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## G = C2×C23⋊2D6order 192 = 26·3

### Direct product of C2 and C23⋊2D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C2×C23⋊2D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C24 — C2×C23⋊2D6
 Lower central C3 — C2×C6 — C2×C23⋊2D6
 Upper central C1 — C23 — C22×D4

Generators and relations for C2×C232D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=e6=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 1960 in 662 conjugacy classes, 143 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C22×C6, C22×C6, C2×C22⋊C4, C22≀C2, C22×D4, C22×D4, C25, D6⋊C4, C6.D4, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C6×D4, C6×D4, S3×C23, S3×C23, C23×C6, C2×C22≀C2, C2×D6⋊C4, C232D6, C2×C6.D4, C22×C3⋊D4, D4×C2×C6, S3×C24, C2×C232D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C3⋊D4, C22×S3, C22≀C2, C22×D4, S3×D4, C2×C3⋊D4, S3×C23, C2×C22≀C2, C232D6, C2×S3×D4, C22×C3⋊D4, C2×C232D6

Smallest permutation representation of C2×C232D6
On 48 points
Generators in S48
(1 32)(2 33)(3 34)(4 35)(5 36)(6 31)(7 20)(8 21)(9 22)(10 23)(11 24)(12 19)(13 39)(14 40)(15 41)(16 42)(17 37)(18 38)(25 48)(26 43)(27 44)(28 45)(29 46)(30 47)
(1 48)(2 12)(3 44)(4 8)(5 46)(6 10)(7 16)(9 18)(11 14)(13 47)(15 43)(17 45)(19 33)(20 42)(21 35)(22 38)(23 31)(24 40)(25 32)(26 41)(27 34)(28 37)(29 36)(30 39)
(1 4)(2 5)(3 6)(7 47)(8 48)(9 43)(10 44)(11 45)(12 46)(13 16)(14 17)(15 18)(19 29)(20 30)(21 25)(22 26)(23 27)(24 28)(31 34)(32 35)(33 36)(37 40)(38 41)(39 42)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 13)(7 44)(8 45)(9 46)(10 47)(11 48)(12 43)(19 26)(20 27)(21 28)(22 29)(23 30)(24 25)(31 39)(32 40)(33 41)(34 42)(35 37)(36 38)
(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 15)(3 14)(4 13)(5 18)(6 17)(7 45)(8 44)(9 43)(10 48)(11 47)(12 46)(19 29)(20 28)(21 27)(22 26)(23 25)(24 30)(31 37)(32 42)(33 41)(34 40)(35 39)(36 38)

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 C3⋊D4 S3×D4 kernel C2×C23⋊2D6 C2×D6⋊C4 C23⋊2D6 C2×C6.D4 C22×C3⋊D4 D4×C2×C6 S3×C24 C22×D4 C22×S3 C22×C6 C22×C4 C2×D4 C24 C23 C22 # reps 1 2 8 1 2 1 1 1 8 4 1 4 2 8 4

Matrix representation of C2×C232D6 in GL5(𝔽13)

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

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,12,0,0,0,0,0,0,11,4,0,0,0,9,2],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,12,1],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,1,0] >;

C2×C232D6 in GAP, Magma, Sage, TeX

C_2\times C_2^3\rtimes_2D_6
% in TeX

G:=Group("C2xC2^3:2D6");
// GroupNames label

G:=SmallGroup(192,1358);
// by ID

G=gap.SmallGroup(192,1358);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,675,297,6278]);
// Polycyclic

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

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