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## G = C3×C8○D4order 96 = 25·3

### Direct product of C3 and C8○D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×C8○D4
 Chief series C1 — C2 — C4 — C12 — C24 — C2×C24 — C3×C8○D4
 Lower central C1 — C2 — C3×C8○D4
 Upper central C1 — C24 — C3×C8○D4

Generators and relations for C3×C8○D4
G = < a,b,c,d | a3=b8=d2=1, c2=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c >

Subgroups: 68 in 62 conjugacy classes, 56 normal (14 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C6, C6 [×3], C8, C8 [×3], C2×C4 [×3], D4 [×3], Q8, C12, C12 [×3], C2×C6 [×3], C2×C8 [×3], M4(2) [×3], C4○D4, C24, C24 [×3], C2×C12 [×3], C3×D4 [×3], C3×Q8, C8○D4, C2×C24 [×3], C3×M4(2) [×3], C3×C4○D4, C3×C8○D4
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], C23, C12 [×4], C2×C6 [×7], C22×C4, C2×C12 [×6], C22×C6, C8○D4, C22×C12, C3×C8○D4

Smallest permutation representation of C3×C8○D4
On 48 points
Generators in S48
(1 9 23)(2 10 24)(3 11 17)(4 12 18)(5 13 19)(6 14 20)(7 15 21)(8 16 22)(25 35 45)(26 36 46)(27 37 47)(28 38 48)(29 39 41)(30 40 42)(31 33 43)(32 34 44)
(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 43 5 47)(2 44 6 48)(3 45 7 41)(4 46 8 42)(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 47)(2 48)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 25)(16 26)(17 39)(18 40)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)

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

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

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

C3×C8○D4 is a maximal subgroup of
C24.99D4  C24.78C23  Q8.8D12  Q8.9D12  Q8.10D12  C24.100D4  C24.54D4  M4(2)⋊28D6  D4.11D12  D4.12D12  D4.13D12  Q8.C36
C3×C8○D4 is a maximal quotient of
D4×C24  Q8×C24

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 C8○D4 C3×C8○D4 kernel C3×C8○D4 C2×C24 C3×M4(2) C3×C4○D4 C8○D4 C3×D4 C3×Q8 C2×C8 M4(2) C4○D4 D4 Q8 C3 C1 # reps 1 3 3 1 2 6 2 6 6 2 12 4 4 8

Matrix representation of C3×C8○D4 in GL3(𝔽73) generated by

 64 0 0 0 1 0 0 0 1
,
 1 0 0 0 10 0 0 0 10
,
 72 0 0 0 0 1 0 72 0
,
 1 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(73))| [64,0,0,0,1,0,0,0,1],[1,0,0,0,10,0,0,0,10],[72,0,0,0,0,72,0,1,0],[1,0,0,0,0,1,0,1,0] >;

C3×C8○D4 in GAP, Magma, Sage, TeX

C_3\times C_8\circ D_4
% in TeX

G:=Group("C3xC8oD4");
// GroupNames label

G:=SmallGroup(96,178);
// by ID

G=gap.SmallGroup(96,178);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-2,144,476,88]);
// Polycyclic

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

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