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## G = C2×Q8.A4order 192 = 26·3

### Direct product of C2 and Q8.A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C2×Q8.A4
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C2×SL2(𝔽3) — C2×C4.A4 — C2×Q8.A4
 Lower central Q8 — C2×Q8.A4
 Upper central C1 — C22 — C2×Q8

Generators and relations for C2×Q8.A4
G = < a,b,c,d,e,f | a2=b4=f3=1, c2=d2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc-1=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ede-1=b2d, fdf-1=b2de, fef-1=d >

Subgroups: 685 in 208 conjugacy classes, 51 normal (10 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C2×C4, C2×C4, D4, Q8, Q8, Q8, C23, C12, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, SL2(𝔽3), C2×C12, C3×Q8, C22×D4, C2×C4○D4, C2×C4○D4, 2+ 1+4, 2+ 1+4, C2×SL2(𝔽3), C4.A4, C6×Q8, C2×2+ 1+4, C2×C4.A4, Q8.A4, C2×Q8.A4
Quotients: C1, C2, C3, C22, C6, C23, A4, C2×C6, C2×A4, C22×C6, C22×A4, Q8.A4, C23×A4, C2×Q8.A4

Smallest permutation representation of C2×Q8.A4
On 48 points
Generators in S48
(1 32)(2 29)(3 30)(4 31)(5 26)(6 27)(7 28)(8 25)(9 33)(10 34)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 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 10 3 12)(2 9 4 11)(5 46 7 48)(6 45 8 47)(13 17 15 19)(14 20 16 18)(21 25 23 27)(22 28 24 26)(29 33 31 35)(30 36 32 34)(37 41 39 43)(38 44 40 42)
(1 11 3 9)(2 12 4 10)(5 46 7 48)(6 47 8 45)(13 14 15 16)(17 20 19 18)(21 27 23 25)(22 28 24 26)(29 36 31 34)(30 33 32 35)(37 38 39 40)(41 44 43 42)
(1 10 3 12)(2 11 4 9)(5 8 7 6)(13 20 15 18)(14 17 16 19)(21 22 23 24)(25 28 27 26)(29 35 31 33)(30 36 32 34)(37 44 39 42)(38 41 40 43)(45 46 47 48)
(1 16 24)(2 13 21)(3 14 22)(4 15 23)(5 34 42)(6 35 43)(7 36 44)(8 33 41)(9 17 25)(10 18 26)(11 19 27)(12 20 28)(29 37 45)(30 38 46)(31 39 47)(32 40 48)

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 3A 3B 4A ··· 4F 4G 4H 6A ··· 6F 12A ··· 12L order 1 2 2 2 2 ··· 2 3 3 4 ··· 4 4 4 6 ··· 6 12 ··· 12 size 1 1 1 1 6 ··· 6 4 4 2 ··· 2 6 6 4 ··· 4 8 ··· 8

38 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 4 4 type + + + + + + + image C1 C2 C2 C3 C6 C6 A4 C2×A4 C2×A4 Q8.A4 Q8.A4 kernel C2×Q8.A4 C2×C4.A4 Q8.A4 C2×2+ 1+4 C2×C4○D4 2+ 1+4 C2×Q8 C2×C4 Q8 C2 C2 # reps 1 3 4 2 6 8 1 3 4 2 4

Matrix representation of C2×Q8.A4 in GL7(ℤ)

 -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 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1
,
 -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 -2 0 0 0 0 1 0 -1 -1 0 0 0 1 0 -1 0 0 0 0 0 1 -1 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 -2 0 0 0 0 0 1 -1 0 0 0 0 0 0 -1 0 1 0 0 0 1 -1 -1 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 -2 0 0 0 1 0 -1 -1 0 0 0 0 1 0 -1 0 0 0 1 0 0 -1
,
 -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 -2 0 0 0 0 0 1 -1 0 0 0 0 0 1 -1 0 -1 0 0 0 0 -1 1 0
,
 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -2 1 1 1 0 0 0 -1 0 1 0 0 0 0 -1 0 0 1 0 0 0 -1 1 0 0

G:=sub<GL(7,Integers())| [-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,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0,0,0,1,0,0,0,-2,-1,-1,-1,0,0,0,0,-1,0,0],[-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,1,0,1,0,0,0,-2,-1,-1,-1,0,0,0,0,0,0,-1,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,1,0,1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,-2,-1,-1,-1],[-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,1,1,0,0,0,0,-2,-1,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,-1,0],[0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-2,-1,-1,-1,0,0,0,1,0,0,1,0,0,0,1,1,0,0,0,0,0,1,0,1,0] >;

C2×Q8.A4 in GAP, Magma, Sage, TeX

C_2\times Q_8.A_4
% in TeX

G:=Group("C2xQ8.A4");
// GroupNames label

G:=SmallGroup(192,1502);
// by ID

G=gap.SmallGroup(192,1502);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,680,1059,520,235,172,404,285,124]);
// Polycyclic

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

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