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## G = C6×C4≀C2order 192 = 26·3

### Direct product of C6 and C4≀C2

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C6×C4≀C2
 Chief series C1 — C2 — C4 — C2×C4 — C2×C12 — C3×M4(2) — C3×C4≀C2 — C6×C4≀C2
 Lower central C1 — C2 — C4 — C6×C4≀C2
 Upper central C1 — C2×C12 — C22×C12 — C6×C4≀C2

Generators and relations for C6×C4≀C2
G = < a,b,c,d | a6=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b-1c >

Subgroups: 274 in 170 conjugacy classes, 82 normal (46 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C12, C12, C2×C6, C2×C6, C42, C42, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×C6, C22×C6, C4≀C2, C2×C42, C2×M4(2), C2×C4○D4, C4×C12, C4×C12, C2×C24, C3×M4(2), C3×M4(2), C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, C3×C4○D4, C2×C4≀C2, C3×C4≀C2, C2×C4×C12, C6×M4(2), C6×C4○D4, C6×C4≀C2
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, C12, C2×C6, C22⋊C4, C22×C4, C2×D4, C2×C12, C3×D4, C22×C6, C4≀C2, C2×C22⋊C4, C3×C22⋊C4, C22×C12, C6×D4, C2×C4≀C2, C3×C4≀C2, C6×C22⋊C4, C6×C4≀C2

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

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C 4D 4E ··· 4N 4O 4P 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 6N 8A 8B 8C 8D 12A ··· 12H 12I ··· 12AB 12AC 12AD 12AE 12AF 24A ··· 24H order 1 2 2 2 2 2 2 2 3 3 4 4 4 4 4 ··· 4 4 4 6 ··· 6 6 6 6 6 6 6 6 6 8 8 8 8 12 ··· 12 12 ··· 12 12 12 12 12 24 ··· 24 size 1 1 1 1 2 2 4 4 1 1 1 1 1 1 2 ··· 2 4 4 1 ··· 1 2 2 2 2 4 4 4 4 4 4 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 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C3 C4 C4 C4 C6 C6 C6 C6 C12 C12 C12 D4 D4 C3×D4 C3×D4 C4≀C2 C3×C4≀C2 kernel C6×C4≀C2 C3×C4≀C2 C2×C4×C12 C6×M4(2) C6×C4○D4 C2×C4≀C2 C6×D4 C6×Q8 C3×C4○D4 C4≀C2 C2×C42 C2×M4(2) C2×C4○D4 C2×D4 C2×Q8 C4○D4 C2×C12 C22×C6 C2×C4 C23 C6 C2 # reps 1 4 1 1 1 2 2 2 4 8 2 2 2 4 4 8 3 1 6 2 8 16

Matrix representation of C6×C4≀C2 in GL5(𝔽73)

 72 0 0 0 0 0 8 0 0 0 0 0 8 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 46 0 0 0 0 0 27
,
 1 0 0 0 0 0 0 72 0 0 0 72 0 0 0 0 0 0 0 46 0 0 0 27 0
,
 46 0 0 0 0 0 1 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 27

G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,46,0,0,0,0,0,27],[1,0,0,0,0,0,0,72,0,0,0,72,0,0,0,0,0,0,0,27,0,0,0,46,0],[46,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,27] >;

C6×C4≀C2 in GAP, Magma, Sage, TeX

C_6\times C_4\wr C_2
% in TeX

G:=Group("C6xC4wrC2");
// GroupNames label

G:=SmallGroup(192,853);
// by ID

G=gap.SmallGroup(192,853);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,4204,2111,172,124]);
// Polycyclic

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

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