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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

Aliases: C3×C8.C22, Q162C6, SD162C6, C12.64D4, M4(2)⋊2C6, C12.49C23, C24.12C22, C8.(C2×C6), (C2×Q8)⋊6C6, (C3×Q16)⋊6C2, (C6×Q8)⋊11C2, C4○D4.4C6, D4.3(C2×C6), (C2×C6).25D4, C4.15(C3×D4), C2.16(C6×D4), C6.79(C2×D4), Q8.6(C2×C6), (C3×SD16)⋊6C2, C4.6(C22×C6), C22.6(C3×D4), (C3×M4(2))⋊4C2, (C2×C12).70C22, (C3×D4).13C22, (C3×Q8).14C22, (C2×C4).11(C2×C6), (C3×C4○D4).5C2, SmallGroup(96,184)

Series: Derived Chief Lower central Upper central

C1C4 — C3×C8.C22
C1C2C4C12C3×D4C3×SD16 — C3×C8.C22
C1C2C4 — C3×C8.C22
C1C6C2×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 2A2B2C3A3B4A4B4C4D4E6A6B6C6D6E6F8A8B12A12B12C12D12E···12J24A24B24C24D
order12223344444666666881212121212···1224242424
size112411224441122444422224···44444

33 irreducible representations

dim111111111111222244
type++++++++-
imageC1C2C2C2C2C2C3C6C6C6C6C6D4D4C3×D4C3×D4C8.C22C3×C8.C22
kernelC3×C8.C22C3×M4(2)C3×SD16C3×Q16C6×Q8C3×C4○D4C8.C22M4(2)SD16Q16C2×Q8C4○D4C12C2×C6C4C22C3C1
# reps112211224422112212

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

2000
0200
0020
0002
,
0622
3632
5340
2154
,
1060
0121
0060
0006
,
1632
1334
2546
1516
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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