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## G = C2×C23.7D6order 192 = 26·3

### Direct product of C2 and C23.7D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C2×C23.7D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C6.D4 — C2×C6.D4 — C2×C23.7D6
 Lower central C3 — C6 — C2×C6 — C2×C23.7D6
 Upper central C1 — C22 — C24 — C22×D4

Generators and relations for C2×C23.7D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=e6=1, f2=cb=bc, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=fbf-1=bd=db, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=cde-1 >

Subgroups: 552 in 210 conjugacy classes, 71 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×6], C22 [×3], C22 [×4], C22 [×18], C6, C6 [×2], C6 [×8], C2×C4 [×2], C2×C4 [×10], D4 [×8], C23 [×3], C23 [×6], C23 [×6], Dic3 [×4], C12 [×2], C2×C6 [×3], C2×C6 [×4], C2×C6 [×18], C22⋊C4 [×6], C22×C4, C22×C4 [×2], C2×D4 [×4], C2×D4 [×4], C24 [×2], C2×Dic3 [×8], C2×C12 [×2], C2×C12 [×2], C3×D4 [×8], C22×C6 [×3], C22×C6 [×6], C22×C6 [×6], C23⋊C4 [×4], C2×C22⋊C4 [×2], C22×D4, C6.D4 [×4], C6.D4 [×2], C22×Dic3 [×2], C22×C12, C6×D4 [×4], C6×D4 [×4], C23×C6 [×2], C2×C23⋊C4, C23.7D6 [×4], C2×C6.D4 [×2], D4×C2×C6, C2×C23.7D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, Dic3 [×4], D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Dic3 [×6], C3⋊D4 [×4], C22×S3, C23⋊C4 [×2], C2×C22⋊C4, C6.D4 [×4], C22×Dic3, C2×C3⋊D4 [×2], C2×C23⋊C4, C23.7D6 [×2], C2×C6.D4, C2×C23.7D6

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

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 2K 3 4A 4B 4C ··· 4J 6A ··· 6G 6H ··· 6O 12A 12B 12C 12D order 1 2 2 2 2 ··· 2 2 2 3 4 4 4 ··· 4 6 ··· 6 6 ··· 6 12 12 12 12 size 1 1 1 1 2 ··· 2 4 4 2 4 4 12 ··· 12 2 ··· 2 4 ··· 4 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 C4 C4 C4 S3 D4 Dic3 Dic3 D6 Dic3 D6 C3⋊D4 C23⋊C4 C23.7D6 kernel C2×C23.7D6 C23.7D6 C2×C6.D4 D4×C2×C6 C22×C12 C6×D4 C23×C6 C22×D4 C22×C6 C22×C4 C2×D4 C2×D4 C24 C24 C23 C6 C2 # reps 1 4 2 1 2 4 2 1 4 1 2 2 1 1 8 2 4

Matrix representation of C2×C23.7D6 in GL8(𝔽13)

 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 7 3 0 0 0 0 0 0 10 6 0 0 0 0 0 0 0 0 6 10 0 0 0 0 0 0 3 7
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 7 3 0 0 0 0 0 0 10 6 0 0 0 0 0 0 0 0 7 3 0 0 0 0 0 0 10 6
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 11 11 0 0 0 0 0 0 2 9 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0
,
 8 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 8 5 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 9 1 0 0 0 0 0 0 0 0 7 3 0 0 0 0 0 0 5 6

G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,7,10,0,0,0,0,0,0,3,6,0,0,0,0,0,0,0,0,6,3,0,0,0,0,0,0,10,7],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,7,10,0,0,0,0,0,0,3,6,0,0,0,0,0,0,0,0,7,10,0,0,0,0,0,0,3,6],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,11,2,0,0,0,0,0,0,11,9,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0],[8,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,8,8,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,12,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,7,5,0,0,0,0,0,0,3,6] >;

C2×C23.7D6 in GAP, Magma, Sage, TeX

C_2\times C_2^3._7D_6
% in TeX

G:=Group("C2xC2^3.7D6");
// GroupNames label

G:=SmallGroup(192,778);
// by ID

G=gap.SmallGroup(192,778);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,422,297,1684,6278]);
// Polycyclic

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

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