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G = C3×C22⋊C8order 96 = 25·3

Direct product of C3 and C22⋊C8

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

Aliases: C3×C22⋊C8, C222C24, C12.65D4, C23.3C12, C6.8M4(2), (C2×C8)⋊1C6, (C2×C6)⋊1C8, (C2×C24)⋊3C2, C2.1(C2×C24), C6.11(C2×C8), (C2×C12).6C4, (C2×C4).3C12, C4.16(C3×D4), (C22×C4).4C6, (C22×C6).3C4, C22.9(C2×C12), (C22×C12).3C2, C2.2(C3×M4(2)), C6.20(C22⋊C4), (C2×C12).135C22, (C2×C4).31(C2×C6), (C2×C6).38(C2×C4), C2.2(C3×C22⋊C4), SmallGroup(96,48)

Series: Derived Chief Lower central Upper central

C1C2 — C3×C22⋊C8
C1C2C4C2×C4C2×C12C2×C24 — C3×C22⋊C8
C1C2 — C3×C22⋊C8
C1C2×C12 — C3×C22⋊C8

Generators and relations for C3×C22⋊C8
 G = < a,b,c,d | a3=b2=c2=d8=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >

2C2
2C2
2C22
2C4
2C22
2C6
2C6
2C2×C4
2C8
2C2×C4
2C8
2C2×C6
2C12
2C2×C6
2C24
2C2×C12
2C24
2C2×C12

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

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

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

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

C3×C22⋊C8 is a maximal subgroup of
C23.35D12  (C22×S3)⋊C8  (C2×Dic3)⋊C8  C22.2D24  Dic3.5M4(2)  Dic3.M4(2)  C24⋊C4⋊C2  C23.39D12  C23.40D12  C23.15D12  C3⋊D4⋊C8  D6⋊M4(2)  D6⋊C8⋊C2  D62M4(2)  Dic3⋊M4(2)  C3⋊C826D4  D12.31D4  D1213D4  D12.32D4  D1214D4  C23.43D12  C22.D24  C23.18D12  Dic614D4  Dic6.32D4  D4×C24

60 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C4D4E4F6A···6F6G6H6I6J8A···8H12A···12H12I12J12K12L24A···24P
order122222334444446···666668···812···121212121224···24
size111122111111221···122222···21···122222···2

60 irreducible representations

dim1111111111112222
type++++
imageC1C2C2C3C4C4C6C6C8C12C12C24D4M4(2)C3×D4C3×M4(2)
kernelC3×C22⋊C8C2×C24C22×C12C22⋊C8C2×C12C22×C6C2×C8C22×C4C2×C6C2×C4C23C22C12C6C4C2
# reps12122242844162244

Matrix representation of C3×C22⋊C8 in GL3(𝔽73) generated by

800
010
001
,
7200
010
04672
,
100
0720
0072
,
6300
04671
0027
G:=sub<GL(3,GF(73))| [8,0,0,0,1,0,0,0,1],[72,0,0,0,1,46,0,0,72],[1,0,0,0,72,0,0,0,72],[63,0,0,0,46,0,0,71,27] >;

C3×C22⋊C8 in GAP, Magma, Sage, TeX

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

G:=Group("C3xC2^2:C8");
// GroupNames label

G:=SmallGroup(96,48);
// by ID

G=gap.SmallGroup(96,48);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-2,144,169,88]);
// Polycyclic

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

Export

Subgroup lattice of C3×C22⋊C8 in TeX

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