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## G = C3×C23.D4order 192 = 26·3

### Direct product of C3 and C23.D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23 — C3×C23.D4
 Chief series C1 — C2 — C22 — C23 — C2×D4 — C6×D4 — C3×C23⋊C4 — C3×C23.D4
 Lower central C1 — C2 — C22 — C23 — C3×C23.D4
 Upper central C1 — C6 — C2×C6 — C6×D4 — C3×C23.D4

Generators and relations for C3×C23.D4
G = < a,b,c,d,e,f | a3=b2=c2=d2=1, e4=d, f2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=bcd, bf=fb, ece-1=fcf-1=cd=dc, de=ed, df=fd, fef-1=be3 >

Subgroups: 162 in 68 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, C23, C12, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, M4(2), C22×C4, C2×D4, C24, C2×C12, C2×C12, C3×D4, C22×C6, C23⋊C4, C4.D4, C22.D4, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, C3×M4(2), C22×C12, C6×D4, C23.D4, C3×C23⋊C4, C3×C4.D4, C3×C22.D4, C3×C23.D4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C12, C2×C6, C22⋊C4, C2×C12, C3×D4, C23⋊C4, C3×C22⋊C4, C23.D4, C3×C23⋊C4, C3×C23.D4

Smallest permutation representation of C3×C23.D4
On 48 points
Generators in S48
(1 27 41)(2 28 42)(3 29 43)(4 30 44)(5 31 45)(6 32 46)(7 25 47)(8 26 48)(9 33 22)(10 34 23)(11 35 24)(12 36 17)(13 37 18)(14 38 19)(15 39 20)(16 40 21)
(2 36)(3 7)(4 34)(6 40)(8 38)(9 13)(10 44)(12 42)(14 48)(16 46)(17 28)(18 22)(19 26)(21 32)(23 30)(25 29)(33 37)(43 47)
(1 39)(2 36)(3 33)(4 38)(5 35)(6 40)(7 37)(8 34)(9 43)(10 48)(11 45)(12 42)(13 47)(14 44)(15 41)(16 46)(17 28)(18 25)(19 30)(20 27)(21 32)(22 29)(23 26)(24 31)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 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 34 36 4)(3 33 7 37)(6 38 40 8)(9 47 13 43)(10 12 44 42)(11 15)(14 16 48 46)(17 30 28 23)(18 29 22 25)(19 21 26 32)(20 24)(35 39)

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

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 type + + + + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 D4 D4 C3×D4 C3×D4 C23⋊C4 C23.D4 C3×C23⋊C4 C3×C23.D4 kernel C3×C23.D4 C3×C23⋊C4 C3×C4.D4 C3×C22.D4 C23.D4 C3×C22⋊C4 C22×C12 C23⋊C4 C4.D4 C22.D4 C22⋊C4 C22×C4 C2×C12 C22×C6 C2×C4 C23 C6 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 2 2 4 4 1 1 2 2 1 2 2 4

Matrix representation of C3×C23.D4 in GL4(𝔽73) generated by

 64 0 0 0 0 64 0 0 0 0 64 0 0 0 0 64
,
 1 0 0 0 0 1 0 0 0 0 72 0 0 0 0 72
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
,
 72 0 0 0 0 72 0 0 0 0 72 0 0 0 0 72
,
 23 50 50 50 23 50 23 23 23 23 23 50 50 50 23 50
,
 1 0 0 0 0 72 0 0 0 0 0 72 0 0 1 0
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,64,0,0,0,0,64],[1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[23,23,23,50,50,50,23,50,50,23,23,23,50,23,50,50],[1,0,0,0,0,72,0,0,0,0,0,1,0,0,72,0] >;

C3×C23.D4 in GAP, Magma, Sage, TeX

C_3\times C_2^3.D_4
% in TeX

G:=Group("C3xC2^3.D4");
// GroupNames label

G:=SmallGroup(192,158);
// by ID

G=gap.SmallGroup(192,158);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,680,1683,1271,375,6053]);
// Polycyclic

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

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