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G = C6×C4≀C2  order 192 = 26·3

Direct product of C6 and C4≀C2

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

Aliases: C6×C4≀C2, C4○D4⋊5C12, D4⋊5(C2×C12), (C6×D4)⋊21C4, (C2×D4)⋊9C12, (C2×C42)⋊9C6, Q8⋊6(C2×C12), (C2×Q8)⋊9C12, (C6×Q8)⋊17C4, C4.72(C6×D4), C42⋊19(C2×C6), (C4×C12)⋊56C22, C12.477(C2×D4), (C2×C12).519D4, M4(2)⋊9(C2×C6), C4.7(C22×C12), C22.12(C6×D4), C23.44(C3×D4), (C2×M4(2))⋊12C6, (C6×M4(2))⋊30C2, (C22×C6).161D4, C12.152(C22×C4), (C2×C12).896C23, C12.117(C22⋊C4), (C3×M4(2))⋊38C22, (C22×C12).584C22, (C2×C4×C12)⋊19C2, (C3×C4○D4)⋊9C4, (C3×D4)⋊25(C2×C4), (C3×Q8)⋊23(C2×C4), (C2×C4).70(C3×D4), (C2×C4).50(C2×C12), (C6×C4○D4).20C2, C4○D4.13(C2×C6), (C2×C4○D4).12C6, (C2×C6).407(C2×D4), C2.23(C6×C22⋊C4), C4.33(C3×C22⋊C4), (C2×C12).271(C2×C4), C6.111(C2×C22⋊C4), (C2×C4).71(C22×C6), C22.6(C3×C22⋊C4), (C22×C4).120(C2×C6), (C3×C4○D4).51C22, (C2×C6).140(C22⋊C4), SmallGroup(192,853)

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 2A2B2C2D2E2F2G3A3B4A4B4C4D4E···4N4O4P6A···6F6G6H6I6J6K6L6M6N8A8B8C8D12A···12H12I···12AB12AC12AD12AE12AF24A···24H
order122222223344444···4446···666666666888812···1212···121212121224···24
size111122441111112···2441···12222444444441···12···244444···4

84 irreducible representations

dim1111111111111111222222
type+++++++
imageC1C2C2C2C2C3C4C4C4C6C6C6C6C12C12C12D4D4C3×D4C3×D4C4≀C2C3×C4≀C2
kernelC6×C4≀C2C3×C4≀C2C2×C4×C12C6×M4(2)C6×C4○D4C2×C4≀C2C6×D4C6×Q8C3×C4○D4C4≀C2C2×C42C2×M4(2)C2×C4○D4C2×D4C2×Q8C4○D4C2×C12C22×C6C2×C4C23C6C2
# reps14111222482224483162816

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

720000
08000
00800
00010
00001
,
10000
072000
007200
000460
000027
,
10000
007200
072000
000046
000270
,
460000
01000
007200
000720
000027

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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