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## G = C6×C8⋊C22order 192 = 26·3

### Direct product of C6 and C8⋊C22

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

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

Generators and relations for C6×C8⋊C22
G = < a,b,c,d | a6=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, cd=dc >

Subgroups: 530 in 298 conjugacy classes, 162 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C12, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, M4(2), D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, C24, C2×C12, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×C6, C22×C6, C2×M4(2), C2×D8, C2×SD16, C8⋊C22, C22×D4, C2×C4○D4, C2×C24, C3×M4(2), C3×D8, C3×SD16, C22×C12, C22×C12, C6×D4, C6×D4, C6×D4, C6×Q8, C3×C4○D4, C3×C4○D4, C23×C6, C2×C8⋊C22, C6×M4(2), C6×D8, C6×SD16, C3×C8⋊C22, D4×C2×C6, C6×C4○D4, C6×C8⋊C22
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C24, C3×D4, C22×C6, C8⋊C22, C22×D4, C6×D4, C23×C6, C2×C8⋊C22, C3×C8⋊C22, D4×C2×C6, C6×C8⋊C22

Smallest permutation representation of C6×C8⋊C22
On 48 points
Generators in S48
(1 15 39 21 25 41)(2 16 40 22 26 42)(3 9 33 23 27 43)(4 10 34 24 28 44)(5 11 35 17 29 45)(6 12 36 18 30 46)(7 13 37 19 31 47)(8 14 38 20 32 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)
(2 4)(3 7)(6 8)(9 13)(10 16)(12 14)(18 20)(19 23)(22 24)(26 28)(27 31)(30 32)(33 37)(34 40)(36 38)(42 44)(43 47)(46 48)
(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)| (1,15,39,21,25,41)(2,16,40,22,26,42)(3,9,33,23,27,43)(4,10,34,24,28,44)(5,11,35,17,29,45)(6,12,36,18,30,46)(7,13,37,19,31,47)(8,14,38,20,32,48), (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), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(18,20)(19,23)(22,24)(26,28)(27,31)(30,32)(33,37)(34,40)(36,38)(42,44)(43,47)(46,48), (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( (1,15,39,21,25,41)(2,16,40,22,26,42)(3,9,33,23,27,43)(4,10,34,24,28,44)(5,11,35,17,29,45)(6,12,36,18,30,46)(7,13,37,19,31,47)(8,14,38,20,32,48), (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), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(18,20)(19,23)(22,24)(26,28)(27,31)(30,32)(33,37)(34,40)(36,38)(42,44)(43,47)(46,48), (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([[(1,15,39,21,25,41),(2,16,40,22,26,42),(3,9,33,23,27,43),(4,10,34,24,28,44),(5,11,35,17,29,45),(6,12,36,18,30,46),(7,13,37,19,31,47),(8,14,38,20,32,48)], [(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)], [(2,4),(3,7),(6,8),(9,13),(10,16),(12,14),(18,20),(19,23),(22,24),(26,28),(27,31),(30,32),(33,37),(34,40),(36,38),(42,44),(43,47),(46,48)], [(1,5),(3,7),(9,13),(11,15),(17,21),(19,23),(25,29),(27,31),(33,37),(35,39),(41,45),(43,47)]])

66 conjugacy classes

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

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 C6 D4 D4 C3×D4 C3×D4 C8⋊C22 C3×C8⋊C22 kernel C6×C8⋊C22 C6×M4(2) C6×D8 C6×SD16 C3×C8⋊C22 D4×C2×C6 C6×C4○D4 C2×C8⋊C22 C2×M4(2) C2×D8 C2×SD16 C8⋊C22 C22×D4 C2×C4○D4 C2×C12 C22×C6 C2×C4 C23 C6 C2 # reps 1 1 2 2 8 1 1 2 2 4 4 16 2 2 3 1 6 2 2 4

Matrix representation of C6×C8⋊C22 in GL6(𝔽73)

 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 71 0 0 0 0 1 1 0 0 0 0 0 0 72 0 2 0 0 0 72 0 1 1 0 0 0 72 1 0 0 0 0 0 1 0
,
 72 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 1 72 0 0 0 0 1 0 0 72 0 0 1 0 72 0
,
 72 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 72 0 1 0 0 0 72 0 0 1

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] >;

C6×C8⋊C22 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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