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## G = C2×C8⋊D6order 192 = 26·3

### Direct product of C2 and C8⋊D6

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
 Upper central C1 — C22 — C22×C4 — C2×M4(2)

Generators and relations for C2×C8⋊D6
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 >

Subgroups: 984 in 298 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, S3, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C2×C8, M4(2), D8, SD16, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C24, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C2×M4(2), C2×D8, C2×SD16, C8⋊C22, C22×D4, C2×C4○D4, C24⋊C2, D24, C2×C24, C3×M4(2), C2×Dic6, S3×C2×C4, C2×D12, C2×D12, C2×D12, C4○D12, C4○D12, C2×C3⋊D4, C22×C12, S3×C23, C2×C8⋊C22, C2×C24⋊C2, C2×D24, C8⋊D6, C6×M4(2), C22×D12, C2×C4○D12, C2×C8⋊D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, D12, C22×S3, C8⋊C22, C22×D4, C2×D12, S3×C23, C2×C8⋊C22, C8⋊D6, C22×D12, C2×C8⋊D6

Smallest permutation representation of C2×C8⋊D6
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)]])

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F ··· 2K 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 24A ··· 24H order 1 2 2 2 2 2 2 ··· 2 3 4 4 4 4 4 4 6 6 6 6 6 8 8 8 8 12 12 12 12 12 12 24 ··· 24 size 1 1 1 1 2 2 12 ··· 12 2 2 2 2 2 12 12 2 2 2 4 4 4 4 4 4 2 2 2 2 4 4 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 D12 D12 C8⋊C22 C8⋊D6 kernel C2×C8⋊D6 C2×C24⋊C2 C2×D24 C8⋊D6 C6×M4(2) C22×D12 C2×C4○D12 C2×M4(2) C2×C12 C22×C6 C2×C8 M4(2) C22×C4 C2×C4 C23 C6 C2 # reps 1 2 2 8 1 1 1 1 3 1 2 4 1 6 2 2 4

Matrix representation of C2×C8⋊D6 in GL6(𝔽73)

 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 1 0 0 0 0 0 0 1 0 0 66 14 0 0 0 0 59 7 0 0
,
 0 1 0 0 0 0 72 72 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 1 72 0 0 0 0 1 0
,
 0 72 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 1 72 0 0 0 0 0 0 7 59 0 0 0 0 66 66

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] >;

C2×C8⋊D6 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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