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G = C2×C12.31D6order 288 = 25·32

Direct product of C2 and C12.31D6

direct product, metabelian, supersoluble, monomial

Aliases: C2×C12.31D6, C3⋊C828D6, C61(C8⋊S3), C12.47(C4×S3), (C3×C6)⋊3M4(2), (C2×C12).302D6, C62.50(C2×C4), C328(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), C32(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

C1C3×C6 — C2×C12.31D6
C1C3C32C3×C6C3×C12C3×C3⋊C8C12.31D6 — C2×C12.31D6
C32C3×C6 — C2×C12.31D6
C1C2×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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