	12	0	0	0	0
0	0	0	12	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

10	9	0	0	0	0	0
9	3	0	0	0	0	0
0	0	10	9	0	0	0
0	0	9	3	0	0	0
0	0	0	0	12	0	0
0	0	0	0	12	0	1
0	0	0	0	12	1	0

0	1	0	0	0	0	0
12	0	0	0	0	0	0
0	0	0	1	0	0	0
0	0	12	0	0	0	0
0	0	0	0	0	1	12
0	0	0	0	1	0	12
0	0	0	0	0	0	12

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	0	12	1
0	0	0	0	0	12	0
0	0	0	0	1	12	0

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	0	1	12
0	0	0	0	1	0	12
0	0	0	0	0	0	12

1	0	0	0	0	0	0
9	3	0	0	0	0	0
0	0	9	0	0	0	0
0	0	3	1	0	0	0
0	0	0	0	0	0	1
0	0	0	0	1	0	0
0	0	0	0	0	1	0

G:=sub<GL(7,GF(13))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[10,9,0,0,0,0,0,9,3,0,0,0,0,0,0,0,10,9,0,0,0,0,0,9,3,0,0,0,0,0,0,0,12,12,12,0,0,0,0,0,0,1,0,0,0,0,0,1,0],[0,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,12,12,12],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,12,12,12,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,12,12,12],[1,9,0,0,0,0,0,0,3,0,0,0,0,0,0,0,9,3,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0] >;

Character table of C₂×Q₈⋊A₄

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	4A	4B	4C	4D	4E	4F	4G	4H	6A	6B	6C	6D	6E	6F
size	1	1	1	1	3	3	3	3	16	16	6	6	6	6	6	6	6	6	16	16	16	16	16	16
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	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	linear of order 2
ρ₃	1	-1	1	-1	-1	1	1	-1	ζ₃	ζ₃²	-1	1	1	1	1	-1	-1	-1	ζ₃²	ζ₃	ζ₆	ζ₆	ζ₆⁵	ζ₆⁵	linear of order 6
ρ₄	1	1	1	1	1	1	1	1	ζ₃²	ζ₃	1	1	1	1	1	1	1	1	ζ₃	ζ₃²	ζ₃	ζ₃	ζ₃²	ζ₃²	linear of order 3
ρ₅	1	-1	1	-1	-1	1	1	-1	ζ₃²	ζ₃	-1	1	1	1	1	-1	-1	-1	ζ₃	ζ₃²	ζ₆⁵	ζ₆⁵	ζ₆	ζ₆	linear of order 6
ρ₆	1	1	1	1	1	1	1	1	ζ₃	ζ₃²	1	1	1	1	1	1	1	1	ζ₃²	ζ₃	ζ₃²	ζ₃²	ζ₃	ζ₃	linear of order 3
ρ₇	2	2	-2	-2	-2	2	-2	2	-1	-1	0	0	0	0	0	0	0	0	1	1	-1	1	-1	1	symplectic lifted from SL₂(𝔽₃), Schur index 2
ρ₈	2	-2	-2	2	2	2	-2	-2	-1	-1	0	0	0	0	0	0	0	0	1	1	1	-1	1	-1	symplectic lifted from SL₂(𝔽₃), Schur index 2
ρ₉	2	-2	-2	2	2	2	-2	-2	ζ₆	ζ₆⁵	0	0	0	0	0	0	0	0	ζ₃	ζ₃²	ζ₃	ζ₆⁵	ζ₃²	ζ₆	complex lifted from SL₂(𝔽₃)
ρ₁₀	2	-2	-2	2	2	2	-2	-2	ζ₆⁵	ζ₆	0	0	0	0	0	0	0	0	ζ₃²	ζ₃	ζ₃²	ζ₆	ζ₃	ζ₆⁵	complex lifted from SL₂(𝔽₃)
ρ₁₁	2	2	-2	-2	-2	2	-2	2	ζ₆	ζ₆⁵	0	0	0	0	0	0	0	0	ζ₃	ζ₃²	ζ₆⁵	ζ₃	ζ₆	ζ₃²	complex lifted from SL₂(𝔽₃)
ρ₁₂	2	2	-2	-2	-2	2	-2	2	ζ₆⁵	ζ₆	0	0	0	0	0	0	0	0	ζ₃²	ζ₃	ζ₆	ζ₃²	ζ₆⁵	ζ₃	complex lifted from SL₂(𝔽₃)
ρ₁₃	3	3	3	3	-1	-1	-1	-1	0	0	-1	-1	3	-1	-1	3	-1	-1	0	0	0	0	0	0	orthogonal lifted from A₄
ρ₁₄	3	-3	3	-3	1	-1	-1	1	0	0	1	-1	-1	3	-1	1	-3	1	0	0	0	0	0	0	orthogonal lifted from C₂×A₄
ρ₁₅	3	3	3	3	-1	-1	-1	-1	0	0	-1	-1	-1	3	-1	-1	3	-1	0	0	0	0	0	0	orthogonal lifted from A₄
ρ₁₆	3	-3	3	-3	1	-1	-1	1	0	0	1	-1	3	-1	-1	-3	1	1	0	0	0	0	0	0	orthogonal lifted from C₂×A₄
ρ₁₇	3	-3	3	-3	-3	3	3	-3	0	0	1	-1	-1	-1	-1	1	1	1	0	0	0	0	0	0	orthogonal lifted from C₂×A₄
ρ₁₈	3	-3	3	-3	1	-1	-1	1	0	0	-3	-1	-1	-1	3	1	1	1	0	0	0	0	0	0	orthogonal lifted from C₂×A₄
ρ₁₉	3	-3	3	-3	1	-1	-1	1	0	0	1	3	-1	-1	-1	1	1	-3	0	0	0	0	0	0	orthogonal lifted from C₂×A₄
ρ₂₀	3	3	3	3	-1	-1	-1	-1	0	0	3	-1	-1	-1	3	-1	-1	-1	0	0	0	0	0	0	orthogonal lifted from A₄
ρ₂₁	3	3	3	3	3	3	3	3	0	0	-1	-1	-1	-1	-1	-1	-1	-1	0	0	0	0	0	0	orthogonal lifted from A₄
ρ₂₂	3	3	3	3	-1	-1	-1	-1	0	0	-1	3	-1	-1	-1	-1	-1	3	0	0	0	0	0	0	orthogonal lifted from A₄
ρ₂₃	6	-6	-6	6	-2	-2	2	2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from Q₈⋊A₄, Schur index 2
ρ₂₄	6	6	-6	-6	2	-2	2	-2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from Q₈⋊A₄, Schur index 2

In GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes A_4

% in TeX

G:=Group("C2xQ8:A4");

// GroupNames label

G:=SmallGroup(192,1506);

// by ID

G=gap.SmallGroup(192,1506);

# by ID

G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,135,262,851,172,1524,285,124]);

// Polycyclic

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

// generators/relations

G = C₂×Q₈⋊A₄ order 192 = 2⁶·3

Direct product of C₂ and Q₈⋊A₄

G = C2×Q8⋊A4 order 192 = 26·3

Direct product of C2 and Q8⋊A4

G = C₂×Q₈⋊A₄ order 192 = 2⁶·3

Direct product of C₂ and Q₈⋊A₄