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G = C2×C8⋊D6  order 192 = 26·3

Direct product of C2 and C8⋊D6

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C8⋊D6, C24⋊2C23, D12⋊5C23, D24⋊9C22, M4(2)⋊17D6, C12.58C24, Dic6⋊5C23, C23.57D12, (C2×C8)⋊4D6, C8⋊2(C22×S3), (C2×D24)⋊14C2, C6⋊1(C8⋊C22), (C2×C24)⋊7C22, (C2×C4).57D12, C4.48(C2×D12), C24⋊C2⋊8C22, C12.238(C2×D4), (C2×C12).203D4, (C2×M4(2))⋊3S3, (C6×M4(2))⋊3C2, C4.55(S3×C23), C6.25(C22×D4), C4○D12⋊18C22, (C22×D12)⋊17C2, (C2×D12)⋊49C22, (C22×C6).118D4, C22.73(C2×D12), (C22×C4).281D6, C2.27(C22×D12), (C2×C12).511C23, (C2×Dic6)⋊57C22, (C3×M4(2))⋊19C22, (C22×C12).266C22, C3⋊1(C2×C8⋊C22), (C2×C24⋊C2)⋊4C2, (C2×C6).62(C2×D4), (C2×C4○D12)⋊26C2, (C2×C4).223(C22×S3), SmallGroup(192,1305)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C12 — C2×C8⋊D6
Chief series
C1 — C3 — C6 — C12 — D12 — C2×D12 — C22×D12 — C2×C8⋊D6
Lower central
C3 — C6 — C12 — C2×C8⋊D6

Subgroups: 984 in 298 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×22], S3 [×6], C6, C6 [×2], C6 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×5], D4 [×17], Q8 [×3], C23, C23 [×11], Dic3 [×2], C12 [×2], C12 [×2], D6 [×20], C2×C6, C2×C6 [×2], C2×C6 [×2], C2×C8 [×2], M4(2) [×4], D8 [×8], SD16 [×8], C22×C4, C22×C4, C2×D4 [×11], C2×Q8, C4○D4 [×6], C24, C24 [×4], Dic6 [×2], Dic6, C4×S3 [×4], D12 [×6], D12 [×7], C2×Dic3, C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×4], C22×S3 [×11], C22×C6, C2×M4(2), C2×D8 [×2], C2×SD16 [×2], C8⋊C22 [×8], C22×D4, C2×C4○D4, C24⋊C2 [×8], D24 [×8], C2×C24 [×2], C3×M4(2) [×4], C2×Dic6, S3×C2×C4, C2×D12, C2×D12 [×6], C2×D12 [×3], C4○D12 [×4], C4○D12 [×2], C2×C3⋊D4, C22×C12, S3×C23, C2×C8⋊C22, C2×C24⋊C2 [×2], C2×D24 [×2], C8⋊D6 [×8], C6×M4(2), C22×D12, C2×C4○D12, C2×C8⋊D6

Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, D12 [×4], C22×S3 [×7], C8⋊C22 [×2], C22×D4, C2×D12 [×6], S3×C23, C2×C8⋊C22, C8⋊D6 [×2], C22×D12, C2×C8⋊D6

Generators and relations
 G = < a,b,c,d | a2=b8=c6=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b-1, dcd=c-1 >

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

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

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

Matrix representation ►G ⊆ GL6(𝔽73)

7200000
0720000
001000
000100
000010
000001
,
7200000
0720000
000010
000001
00661400
0059700
,
010000
72720000
0072100
0072000
0000172
000010
,
0720000
7200000
001000
0017200
0000759
00006666

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,66,59,0,0,0,0,14,7,0,0,1,0,0,0,0,0,0,1,0,0],[0,72,0,0,0,0,1,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,72,0],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,7,66,0,0,0,0,59,66] >;
 

42 conjugacy classes

class 1 2A2B2C2D2E2F···2K 3 4A4B4C4D4E4F6A6B6C6D6E8A8B8C8D12A12B12C12D12E12F24A···24H
order1222222···2344444466666888812121212121224···24
size11112212···122222212122224444442222444···4

42 irreducible representations

dim11111112222222244
type+++++++++++++++++
imageC1C2C2C2C2C2C2S3D4D4D6D6D6D12D12C8⋊C22C8⋊D6
kernelC2×C8⋊D6C2×C24⋊C2C2×D24C8⋊D6C6×M4(2)C22×D12C2×C4○D12C2×M4(2)C2×C12C22×C6C2×C8M4(2)C22×C4C2×C4C23C6C2
# reps12281111312416224

In GAP, Magma, Sage, TeX 

C_2\times C_8\rtimes D_6
 
% in TeX
 
G:=Group("C2xC8:D6");
 
// GroupNames label
 
G:=SmallGroup(192,1305);
 
// by ID
 
G=gap.SmallGroup(192,1305);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,675,297,80,1684,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^8=c^6=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

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