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## G = C3×M4(2)⋊4C4order 192 = 26·3

### Direct product of C3 and M4(2)⋊4C4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×M4(2)⋊4C4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C12 — C3×C42⋊C2 — C3×M4(2)⋊4C4
 Lower central C1 — C2 — C22 — C3×M4(2)⋊4C4
 Upper central C1 — C12 — C22×C12 — C3×M4(2)⋊4C4

Generators and relations for C3×M4(2)⋊4C4
G = < a,b,c,d | a3=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b5, dbd-1=bc, dcd-1=b4c >

Subgroups: 138 in 90 conjugacy classes, 54 normal (42 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C24, C2×C12, C2×C12, C22×C6, C42⋊C2, C2×M4(2), C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×C24, C2×C24, C3×M4(2), C3×M4(2), C22×C12, M4(2)⋊4C4, C3×C42⋊C2, C6×M4(2), C3×M4(2)⋊4C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C12, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C12, C3×D4, C3×Q8, C2.C42, C4×C12, C3×C22⋊C4, C3×C4⋊C4, M4(2)⋊4C4, C3×C2.C42, C3×M4(2)⋊4C4

Smallest permutation representation of C3×M4(2)⋊4C4
On 48 points
Generators in S48
(1 38 11)(2 39 12)(3 40 13)(4 33 14)(5 34 15)(6 35 16)(7 36 9)(8 37 10)(17 26 44)(18 27 45)(19 28 46)(20 29 47)(21 30 48)(22 31 41)(23 32 42)(24 25 43)
(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 22)(2 19)(3 24)(4 21)(5 18)(6 23)(7 20)(8 17)(9 47)(10 44)(11 41)(12 46)(13 43)(14 48)(15 45)(16 42)(25 40)(26 37)(27 34)(28 39)(29 36)(30 33)(31 38)(32 35)
(2 23 6 19)(3 7)(4 21 8 17)(9 13)(10 44 14 48)(12 42 16 46)(18 22)(26 33 30 37)(27 31)(28 39 32 35)(36 40)(41 45)

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

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

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

66 conjugacy classes

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

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + - image C1 C2 C2 C3 C4 C4 C4 C6 C6 C12 C12 C12 D4 Q8 C3×D4 C3×Q8 M4(2)⋊4C4 C3×M4(2)⋊4C4 kernel C3×M4(2)⋊4C4 C3×C42⋊C2 C6×M4(2) M4(2)⋊4C4 C3×C22⋊C4 C2×C24 C3×M4(2) C42⋊C2 C2×M4(2) C22⋊C4 C2×C8 M4(2) C2×C12 C2×C12 C2×C4 C2×C4 C3 C1 # reps 1 1 2 2 4 4 4 2 4 8 8 8 3 1 6 2 2 4

Matrix representation of C3×M4(2)⋊4C4 in GL4(𝔽73) generated by

 8 0 0 0 0 8 0 0 0 0 8 0 0 0 0 8
,
 0 0 27 46 0 0 0 46 1 72 0 0 2 72 0 0
,
 72 0 0 0 71 1 0 0 0 0 72 0 0 0 71 1
,
 1 72 0 0 0 72 0 0 0 0 72 1 0 0 71 1
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[0,0,1,2,0,0,72,72,27,0,0,0,46,46,0,0],[72,71,0,0,0,1,0,0,0,0,72,71,0,0,0,1],[1,0,0,0,72,72,0,0,0,0,72,71,0,0,1,1] >;

C3×M4(2)⋊4C4 in GAP, Magma, Sage, TeX

C_3\times M_4(2)\rtimes_4C_4
% in TeX

G:=Group("C3xM4(2):4C4");
// GroupNames label

G:=SmallGroup(192,150);
// by ID

G=gap.SmallGroup(192,150);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,344,3027,2111,124]);
// Polycyclic

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

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