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## G = C2×D4.A4order 192 = 26·3

### Direct product of C2 and D4.A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C2×D4.A4
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C2×SL2(𝔽3) — C22×SL2(𝔽3) — C2×D4.A4
 Lower central Q8 — C2×D4.A4
 Upper central C1 — C22 — C2×D4

Generators and relations for C2×D4.A4
G = < a,b,c,d,e,f | a2=b4=c2=f3=1, d2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ede-1=b2d, fdf-1=b2de, fef-1=d >

Subgroups: 573 in 204 conjugacy classes, 51 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C12, C2×C6, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4○D4, SL2(𝔽3), C2×C12, C3×D4, C22×C6, C22×Q8, C22×Q8, C2×C4○D4, C2×C4○D4, 2- 1+4, 2- 1+4, C2×SL2(𝔽3), C2×SL2(𝔽3), C4.A4, C6×D4, C2×2- 1+4, C22×SL2(𝔽3), C2×C4.A4, D4.A4, C2×D4.A4
Quotients: C1, C2, C3, C22, C6, C23, A4, C2×C6, C2×A4, C22×C6, C22×A4, D4.A4, C23×A4, C2×D4.A4

Smallest permutation representation of C2×D4.A4
On 32 points
Generators in S32
(1 24)(2 21)(3 22)(4 23)(5 25)(6 26)(7 27)(8 28)(9 19)(10 20)(11 17)(12 18)(13 29)(14 30)(15 31)(16 32)
(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)
(2 4)(5 7)(9 11)(14 16)(17 19)(21 23)(25 27)(30 32)
(1 18 3 20)(2 19 4 17)(5 14 7 16)(6 15 8 13)(9 23 11 21)(10 24 12 22)(25 30 27 32)(26 31 28 29)
(1 15 3 13)(2 16 4 14)(5 19 7 17)(6 20 8 18)(9 27 11 25)(10 28 12 26)(21 32 23 30)(22 29 24 31)
(5 19 16)(6 20 13)(7 17 14)(8 18 15)(9 32 25)(10 29 26)(11 30 27)(12 31 28)

G:=sub<Sym(32)| (1,24)(2,21)(3,22)(4,23)(5,25)(6,26)(7,27)(8,28)(9,19)(10,20)(11,17)(12,18)(13,29)(14,30)(15,31)(16,32), (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), (2,4)(5,7)(9,11)(14,16)(17,19)(21,23)(25,27)(30,32), (1,18,3,20)(2,19,4,17)(5,14,7,16)(6,15,8,13)(9,23,11,21)(10,24,12,22)(25,30,27,32)(26,31,28,29), (1,15,3,13)(2,16,4,14)(5,19,7,17)(6,20,8,18)(9,27,11,25)(10,28,12,26)(21,32,23,30)(22,29,24,31), (5,19,16)(6,20,13)(7,17,14)(8,18,15)(9,32,25)(10,29,26)(11,30,27)(12,31,28)>;

G:=Group( (1,24)(2,21)(3,22)(4,23)(5,25)(6,26)(7,27)(8,28)(9,19)(10,20)(11,17)(12,18)(13,29)(14,30)(15,31)(16,32), (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), (2,4)(5,7)(9,11)(14,16)(17,19)(21,23)(25,27)(30,32), (1,18,3,20)(2,19,4,17)(5,14,7,16)(6,15,8,13)(9,23,11,21)(10,24,12,22)(25,30,27,32)(26,31,28,29), (1,15,3,13)(2,16,4,14)(5,19,7,17)(6,20,8,18)(9,27,11,25)(10,28,12,26)(21,32,23,30)(22,29,24,31), (5,19,16)(6,20,13)(7,17,14)(8,18,15)(9,32,25)(10,29,26)(11,30,27)(12,31,28) );

G=PermutationGroup([[(1,24),(2,21),(3,22),(4,23),(5,25),(6,26),(7,27),(8,28),(9,19),(10,20),(11,17),(12,18),(13,29),(14,30),(15,31),(16,32)], [(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)], [(2,4),(5,7),(9,11),(14,16),(17,19),(21,23),(25,27),(30,32)], [(1,18,3,20),(2,19,4,17),(5,14,7,16),(6,15,8,13),(9,23,11,21),(10,24,12,22),(25,30,27,32),(26,31,28,29)], [(1,15,3,13),(2,16,4,14),(5,19,7,17),(6,20,8,18),(9,27,11,25),(10,28,12,26),(21,32,23,30),(22,29,24,31)], [(5,19,16),(6,20,13),(7,17,14),(8,18,15),(9,32,25),(10,29,26),(11,30,27),(12,31,28)]])

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 4 4 type + + + + + + + + - image C1 C2 C2 C2 C3 C6 C6 C6 A4 C2×A4 C2×A4 C2×A4 D4.A4 D4.A4 kernel C2×D4.A4 C22×SL2(𝔽3) C2×C4.A4 D4.A4 C2×2- 1+4 C22×Q8 C2×C4○D4 2- 1+4 C2×D4 C2×C4 D4 C23 C2 C2 # reps 1 2 1 4 2 4 2 8 1 1 4 2 2 4

Matrix representation of C2×D4.A4 in GL7(𝔽13)

 12 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 0 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 1 0 0 0 0 0 0 0 1 0 0
,
 12 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 12 0 0 0 0 0 0 0 12
,
 12 12 12 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 9 3 0 0 0 0 0 3 4 0 0 0 0 0 0 0 9 3 0 0 0 0 0 3 4
,
 0 1 0 0 0 0 0 1 0 0 0 0 0 0 12 12 12 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 0 0 0 0 0 12 0
,
 1 0 0 0 0 0 0 12 12 12 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 9 3 0 0 0 0 0 0 0 1 0 0 0 0 0 0 9 3

G:=sub<GL(7,GF(13))| [12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,12,0,0,0,0,0,0,0,12,0,0],[12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,1,0,0,0,0,12,1,0,0,0,0,0,0,0,0,9,3,0,0,0,0,0,3,4,0,0,0,0,0,0,0,9,3,0,0,0,0,0,3,4],[0,1,12,0,0,0,0,1,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,0],[1,12,0,0,0,0,0,0,12,1,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,9,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,9,0,0,0,0,0,0,3] >;

C2×D4.A4 in GAP, Magma, Sage, TeX

C_2\times D_4.A_4
% in TeX

G:=Group("C2xD4.A4");
// GroupNames label

G:=SmallGroup(192,1503);
// by ID

G=gap.SmallGroup(192,1503);
# by ID

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

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