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## G = C3×D8⋊2C4order 192 = 26·3

### Direct product of C3 and D8⋊2C4

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C3×D8⋊2C4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C2×C24 — C3×C4.Q8 — C3×D8⋊2C4
 Lower central C1 — C2 — C4 — C8 — C3×D8⋊2C4
 Upper central C1 — C6 — C2×C12 — C2×C24 — C3×D8⋊2C4

Generators and relations for C3×D82C4
G = < a,b,c,d | a3=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b3, dcd-1=b5c >

Subgroups: 130 in 58 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C12, C12, C2×C6, C2×C6, C16, C4⋊C4, C2×C8, D8, SD16, Q16, C4○D4, C24, C2×C12, C2×C12, C3×D4, C3×Q8, C4.Q8, M5(2), C4○D8, C48, C3×C4⋊C4, C2×C24, C3×D8, C3×SD16, C3×Q16, C3×C4○D4, D82C4, C3×C4.Q8, C3×M5(2), C3×C4○D8, C3×D82C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C12, C2×C6, C22⋊C4, D8, SD16, C2×C12, C3×D4, D4⋊C4, C3×C22⋊C4, C3×D8, C3×SD16, D82C4, C3×D4⋊C4, C3×D82C4

Smallest permutation representation of C3×D82C4
On 48 points
Generators in S48
(1 23 15)(2 24 16)(3 17 9)(4 18 10)(5 19 11)(6 20 12)(7 21 13)(8 22 14)(25 41 33)(26 42 34)(27 43 35)(28 44 36)(29 45 37)(30 46 38)(31 47 39)(32 48 40)
(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 32)(2 31)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 40)(16 39)(17 46)(18 45)(19 44)(20 43)(21 42)(22 41)(23 48)(24 47)
(2 4)(3 7)(6 8)(9 13)(10 16)(12 14)(17 21)(18 24)(20 22)(25 26 29 30)(27 32 31 28)(33 34 37 38)(35 40 39 36)(41 42 45 46)(43 48 47 44)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 8A 8B 8C 12A 12B 12C 12D 12E ··· 12J 16A 16B 16C 16D 24A 24B 24C 24D 24E 24F 48A ··· 48H order 1 2 2 2 3 3 4 4 4 4 4 6 6 6 6 6 6 8 8 8 12 12 12 12 12 ··· 12 16 16 16 16 24 24 24 24 24 24 48 ··· 48 size 1 1 2 8 1 1 2 2 8 8 8 1 1 2 2 8 8 2 2 4 2 2 2 2 8 ··· 8 4 4 4 4 2 2 2 2 4 4 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 D4 D4 SD16 D8 C3×D4 C3×D4 C3×SD16 C3×D8 D8⋊2C4 C3×D8⋊2C4 kernel C3×D8⋊2C4 C3×C4.Q8 C3×M5(2) C3×C4○D8 D8⋊2C4 C3×D8 C3×Q16 C4.Q8 M5(2) C4○D8 D8 Q16 C24 C2×C12 C12 C2×C6 C8 C2×C4 C4 C22 C3 C1 # reps 1 1 1 1 2 2 2 2 2 2 4 4 1 1 2 2 2 2 4 4 2 4

Matrix representation of C3×D82C4 in GL4(𝔽97) generated by

 61 0 0 0 0 61 0 0 0 0 61 0 0 0 0 61
,
 57 40 0 0 57 57 0 0 0 0 40 40 0 0 57 40
,
 0 0 40 40 0 0 57 40 57 40 0 0 57 57 0 0
,
 1 0 0 0 0 96 0 0 0 0 40 57 0 0 57 57
G:=sub<GL(4,GF(97))| [61,0,0,0,0,61,0,0,0,0,61,0,0,0,0,61],[57,57,0,0,40,57,0,0,0,0,40,57,0,0,40,40],[0,0,57,57,0,0,40,57,40,57,0,0,40,40,0,0],[1,0,0,0,0,96,0,0,0,0,40,57,0,0,57,57] >;

C3×D82C4 in GAP, Magma, Sage, TeX

C_3\times D_8\rtimes_2C_4
% in TeX

G:=Group("C3xD8:2C4");
// GroupNames label

G:=SmallGroup(192,166);
// by ID

G=gap.SmallGroup(192,166);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,1683,2194,136,2111,6053,3036,124]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^3,d*c*d^-1=b^5*c>;
// generators/relations

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