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G = C2×C12.46D4order 192 = 26·3

Direct product of C2 and C12.46D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C2×C12.46D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C22×D12 — C2×C12.46D4
 Lower central C3 — C6 — C2×C6 — C2×C12.46D4
 Upper central C1 — C22 — C22×C4 — C2×M4(2)

Generators and relations for C2×C12.46D4
G = < a,b,c,d,e | a2=b8=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b5, bd=db, ebe=bc, cd=dc, ce=ec, ede=d-1 >

Subgroups: 664 in 186 conjugacy classes, 63 normal (39 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C6, C6, C8, C2×C4, D4, C23, C23, C12, D6, C2×C6, C2×C6, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C24, C3⋊C8, C24, D12, C2×C12, C22×S3, C22×S3, C22×C6, C4.D4, C2×M4(2), C2×M4(2), C22×D4, C2×C3⋊C8, C4.Dic3, C4.Dic3, C2×C24, C3×M4(2), C3×M4(2), C2×D12, C2×D12, C22×C12, S3×C23, C2×C4.D4, C12.46D4, C2×C4.Dic3, C6×M4(2), C22×D12, C2×C12.46D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, D12, C3⋊D4, C22×S3, C4.D4, C2×C22⋊C4, D6⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C2×C4.D4, C12.46D4, C2×D6⋊C4, C2×C12.46D4

Smallest permutation representation of C2×C12.46D4
On 48 points
Generators in S48
(1 16)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(17 31)(18 32)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(33 48)(34 41)(35 42)(36 43)(37 44)(38 45)(39 46)(40 47)
(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 16)(2 13)(3 10)(4 15)(5 12)(6 9)(7 14)(8 11)(17 31)(18 28)(19 25)(20 30)(21 27)(22 32)(23 29)(24 26)(33 44)(34 41)(35 46)(36 43)(37 48)(38 45)(39 42)(40 47)
(1 47 23)(2 48 24)(3 41 17)(4 42 18)(5 43 19)(6 44 20)(7 45 21)(8 46 22)(9 33 30)(10 34 31)(11 35 32)(12 36 25)(13 37 26)(14 38 27)(15 39 28)(16 40 29)
(1 10)(2 8)(3 16)(4 6)(5 14)(7 12)(9 15)(11 13)(17 40)(18 44)(19 38)(20 42)(21 36)(22 48)(23 34)(24 46)(25 45)(26 35)(27 43)(28 33)(29 41)(30 39)(31 47)(32 37)

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 S3 D4 D6 D6 C4×S3 D12 C3⋊D4 C4×S3 C4.D4 C12.46D4 kernel C2×C12.46D4 C12.46D4 C2×C4.Dic3 C6×M4(2) C22×D12 C2×D12 S3×C23 C2×M4(2) C2×C12 M4(2) C22×C4 C2×C4 C2×C4 C2×C4 C23 C6 C2 # reps 1 4 1 1 1 4 4 1 4 2 1 2 4 4 2 2 4

Matrix representation of C2×C12.46D4 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 66 14 0 0 0 0 59 7 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 0 1 0 0 0 0 72 72 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 72 1 0 0 0 0 72 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 59 7 0 0 0 0 66 14 0 0 0 0 0 0 59 7 0 0 0 0 66 14

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,66,59,0,0,0,0,14,7,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,72,0,0,0,0,1,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,59,66,0,0,0,0,7,14,0,0,0,0,0,0,59,66,0,0,0,0,7,14] >;

C2×C12.46D4 in GAP, Magma, Sage, TeX

C_2\times C_{12}._{46}D_4
% in TeX

G:=Group("C2xC12.46D4");
// GroupNames label

G:=SmallGroup(192,689);
// by ID

G=gap.SmallGroup(192,689);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,422,58,1123,136,438,6278]);
// Polycyclic

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

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