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## G = C3×C8.C22order 96 = 25·3

### Direct product of C3 and C8.C22

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C3×C8.C22
 Chief series C1 — C2 — C4 — C12 — C3×D4 — C3×SD16 — C3×C8.C22
 Lower central C1 — C2 — C4 — C3×C8.C22
 Upper central C1 — C6 — C2×C12 — C3×C8.C22

Generators and relations for C3×C8.C22
G = < a,b,c,d | a3=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, dcd=b4c >

Subgroups: 84 in 60 conjugacy classes, 40 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Q8, C12, C12, C2×C6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C3×Q8, C8.C22, C3×M4(2), C3×SD16, C3×Q16, C6×Q8, C3×C4○D4, C3×C8.C22
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C3×D4, C22×C6, C8.C22, C6×D4, C3×C8.C22

Smallest permutation representation of C3×C8.C22
On 48 points
Generators in S48
(1 14 18)(2 15 19)(3 16 20)(4 9 21)(5 10 22)(6 11 23)(7 12 24)(8 13 17)(25 34 47)(26 35 48)(27 36 41)(28 37 42)(29 38 43)(30 39 44)(31 40 45)(32 33 46)
(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 15)(11 13)(12 16)(17 23)(19 21)(20 24)(26 28)(27 31)(30 32)(33 39)(35 37)(36 40)(41 45)(42 48)(44 46)
(1 41)(2 46)(3 43)(4 48)(5 45)(6 42)(7 47)(8 44)(9 26)(10 31)(11 28)(12 25)(13 30)(14 27)(15 32)(16 29)(17 39)(18 36)(19 33)(20 38)(21 35)(22 40)(23 37)(24 34)

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

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

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

C3×C8.C22 is a maximal subgroup of   D12.39D4  M4(2).15D6  M4(2).16D6  D12.40D4  D24⋊C22  C24.C23  SD16.D6

33 conjugacy classes

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

33 irreducible representations

 dim 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 C3 C6 C6 C6 C6 C6 D4 D4 C3×D4 C3×D4 C8.C22 C3×C8.C22 kernel C3×C8.C22 C3×M4(2) C3×SD16 C3×Q16 C6×Q8 C3×C4○D4 C8.C22 M4(2) SD16 Q16 C2×Q8 C4○D4 C12 C2×C6 C4 C22 C3 C1 # reps 1 1 2 2 1 1 2 2 4 4 2 2 1 1 2 2 1 2

Matrix representation of C3×C8.C22 in GL4(𝔽7) generated by

 2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2
,
 0 6 2 2 3 6 3 2 5 3 4 0 2 1 5 4
,
 1 0 6 0 0 1 2 1 0 0 6 0 0 0 0 6
,
 1 6 3 2 1 3 3 4 2 5 4 6 1 5 1 6
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[0,3,5,2,6,6,3,1,2,3,4,5,2,2,0,4],[1,0,0,0,0,1,0,0,6,2,6,0,0,1,0,6],[1,1,2,1,6,3,5,5,3,3,4,1,2,4,6,6] >;

C3×C8.C22 in GAP, Magma, Sage, TeX

C_3\times C_8.C_2^2
% in TeX

G:=Group("C3xC8.C2^2");
// GroupNames label

G:=SmallGroup(96,184);
// by ID

G=gap.SmallGroup(96,184);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-2,313,295,938,2164,1090,88]);
// Polycyclic

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

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