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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, D82C6, C246C22, SD161C6, C12.63D4, M4(2)⋊1C6, C12.48C23, C8⋊(C2×C6), C4○D44C6, (C2×D4)⋊5C6, D42(C2×C6), (C3×D8)⋊6C2, Q83(C2×C6), (C6×D4)⋊14C2, C6.78(C2×D4), (C2×C6).24D4, C4.14(C3×D4), C2.15(C6×D4), (C3×SD16)⋊5C2, C4.5(C22×C6), C22.5(C3×D4), (C3×D4)⋊11C22, (C3×M4(2))⋊3C2, (C3×Q8)⋊10C22, (C2×C12).69C22, (C3×C4○D4)⋊7C2, (C2×C4).10(C2×C6), SmallGroup(96,183)

Series: Derived Chief Lower central Upper central

C1C4 — C3×C8⋊C22
C1C2C4C12C3×D4C3×D8 — 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, cd=dc >

Subgroups: 116 in 68 conjugacy classes, 40 normal (24 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4, C22, C22 [×5], C6, C6 [×4], C8 [×2], C2×C4, C2×C4, D4, D4 [×2], D4 [×2], Q8, C23, C12 [×2], C12, C2×C6, C2×C6 [×5], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C24 [×2], C2×C12, C2×C12, C3×D4, C3×D4 [×2], C3×D4 [×2], C3×Q8, C22×C6, C8⋊C22, C3×M4(2), C3×D8 [×2], C3×SD16 [×2], C6×D4, 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

Permutation representations of C3×C8⋊C22
On 24 points - transitive group 24T114
Generators in S24
(1 15 19)(2 16 20)(3 9 21)(4 10 22)(5 11 23)(6 12 24)(7 13 17)(8 14 18)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(2 4)(3 7)(6 8)(9 13)(10 16)(12 14)(17 21)(18 24)(20 22)
(1 5)(3 7)(9 13)(11 15)(17 21)(19 23)

G:=sub<Sym(24)| (1,15,19)(2,16,20)(3,9,21)(4,10,22)(5,11,23)(6,12,24)(7,13,17)(8,14,18), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(17,21)(18,24)(20,22), (1,5)(3,7)(9,13)(11,15)(17,21)(19,23)>;

G:=Group( (1,15,19)(2,16,20)(3,9,21)(4,10,22)(5,11,23)(6,12,24)(7,13,17)(8,14,18), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(17,21)(18,24)(20,22), (1,5)(3,7)(9,13)(11,15)(17,21)(19,23) );

G=PermutationGroup([(1,15,19),(2,16,20),(3,9,21),(4,10,22),(5,11,23),(6,12,24),(7,13,17),(8,14,18)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(2,4),(3,7),(6,8),(9,13),(10,16),(12,14),(17,21),(18,24),(20,22)], [(1,5),(3,7),(9,13),(11,15),(17,21),(19,23)])

G:=TransitiveGroup(24,114);

C3×C8⋊C22 is a maximal subgroup of   D1218D4  M4(2).D6  M4(2).13D6  D12.38D4  D84D6  D85D6  D86D6

33 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C6A6B6C6D6E···6J8A8B12A12B12C12D12E12F24A24B24C24D
order1222223344466666···68812121212121224242424
size1124441122411224···4442222444444

33 irreducible representations

dim111111111111222244
type+++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6D4D4C3×D4C3×D4C8⋊C22C3×C8⋊C22
kernelC3×C8⋊C22C3×M4(2)C3×D8C3×SD16C6×D4C3×C4○D4C8⋊C22M4(2)D8SD16C2×D4C4○D4C12C2×C6C4C22C3C1
# reps112211224422112212

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

4000
0400
0040
0004
,
4104
4330
2564
3321
,
6032
0622
0010
0001
,
0632
6042
0060
0001
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[4,4,2,3,1,3,5,3,0,3,6,2,4,0,4,1],[6,0,0,0,0,6,0,0,3,2,1,0,2,2,0,1],[0,6,0,0,6,0,0,0,3,4,6,0,2,2,0,1] >;

C3×C8⋊C22 in GAP, Magma, Sage, TeX

C_3\times C_8\rtimes C_2^2
% in TeX

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

G:=SmallGroup(96,183);
// by ID

G=gap.SmallGroup(96,183);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-2,313,938,2164,1090,88]);
// 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,c*d=d*c>;
// generators/relations

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