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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 [×2], C3, C4 [×2], C4 [×3], C22, C22, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4, D4, Q8, Q8 [×2], Q8, C12 [×2], C12 [×3], C2×C6, C2×C6, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C24 [×2], C2×C12, C2×C12 [×2], C3×D4, C3×D4, C3×Q8, C3×Q8 [×2], C3×Q8, C8.C22, C3×M4(2), C3×SD16 [×2], C3×Q16 [×2], C6×Q8, C3×C4○D4, C3×C8.C22
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, C2×C6 [×7], C2×D4, C3×D4 [×2], C22×C6, C8.C22, C6×D4, C3×C8.C22

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

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

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

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

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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