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## G = C3×M4(2).8C22order 192 = 26·3

### Direct product of C3 and M4(2).8C22

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×M4(2).8C22
 Chief series C1 — C2 — C4 — C2×C4 — C2×C12 — C3×M4(2) — C3×C4.D4 — C3×M4(2).8C22
 Lower central C1 — C2 — C22 — C3×M4(2).8C22
 Upper central C1 — C12 — C22×C12 — C3×M4(2).8C22

Generators and relations for C3×M4(2).8C22
G = < a,b,c,d,e | a3=b8=c2=d2=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b5, dbd=bc, cd=dc, ece-1=b4c, ede-1=b4cd >

Subgroups: 242 in 150 conjugacy classes, 78 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C12, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C22×C6, C4.D4, C4.10D4, C2×M4(2), C2×C4○D4, C2×C24, C3×M4(2), C3×M4(2), C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, M4(2).8C22, C3×C4.D4, C3×C4.10D4, C6×M4(2), C6×C4○D4, C3×M4(2).8C22
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, C12, C2×C6, C22⋊C4, C22×C4, C2×D4, C2×C12, C3×D4, C22×C6, C2×C22⋊C4, C3×C22⋊C4, C22×C12, C6×D4, M4(2).8C22, C6×C22⋊C4, C3×M4(2).8C22

Smallest permutation representation of C3×M4(2).8C22
On 48 points
Generators in S48
(1 33 22)(2 34 23)(3 35 24)(4 36 17)(5 37 18)(6 38 19)(7 39 20)(8 40 21)(9 44 30)(10 45 31)(11 46 32)(12 47 25)(13 48 26)(14 41 27)(15 42 28)(16 43 29)
(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)
(2 6)(4 8)(9 13)(11 15)(17 21)(19 23)(26 30)(28 32)(34 38)(36 40)(42 46)(44 48)
(1 41)(2 46)(3 47)(4 44)(5 45)(6 42)(7 43)(8 48)(9 17)(10 18)(11 23)(12 24)(13 21)(14 22)(15 19)(16 20)(25 35)(26 40)(27 33)(28 38)(29 39)(30 36)(31 37)(32 34)
(1 8 3 2 5 4 7 6)(9 16 11 10 13 12 15 14)(17 20 19 22 21 24 23 18)(25 28 27 30 29 32 31 26)(33 40 35 34 37 36 39 38)(41 44 43 46 45 48 47 42)

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3A 3B 4A 4B 4C 4D 4E 4F 4G 6A 6B 6C ··· 6H 6I 6J 6K 6L 8A ··· 8H 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N 24A ··· 24P order 1 2 2 2 2 2 2 3 3 4 4 4 4 4 4 4 6 6 6 ··· 6 6 6 6 6 8 ··· 8 12 12 12 12 12 ··· 12 12 12 12 12 24 ··· 24 size 1 1 2 2 2 4 4 1 1 1 1 2 2 2 4 4 1 1 2 ··· 2 4 4 4 4 4 ··· 4 1 1 1 1 2 ··· 2 4 4 4 4 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + + image C1 C2 C2 C2 C2 C3 C4 C4 C6 C6 C6 C6 C12 C12 D4 C3×D4 M4(2).8C22 C3×M4(2).8C22 kernel C3×M4(2).8C22 C3×C4.D4 C3×C4.10D4 C6×M4(2) C6×C4○D4 M4(2).8C22 C22×C12 C6×D4 C4.D4 C4.10D4 C2×M4(2) C2×C4○D4 C22×C4 C2×D4 C2×C12 C2×C4 C3 C1 # reps 1 2 2 2 1 2 4 4 4 4 4 2 8 8 4 8 2 4

Matrix representation of C3×M4(2).8C22 in GL4(𝔽73) generated by

 8 0 0 0 0 8 0 0 0 0 8 0 0 0 0 8
,
 0 1 46 14 0 1 0 14 72 72 0 72 0 71 0 72
,
 1 0 0 1 0 1 0 1 0 0 72 0 0 0 0 72
,
 0 1 1 0 1 0 1 0 0 0 72 0 0 0 71 1
,
 0 46 1 59 0 46 0 59 27 27 0 27 0 54 0 27
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[0,0,72,0,1,1,72,71,46,0,0,0,14,14,72,72],[1,0,0,0,0,1,0,0,0,0,72,0,1,1,0,72],[0,1,0,0,1,0,0,0,1,1,72,71,0,0,0,1],[0,0,27,0,46,46,27,54,1,0,0,0,59,59,27,27] >;

C3×M4(2).8C22 in GAP, Magma, Sage, TeX

C_3\times M_4(2)._8C_2^2
% in TeX

G:=Group("C3xM4(2).8C2^2");
// GroupNames label

G:=SmallGroup(192,846);
// by ID

G=gap.SmallGroup(192,846);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,520,4204,3036,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^8=c^2=d^2=1,e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^5,d*b*d=b*c,c*d=d*c,e*c*e^-1=b^4*c,e*d*e^-1=b^4*c*d>;
// generators/relations

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