Copied to
clipboard

## G = C2×C23.6D6order 192 = 26·3

### Direct product of C2 and C23.6D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C2×C23.6D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C2×C3⋊D4 — C22×C3⋊D4 — C2×C23.6D6
 Lower central C3 — C6 — C2×C6 — C2×C23.6D6
 Upper central C1 — C22 — C24 — C2×C22⋊C4

Generators and relations for C2×C23.6D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=b, f2=bcd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, ece-1=fcf-1=cd=dc, de=ed, df=fd, fef-1=cde5 >

Subgroups: 680 in 210 conjugacy classes, 63 normal (41 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×6], C22 [×7], C22 [×18], S3 [×2], C6, C6 [×2], C6 [×6], C2×C4 [×12], D4 [×8], C23 [×7], C23 [×8], Dic3 [×4], C12 [×2], D6 [×8], C2×C6 [×7], C2×C6 [×10], C22⋊C4 [×2], C22⋊C4 [×4], C22×C4 [×3], C2×D4 [×8], C24, C24, C2×Dic3 [×2], C2×Dic3 [×6], C3⋊D4 [×8], C2×C12 [×4], C22×S3 [×2], C22×S3 [×4], C22×C6 [×7], C22×C6 [×2], C23⋊C4 [×4], C2×C22⋊C4, C2×C22⋊C4, C22×D4, C6.D4 [×2], C6.D4, C3×C22⋊C4 [×2], C3×C22⋊C4, C22×Dic3, C22×Dic3, C2×C3⋊D4 [×4], C2×C3⋊D4 [×4], C22×C12, S3×C23, C23×C6, C2×C23⋊C4, C23.6D6 [×4], C2×C6.D4, C6×C22⋊C4, C22×C3⋊D4, C2×C23.6D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C22×S3, C23⋊C4 [×2], C2×C22⋊C4, D6⋊C4 [×4], S3×C2×C4, C2×D12, C2×C3⋊D4, C2×C23⋊C4, C23.6D6 [×2], C2×D6⋊C4, C2×C23.6D6

Smallest permutation representation of C2×C23.6D6
On 48 points
Generators in S48
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 13)(11 14)(12 15)(25 46)(26 47)(27 48)(28 37)(29 38)(30 39)(31 40)(32 41)(33 42)(34 43)(35 44)(36 45)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(1 16)(2 36)(3 18)(4 26)(5 20)(6 28)(7 22)(8 30)(9 24)(10 32)(11 14)(12 34)(13 41)(15 43)(17 45)(19 47)(21 37)(23 39)(25 46)(27 48)(29 38)(31 40)(33 42)(35 44)
(1 44)(2 45)(3 46)(4 47)(5 48)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 43)(13 32)(14 33)(15 34)(16 35)(17 36)(18 25)(19 26)(20 27)(21 28)(22 29)(23 30)(24 31)
(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 21 29 12)(2 42 23 20)(3 26 31 41)(4 9 13 25)(5 17 33 8)(6 38 15 16)(7 34 35 37)(10 46 19 24)(11 30 27 45)(14 39 48 36)(18 47 40 32)(22 43 44 28)

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 S3 D4 D6 D6 C4×S3 D12 C3⋊D4 C23⋊C4 C23.6D6 kernel C2×C23.6D6 C23.6D6 C2×C6.D4 C6×C22⋊C4 C22×C3⋊D4 C22×Dic3 C2×C3⋊D4 S3×C23 C2×C22⋊C4 C22×C6 C22⋊C4 C24 C23 C23 C23 C6 C2 # reps 1 4 1 1 1 2 4 2 1 4 2 1 4 4 4 2 4

Matrix representation of C2×C23.6D6 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 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 2 4 0 0 0 0 9 11 0 0 0 0 0 0 2 4 0 0 0 0 9 11
,
 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 12 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 12 12 0 0 9 11 0 0 0 0 2 11 0 0
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 1 0 0 11 2 0 0 0 0 4 2 0 0

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,9,0,0,0,0,4,11,0,0,0,0,0,0,2,9,0,0,0,0,4,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,1,0,0,0,0,12,0,0,0,0,0,0,0,0,0,9,2,0,0,0,0,11,11,0,0,0,12,0,0,0,0,1,12,0,0],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,0,11,4,0,0,0,0,2,2,0,0,12,0,0,0,0,0,12,1,0,0] >;

C2×C23.6D6 in GAP, Magma, Sage, TeX

C_2\times C_2^3._6D_6
% in TeX

G:=Group("C2xC2^3.6D6");
// GroupNames label

G:=SmallGroup(192,513);
// by ID

G=gap.SmallGroup(192,513);
# by ID

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

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=b,f^2=b*c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,e*c*e^-1=f*c*f^-1=c*d=d*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*d*e^5>;
// generators/relations

׿
×
𝔽