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## G = C6×C22⋊C4order 96 = 25·3

### Direct product of C6 and C22⋊C4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C6×C22⋊C4
 Chief series C1 — C2 — C22 — C2×C6 — C2×C12 — C3×C22⋊C4 — C6×C22⋊C4
 Lower central C1 — C2 — C6×C22⋊C4
 Upper central C1 — C22×C6 — C6×C22⋊C4

Generators and relations for C6×C22⋊C4
G = < a,b,c,d | a6=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >

Subgroups: 188 in 132 conjugacy classes, 76 normal (12 characteristic)
C1, C2, C2 [×6], C2 [×4], C3, C4 [×4], C22, C22 [×10], C22 [×12], C6, C6 [×6], C6 [×4], C2×C4 [×4], C2×C4 [×4], C23, C23 [×6], C23 [×4], C12 [×4], C2×C6, C2×C6 [×10], C2×C6 [×12], C22⋊C4 [×4], C22×C4 [×2], C24, C2×C12 [×4], C2×C12 [×4], C22×C6, C22×C6 [×6], C22×C6 [×4], C2×C22⋊C4, C3×C22⋊C4 [×4], C22×C12 [×2], C23×C6, C6×C22⋊C4
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], D4 [×4], C23, C12 [×4], C2×C6 [×7], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×C12 [×6], C3×D4 [×4], C22×C6, C2×C22⋊C4, C3×C22⋊C4 [×4], C22×C12, C6×D4 [×2], C6×C22⋊C4

Smallest permutation representation of C6×C22⋊C4
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 26)(2 27)(3 28)(4 29)(5 30)(6 25)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 20)(14 21)(15 22)(16 23)(17 24)(18 19)(31 39)(32 40)(33 41)(34 42)(35 37)(36 38)
(1 39)(2 40)(3 41)(4 42)(5 37)(6 38)(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 36)(26 31)(27 32)(28 33)(29 34)(30 35)
(1 11 26 17)(2 12 27 18)(3 7 28 13)(4 8 29 14)(5 9 30 15)(6 10 25 16)(19 32 48 40)(20 33 43 41)(21 34 44 42)(22 35 45 37)(23 36 46 38)(24 31 47 39)

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

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

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

60 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3A 3B 4A ··· 4H 6A ··· 6N 6O ··· 6V 12A ··· 12P order 1 2 ··· 2 2 2 2 2 3 3 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 2 2 2 1 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 D4 C3×D4 kernel C6×C22⋊C4 C3×C22⋊C4 C22×C12 C23×C6 C2×C22⋊C4 C22×C6 C22⋊C4 C22×C4 C24 C23 C2×C6 C22 # reps 1 4 2 1 2 8 8 4 2 16 4 8

Matrix representation of C6×C22⋊C4 in GL4(𝔽13) generated by

 12 0 0 0 0 10 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 12 0 0 0 0 1 0 0 0 0 12
,
 1 0 0 0 0 1 0 0 0 0 12 0 0 0 0 12
,
 5 0 0 0 0 1 0 0 0 0 0 12 0 0 12 0
G:=sub<GL(4,GF(13))| [12,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,12,0,0,0,0,1,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[5,0,0,0,0,1,0,0,0,0,0,12,0,0,12,0] >;

C6×C22⋊C4 in GAP, Magma, Sage, TeX

C_6\times C_2^2\rtimes C_4
% in TeX

G:=Group("C6xC2^2:C4");
// GroupNames label

G:=SmallGroup(96,162);
// by ID

G=gap.SmallGroup(96,162);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-2,288,313]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^2=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations

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