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G = C6×C8⋊C22  order 192 = 26·3

Direct product of C6 and C8⋊C22

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

Aliases: C6×C8⋊C22, C24⋊7C23, C12.82C24, C8⋊(C22×C6), D8⋊3(C2×C6), (C2×D8)⋊11C6, (C6×D8)⋊25C2, C4.66(C6×D4), (C2×SD16)⋊4C6, SD16⋊1(C2×C6), D4⋊2(C22×C6), C4.5(C23×C6), Q8⋊3(C22×C6), (C2×C24)⋊21C22, (C6×SD16)⋊15C2, (C2×C12).525D4, C12.329(C2×D4), (C6×D4)⋊66C22, (C3×D8)⋊19C22, (C22×D4)⋊14C6, (C3×D4)⋊13C23, (C2×M4(2))⋊3C6, (C6×M4(2))⋊8C2, M4(2)⋊3(C2×C6), (C3×Q8)⋊12C23, (C6×Q8)⋊54C22, C23.55(C3×D4), C22.23(C6×D4), (C22×C6).172D4, C6.203(C22×D4), (C2×C12).975C23, (C3×SD16)⋊17C22, (C3×M4(2))⋊24C22, (C22×C12).465C22, (C2×C8)⋊2(C2×C6), (D4×C2×C6)⋊26C2, C2.27(D4×C2×C6), C4○D4⋊6(C2×C6), (C2×C4○D4)⋊15C6, (C6×C4○D4)⋊27C2, (C2×D4)⋊15(C2×C6), (C2×Q8)⋊16(C2×C6), (C2×C4).136(C3×D4), (C2×C6).419(C2×D4), (C3×C4○D4)⋊24C22, (C22×C4).81(C2×C6), (C2×C4).45(C22×C6), SmallGroup(192,1462)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C4 — C6×C8⋊C22
Chief series
C1 — C2 — C4 — C12 — C3×D4 — C3×D8 — C3×C8⋊C22 — C6×C8⋊C22
Lower central
C1 — C2 — C4 — C6×C8⋊C22
Upper central
C1 — C2×C6 — C22×C12 — C6×C8⋊C22

Subgroups: 530 in 298 conjugacy classes, 162 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×22], C6, C6 [×2], C6 [×8], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×5], D4 [×6], D4 [×11], Q8 [×2], Q8, C23, C23 [×11], C12 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×22], C2×C8 [×2], M4(2) [×4], D8 [×8], SD16 [×8], C22×C4, C22×C4, C2×D4, C2×D4 [×6], C2×D4 [×4], C2×Q8, C4○D4 [×4], C4○D4 [×2], C24, C24 [×4], C2×C12 [×2], C2×C12 [×4], C2×C12 [×5], C3×D4 [×6], C3×D4 [×11], C3×Q8 [×2], C3×Q8, C22×C6, C22×C6 [×11], C2×M4(2), C2×D8 [×2], C2×SD16 [×2], C8⋊C22 [×8], C22×D4, C2×C4○D4, C2×C24 [×2], C3×M4(2) [×4], C3×D8 [×8], C3×SD16 [×8], C22×C12, C22×C12, C6×D4, C6×D4 [×6], C6×D4 [×4], C6×Q8, C3×C4○D4 [×4], C3×C4○D4 [×2], C23×C6, C2×C8⋊C22, C6×M4(2), C6×D8 [×2], C6×SD16 [×2], C3×C8⋊C22 [×8], D4×C2×C6, C6×C4○D4, C6×C8⋊C22

Quotients:
C1, C2 [×15], C3, C22 [×35], C6 [×15], D4 [×4], C23 [×15], C2×C6 [×35], C2×D4 [×6], C24, C3×D4 [×4], C22×C6 [×15], C8⋊C22 [×2], C22×D4, C6×D4 [×6], C23×C6, C2×C8⋊C22, C3×C8⋊C22 [×2], D4×C2×C6, C6×C8⋊C22

Generators and relations
 G = < a,b,c,d | a6=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, cd=dc >

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

(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)

(2 4)(3 7)(6 8)(9 13)(10 16)(12 14)(17 21)(18 24)(20 22)(25 29)(26 32)(28 30)(33 37)(34 40)(36 38)(41 45)(42 48)(44 46)

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

Matrix representation ►G ⊆ GL6(𝔽73)

6500000
0650000
0072000
0007200
0000720
0000072
,
72710000
110000
0072020
0072011
0007210
000010
,
7200000
110000
001000
0017200
0010072
0010720
,
7200000
0720000
0072000
0007200
0072010
0072001

G:=sub<GL(6,GF(73))| [65,0,0,0,0,0,0,65,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,1,0,0,0,0,71,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,72,0,0,0,2,1,1,1,0,0,0,1,0,0],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,1,1,1,1,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,72,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,72,72,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
 

66 conjugacy classes

class 1 2A2B2C2D2E2F···2K3A3B4A4B4C4D4E4F6A···6F6G6H6I6J6K···6V8A8B8C8D12A···12H12I12J12K12L24A···24H
order1222222···2334444446···666666···6888812···121212121224···24
size1111224···4112222441···122224···444442···244444···4

66 irreducible representations

dim11111111111111222244
type++++++++++
imageC1C2C2C2C2C2C2C3C6C6C6C6C6C6D4D4C3×D4C3×D4C8⋊C22C3×C8⋊C22
kernelC6×C8⋊C22C6×M4(2)C6×D8C6×SD16C3×C8⋊C22D4×C2×C6C6×C4○D4C2×C8⋊C22C2×M4(2)C2×D8C2×SD16C8⋊C22C22×D4C2×C4○D4C2×C12C22×C6C2×C4C23C6C2
# reps112281122441622316224

In GAP, Magma, Sage, TeX 

C_6\times C_8\rtimes C_2^2
 
% in TeX
 
G:=Group("C6xC8:C2^2");
 
// GroupNames label
 
G:=SmallGroup(192,1462);
 
// by ID
 
G=gap.SmallGroup(192,1462);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,2102,6053,3036,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,c*d=d*c>;
 
// generators/relations
 

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