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G = C2×C12.31D6  order 288 = 25·32

Direct product of C2 and C12.31D6

direct product, metabelian, supersoluble, monomial

Aliases: C2×C12.31D6, C3⋊C8⋊28D6, C6⋊1(C8⋊S3), C12.47(C4×S3), (C3×C6)⋊3M4(2), (C2×C12).302D6, C62.50(C2×C4), C32⋊8(C2×M4(2)), C12.148(C22×S3), (C6×C12).207C22, (C3×C12).149C23, C4.16(C6.D6), C22.13(C6.D6), (C6×C3⋊C8)⋊20C2, (C2×C3⋊C8)⋊12S3, C4.95(C2×S32), C6.28(S3×C2×C4), C3⋊2(C2×C8⋊S3), (C2×C4).135S32, (C4×C3⋊S3).13C4, (C3×C3⋊C8)⋊37C22, (C2×C6).30(C4×S3), (C3×C12).88(C2×C4), C2.6(C2×C6.D6), (C22×C3⋊S3).11C4, (C4×C3⋊S3).90C22, (C2×C3⋊Dic3).20C4, C3⋊Dic3.42(C2×C4), (C3×C6).45(C22×C4), (C2×C4×C3⋊S3).16C2, (C2×C3⋊S3).36(C2×C4), SmallGroup(288,468)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — C2×C12.31D6
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — C3×C3⋊C8 — C12.31D6 — C2×C12.31D6
Lower central
C32 — C3×C6 — C2×C12.31D6
Upper central
C1 — C2×C4

Generators and relations for C2×C12.31D6
 G = < a,b,c,d | a2=b12=d2=1, c6=b3, ab=ba, ac=ca, ad=da, cbc-1=dbd=b5, dcd=c5 >

Subgroups: 594 in 163 conjugacy classes, 60 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], S3 [×8], C6 [×6], C6 [×3], C8 [×4], C2×C4, C2×C4 [×5], C23, C32, Dic3 [×6], C12 [×4], C12 [×2], D6 [×14], C2×C6 [×2], C2×C6, C2×C8 [×2], M4(2) [×4], C22×C4, C3⋊S3 [×2], C3×C6, C3×C6 [×2], C3⋊C8 [×4], C24 [×4], C4×S3 [×12], C2×Dic3 [×3], C2×C12 [×2], C2×C12, C22×S3 [×3], C2×M4(2), C3⋊Dic3 [×2], C3×C12 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C8⋊S3 [×8], C2×C3⋊C8 [×2], C2×C24 [×2], S3×C2×C4 [×3], C3×C3⋊C8 [×4], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C2×C8⋊S3 [×2], C12.31D6 [×4], C6×C3⋊C8 [×2], C2×C4×C3⋊S3, C2×C12.31D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], C23, D6 [×6], M4(2) [×2], C22×C4, C4×S3 [×4], C22×S3 [×2], C2×M4(2), S32, C8⋊S3 [×4], S3×C2×C4 [×2], C6.D6 [×2], C2×S32, C2×C8⋊S3 [×2], C12.31D6 [×2], C2×C6.D6, C2×C12.31D6

Smallest permutation representation of C2×C12.31D6
►On 48 points
Generators in S48
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)

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

(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)

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

60 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D4E4F6A···6F6G6H6I8A···8H12A···12H12I12J12K12L24A···24P
order1222223334444446···66668···812···121212121224···24
size11111818224111118182···24446···62···244446···6

60 irreducible representations

dim1111111222222244444
type+++++++++++
imageC1C2C2C2C4C4C4S3D6D6M4(2)C4×S3C4×S3C8⋊S3S32C6.D6C2×S32C6.D6C12.31D6
kernelC2×C12.31D6C12.31D6C6×C3⋊C8C2×C4×C3⋊S3C4×C3⋊S3C2×C3⋊Dic3C22×C3⋊S3C2×C3⋊C8C3⋊C8C2×C12C3×C6C12C2×C6C6C2×C4C4C4C22C2
# reps14214222424441611114

Matrix representation of C2×C12.31D6 ►in GL6(𝔽73)

7200000
0720000
001000
000100
000010
000001
,
72720000
100000
0046000
0004600
0000720
0000072
,
100000
72720000
0001800
0035000
00004627
0000460
,
100000
72720000
0072000
000100
0000720
0000721

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,0,35,0,0,0,0,18,0,0,0,0,0,0,0,46,46,0,0,0,0,27,0],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,1] >;
 

C2×C12.31D6 in GAP, Magma, Sage, TeX 

C_2\times C_{12}._{31}D_6
 
% in TeX
 
G:=Group("C2xC12.31D6");
 
// GroupNames label
 
G:=SmallGroup(288,468);
 
// by ID
 
G=gap.SmallGroup(288,468);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,253,64,80,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^12=d^2=1,c^6=b^3,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^5,d*c*d=c^5>;
 
// generators/relations
 

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