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## G = C6×C22≀C2order 192 = 26·3

### Direct product of C6 and C22≀C2

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C6×C22≀C2
 Chief series C1 — C2 — C22 — C2×C6 — C22×C6 — C6×D4 — C3×C22≀C2 — C6×C22≀C2
 Lower central C1 — C22 — C6×C22≀C2
 Upper central C1 — C22×C6 — C6×C22≀C2

Generators and relations for C6×C22≀C2
G = < a,b,c,d,e,f | a6=b2=c2=d2=e2=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf=bd=db, be=eb, cd=dc, fcf=ce=ec, de=ed, df=fd, ef=fe >

Subgroups: 1138 in 662 conjugacy classes, 210 normal (12 characteristic)
C1, C2 [×7], C2 [×14], C3, C4 [×6], C22, C22 [×18], C22 [×78], C6 [×7], C6 [×14], C2×C4 [×6], C2×C4 [×6], D4 [×24], C23, C23 [×20], C23 [×74], C12 [×6], C2×C6, C2×C6 [×18], C2×C6 [×78], C22⋊C4 [×12], C22×C4 [×3], C2×D4 [×12], C2×D4 [×12], C24, C24 [×7], C24 [×12], C2×C12 [×6], C2×C12 [×6], C3×D4 [×24], C22×C6, C22×C6 [×20], C22×C6 [×74], C2×C22⋊C4 [×3], C22≀C2 [×8], C22×D4 [×3], C25, C3×C22⋊C4 [×12], C22×C12 [×3], C6×D4 [×12], C6×D4 [×12], C23×C6, C23×C6 [×7], C23×C6 [×12], C2×C22≀C2, C6×C22⋊C4 [×3], C3×C22≀C2 [×8], D4×C2×C6 [×3], C24×C6, C6×C22≀C2
Quotients: C1, C2 [×15], C3, C22 [×35], C6 [×15], D4 [×12], C23 [×15], C2×C6 [×35], C2×D4 [×18], C24, C3×D4 [×12], C22×C6 [×15], C22≀C2 [×4], C22×D4 [×3], C6×D4 [×18], C23×C6, C2×C22≀C2, C3×C22≀C2 [×4], D4×C2×C6 [×3], C6×C22≀C2

Smallest permutation representation of C6×C22≀C2
On 48 points
Generators in S48
(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 29)(2 30)(3 25)(4 26)(5 27)(6 28)(7 16)(8 17)(9 18)(10 13)(11 14)(12 15)(19 45)(20 46)(21 47)(22 48)(23 43)(24 44)(31 39)(32 40)(33 41)(34 42)(35 37)(36 38)
(1 40)(2 41)(3 42)(4 37)(5 38)(6 39)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 34)(26 35)(27 36)(28 31)(29 32)(30 33)(43 46)(44 47)(45 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 43)(14 44)(15 45)(16 46)(17 47)(18 48)(25 42)(26 37)(27 38)(28 39)(29 40)(30 41)
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(19 48)(20 43)(21 44)(22 45)(23 46)(24 47)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)
(1 17)(2 18)(3 13)(4 14)(5 15)(6 16)(7 39)(8 40)(9 41)(10 42)(11 37)(12 38)(19 27)(20 28)(21 29)(22 30)(23 25)(24 26)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)

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

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

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

84 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2S 2T 2U 3A 3B 4A ··· 4F 6A ··· 6N 6O ··· 6AL 6AM 6AN 6AO 6AP 12A ··· 12L order 1 2 ··· 2 2 ··· 2 2 2 3 3 4 ··· 4 6 ··· 6 6 ··· 6 6 6 6 6 12 ··· 12 size 1 1 ··· 1 2 ··· 2 4 4 1 1 4 ··· 4 1 ··· 1 2 ··· 2 4 4 4 4 4 ··· 4

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 D4 C3×D4 kernel C6×C22≀C2 C6×C22⋊C4 C3×C22≀C2 D4×C2×C6 C24×C6 C2×C22≀C2 C2×C22⋊C4 C22≀C2 C22×D4 C25 C22×C6 C23 # reps 1 3 8 3 1 2 6 16 6 2 12 24

Matrix representation of C6×C22≀C2 in GL6(𝔽13)

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

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

C6×C22≀C2 in GAP, Magma, Sage, TeX

C_6\times C_2^2\wr C_2
% in TeX

G:=Group("C6xC2^2wrC2");
// GroupNames label

G:=SmallGroup(192,1410);
// by ID

G=gap.SmallGroup(192,1410);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,2102]);
// Polycyclic

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

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