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## G = C3×C4○D8order 96 = 25·3

### Direct product of C3 and C4○D8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C3×C4○D8
 Chief series C1 — C2 — C4 — C12 — C3×D4 — C3×D8 — C3×C4○D8
 Lower central C1 — C2 — C4 — C3×C4○D8
 Upper central C1 — C12 — C2×C12 — C3×C4○D8

Generators and relations for C3×C4○D8
G = < a,b,c,d | a3=b4=d2=1, c4=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c3 >

Subgroups: 92 in 62 conjugacy classes, 40 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C12, C12, C2×C6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C4○D8, C2×C24, C3×D8, C3×SD16, C3×Q16, C3×C4○D4, C3×C4○D8
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C3×D4, C22×C6, C4○D8, C6×D4, C3×C4○D8

Smallest permutation representation of C3×C4○D8
On 48 points
Generators in S48
(1 12 19)(2 13 20)(3 14 21)(4 15 22)(5 16 23)(6 9 24)(7 10 17)(8 11 18)(25 36 48)(26 37 41)(27 38 42)(28 39 43)(29 40 44)(30 33 45)(31 34 46)(32 35 47)
(1 41 5 45)(2 42 6 46)(3 43 7 47)(4 44 8 48)(9 31 13 27)(10 32 14 28)(11 25 15 29)(12 26 16 30)(17 35 21 39)(18 36 22 40)(19 37 23 33)(20 38 24 34)
(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 8)(3 7)(4 6)(9 15)(10 14)(11 13)(17 21)(18 20)(22 24)(25 27)(28 32)(29 31)(34 40)(35 39)(36 38)(42 48)(43 47)(44 46)

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

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

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

C3×C4○D8 is a maximal subgroup of
D82Dic3  C24.41D4  Q16⋊D6  Q16.D6  D8.9D6  D85Dic3  D84Dic3  SD16⋊D6  D815D6  D811D6  D8.10D6
C3×C4○D8 is a maximal quotient of
C12×D8  C12×SD16  C12×Q16

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D4 D4 C3×D4 C3×D4 C4○D8 C3×C4○D8 kernel C3×C4○D8 C2×C24 C3×D8 C3×SD16 C3×Q16 C3×C4○D4 C4○D8 C2×C8 D8 SD16 Q16 C4○D4 C12 C2×C6 C4 C22 C3 C1 # reps 1 1 1 2 1 2 2 2 2 4 2 4 1 1 2 2 4 8

Matrix representation of C3×C4○D8 in GL2(𝔽73) generated by

 64 0 0 64
,
 46 0 0 46
,
 16 57 16 16
,
 1 0 0 72
G:=sub<GL(2,GF(73))| [64,0,0,64],[46,0,0,46],[16,16,57,16],[1,0,0,72] >;

C3×C4○D8 in GAP, Magma, Sage, TeX

C_3\times C_4\circ D_8
% in TeX

G:=Group("C3xC4oD8");
// GroupNames label

G:=SmallGroup(96,182);
// by ID

G=gap.SmallGroup(96,182);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-2,313,230,2164,1090,88]);
// Polycyclic

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

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