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## G = C2×Q8⋊3Dic3order 192 = 26·3

### Direct product of C2 and Q8⋊3Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C2×Q8⋊3Dic3
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4.Dic3 — C2×C4.Dic3 — C2×Q8⋊3Dic3
 Lower central C3 — C6 — C12 — C2×Q8⋊3Dic3
 Upper central C1 — C2×C4 — C22×C4 — C2×C4○D4

Generators and relations for C2×Q83Dic3
G = < a,b,c,d,e | a2=b4=d6=1, c2=b2, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=b2c, ece-1=b-1c, ede-1=d-1 >

Subgroups: 360 in 170 conjugacy classes, 71 normal (43 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×C6, C22×C6, C4≀C2, C2×C42, C2×M4(2), C2×C4○D4, C2×C3⋊C8, C4.Dic3, C4.Dic3, C4×Dic3, C4×Dic3, C22×Dic3, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, C3×C4○D4, C2×C4≀C2, Q83Dic3, C2×C4.Dic3, C2×C4×Dic3, C6×C4○D4, C2×Q83Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22⋊C4, C22×C4, C2×D4, C2×Dic3, C3⋊D4, C22×S3, C4≀C2, C2×C22⋊C4, C6.D4, C22×Dic3, C2×C3⋊D4, C2×C4≀C2, Q83Dic3, C2×C6.D4, C2×Q83Dic3

Smallest permutation representation of C2×Q83Dic3
On 48 points
Generators in S48
(1 19)(2 20)(3 21)(4 18)(5 16)(6 17)(7 14)(8 15)(9 13)(10 23)(11 24)(12 22)(25 44)(26 45)(27 46)(28 47)(29 48)(30 43)(31 40)(32 41)(33 42)(34 37)(35 38)(36 39)
(1 5 12 9)(2 6 10 7)(3 4 11 8)(13 19 16 22)(14 20 17 23)(15 21 18 24)(25 41 28 38)(26 42 29 39)(27 37 30 40)(31 46 34 43)(32 47 35 44)(33 48 36 45)
(1 39 12 42)(2 37 10 40)(3 41 11 38)(4 25 8 28)(5 29 9 26)(6 27 7 30)(13 45 16 48)(14 43 17 46)(15 47 18 44)(19 36 22 33)(20 34 23 31)(21 32 24 35)
(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 3)(4 6)(7 8)(10 11)(14 15)(17 18)(20 21)(23 24)(25 40 28 37)(26 39 29 42)(27 38 30 41)(31 47 34 44)(32 46 35 43)(33 45 36 48)

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 type + + + + + + + + + - - - + image C1 C2 C2 C2 C2 C4 C4 C4 S3 D4 D4 D6 Dic3 Dic3 Dic3 D6 C3⋊D4 C3⋊D4 C4≀C2 Q8⋊3Dic3 kernel C2×Q8⋊3Dic3 Q8⋊3Dic3 C2×C4.Dic3 C2×C4×Dic3 C6×C4○D4 C6×D4 C6×Q8 C3×C4○D4 C2×C4○D4 C2×C12 C22×C6 C22×C4 C2×D4 C2×Q8 C4○D4 C4○D4 C2×C4 C23 C6 C2 # reps 1 4 1 1 1 2 2 4 1 3 1 1 1 1 2 2 6 2 8 4

Matrix representation of C2×Q83Dic3 in GL5(𝔽73)

 72 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 46 27 0 0 0 0 27 0 0 0 0 0 72 0 0 0 0 0 72
,
 1 0 0 0 0 0 27 0 0 0 0 54 46 0 0 0 0 0 43 13 0 0 0 60 30
,
 72 0 0 0 0 0 72 1 0 0 0 0 1 0 0 0 0 0 72 72 0 0 0 1 0
,
 27 0 0 0 0 0 27 60 0 0 0 0 1 0 0 0 0 0 72 0 0 0 0 1 1

G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,46,0,0,0,0,27,27,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,27,54,0,0,0,0,46,0,0,0,0,0,43,60,0,0,0,13,30],[72,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,72,1,0,0,0,72,0],[27,0,0,0,0,0,27,0,0,0,0,60,1,0,0,0,0,0,72,1,0,0,0,0,1] >;

C2×Q83Dic3 in GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes_3{\rm Dic}_3
% in TeX

G:=Group("C2xQ8:3Dic3");
// GroupNames label

G:=SmallGroup(192,794);
// by ID

G=gap.SmallGroup(192,794);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,422,136,1684,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=d^6=1,c^2=b^2,e^2=d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^2*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^-1>;
// generators/relations

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