Copied to
clipboard

## G = C2×Q8×A4order 192 = 26·3

### Direct product of C2, Q8 and A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23 — C2×Q8×A4
 Chief series C1 — C22 — C23 — C2×A4 — C22×A4 — C2×C4×A4 — C2×Q8×A4
 Lower central C22 — C23 — 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=d2=e2=f3=1, c2=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, fdf-1=de=ed, fef-1=d >

Subgroups: 520 in 205 conjugacy classes, 57 normal (12 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C2×C4, C2×C4, Q8, Q8, C23, C23, C23, C12, A4, C2×C6, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, C2×C12, C3×Q8, C2×A4, C2×A4, C23×C4, C22×Q8, C22×Q8, C4×A4, C6×Q8, C22×A4, Q8×C23, C2×C4×A4, Q8×A4, C2×Q8×A4
Quotients: C1, C2, C3, C22, C6, Q8, C23, A4, C2×C6, C2×Q8, C3×Q8, C2×A4, C22×C6, C6×Q8, C22×A4, Q8×A4, C23×A4, C2×Q8×A4

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

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

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

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

40 conjugacy classes

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

40 irreducible representations

 dim 1 1 1 1 1 1 2 2 3 3 3 6 type + + + - + + + - image C1 C2 C2 C3 C6 C6 Q8 C3×Q8 A4 C2×A4 C2×A4 Q8×A4 kernel C2×Q8×A4 C2×C4×A4 Q8×A4 Q8×C23 C23×C4 C22×Q8 C2×A4 C23 C2×Q8 C2×C4 Q8 C2 # reps 1 3 4 2 6 8 2 4 1 3 4 2

Matrix representation of C2×Q8×A4 in GL5(𝔽13)

 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 11 0 0 0 1 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 5 0 0 0 0 5 8 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1
,
 9 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,1,0,0,0,11,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[5,5,0,0,0,0,8,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1],[9,0,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0] >;

C2×Q8×A4 in GAP, Magma, Sage, TeX

C_2\times Q_8\times A_4
% in TeX

G:=Group("C2xQ8xA4");
// GroupNames label

G:=SmallGroup(192,1499);
// by ID

G=gap.SmallGroup(192,1499);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,2,176,303,142,530,909]);
// 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,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,f*d*f^-1=d*e=e*d,f*e*f^-1=d>;
// generators/relations

׿
×
𝔽