Copied to
clipboard

G = C2×C62.C4order 288 = 25·32

Direct product of C2 and C62.C4

direct product, metabelian, soluble, monomial

Aliases: C2×C62.C4, (C3×C6)⋊4M4(2), (C2×C62).7C4, C62.20(C2×C4), C329(C2×M4(2)), C322C89C22, C23.3(C32⋊C4), C3⋊Dic3.35C23, (C2×C322C8)⋊9C2, (C2×C3⋊Dic3).24C4, C3⋊Dic3.55(C2×C4), (C3×C6).35(C22×C4), C22.20(C2×C32⋊C4), C2.12(C22×C32⋊C4), (C22×C3⋊Dic3).14C2, (C2×C3⋊Dic3).176C22, SmallGroup(288,940)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C2×C62.C4
C1C32C3×C6C3⋊Dic3C322C8C2×C322C8 — C2×C62.C4
C32C3×C6 — C2×C62.C4
C1C22C23

Generators and relations for C2×C62.C4
 G = < a,b,c,d | a2=b6=c6=1, d4=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c, dcd-1=b4c >

Subgroups: 448 in 122 conjugacy classes, 38 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×2], C4 [×4], C22, C22 [×2], C22 [×2], C6 [×14], C8 [×4], C2×C4 [×6], C23, C32, Dic3 [×8], C2×C6 [×14], C2×C8 [×2], M4(2) [×4], C22×C4, C3×C6, C3×C6 [×2], C3×C6 [×2], C2×Dic3 [×12], C22×C6 [×2], C2×M4(2), C3⋊Dic3 [×2], C3⋊Dic3 [×2], C62, C62 [×2], C62 [×2], C22×Dic3 [×2], C322C8 [×4], C2×C3⋊Dic3 [×2], C2×C3⋊Dic3 [×4], C2×C62, C2×C322C8 [×2], C62.C4 [×4], C22×C3⋊Dic3, C2×C62.C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, M4(2) [×2], C22×C4, C2×M4(2), C32⋊C4, C2×C32⋊C4 [×3], C62.C4 [×2], C22×C32⋊C4, C2×C62.C4

Smallest permutation representation of C2×C62.C4
On 48 points
Generators in S48
(1 24)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 40)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(25 45)(26 46)(27 47)(28 48)(29 41)(30 42)(31 43)(32 44)
(1 46 40)(2 37 47 6 33 43)(3 34 48)(4 45 35 8 41 39)(5 42 36)(7 38 44)(9 24 26)(10 31 17 14 27 21)(11 28 18)(12 23 29 16 19 25)(13 20 30)(15 32 22)
(1 5)(2 43 33 6 47 37)(3 7)(4 39 41 8 35 45)(9 13)(10 21 27 14 17 31)(11 15)(12 25 19 16 29 23)(18 22)(20 24)(26 30)(28 32)(34 38)(36 40)(42 46)(44 48)
(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)| (1,24)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,40)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(25,45)(26,46)(27,47)(28,48)(29,41)(30,42)(31,43)(32,44), (1,46,40)(2,37,47,6,33,43)(3,34,48)(4,45,35,8,41,39)(5,42,36)(7,38,44)(9,24,26)(10,31,17,14,27,21)(11,28,18)(12,23,29,16,19,25)(13,20,30)(15,32,22), (1,5)(2,43,33,6,47,37)(3,7)(4,39,41,8,35,45)(9,13)(10,21,27,14,17,31)(11,15)(12,25,19,16,29,23)(18,22)(20,24)(26,30)(28,32)(34,38)(36,40)(42,46)(44,48), (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( (1,24)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,40)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(25,45)(26,46)(27,47)(28,48)(29,41)(30,42)(31,43)(32,44), (1,46,40)(2,37,47,6,33,43)(3,34,48)(4,45,35,8,41,39)(5,42,36)(7,38,44)(9,24,26)(10,31,17,14,27,21)(11,28,18)(12,23,29,16,19,25)(13,20,30)(15,32,22), (1,5)(2,43,33,6,47,37)(3,7)(4,39,41,8,35,45)(9,13)(10,21,27,14,17,31)(11,15)(12,25,19,16,29,23)(18,22)(20,24)(26,30)(28,32)(34,38)(36,40)(42,46)(44,48), (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([(1,24),(2,17),(3,18),(4,19),(5,20),(6,21),(7,22),(8,23),(9,40),(10,33),(11,34),(12,35),(13,36),(14,37),(15,38),(16,39),(25,45),(26,46),(27,47),(28,48),(29,41),(30,42),(31,43),(32,44)], [(1,46,40),(2,37,47,6,33,43),(3,34,48),(4,45,35,8,41,39),(5,42,36),(7,38,44),(9,24,26),(10,31,17,14,27,21),(11,28,18),(12,23,29,16,19,25),(13,20,30),(15,32,22)], [(1,5),(2,43,33,6,47,37),(3,7),(4,39,41,8,35,45),(9,13),(10,21,27,14,17,31),(11,15),(12,25,19,16,29,23),(18,22),(20,24),(26,30),(28,32),(34,38),(36,40),(42,46),(44,48)], [(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)])

36 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C4D4E4F6A···6N8A···8H
order122222334444446···68···8
size11112244999918184···418···18

36 irreducible representations

dim1111112444
type++++++-
imageC1C2C2C2C4C4M4(2)C32⋊C4C2×C32⋊C4C62.C4
kernelC2×C62.C4C2×C322C8C62.C4C22×C3⋊Dic3C2×C3⋊Dic3C2×C62C3×C6C23C22C2
# reps1241624268

Matrix representation of C2×C62.C4 in GL6(𝔽73)

100000
010000
0072000
0007200
0000720
0000072
,
100000
68720000
0072100
0072000
0000172
000010
,
7200000
0720000
0072000
0007200
000001
0000721
,
47480000
61260000
0000720
0000072
00196800
00145400

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,68,0,0,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,72,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,1],[47,61,0,0,0,0,48,26,0,0,0,0,0,0,0,0,19,14,0,0,0,0,68,54,0,0,72,0,0,0,0,0,0,72,0,0] >;

C2×C62.C4 in GAP, Magma, Sage, TeX

C_2\times C_6^2.C_4
% in TeX

G:=Group("C2xC6^2.C4");
// GroupNames label

G:=SmallGroup(288,940);
// by ID

G=gap.SmallGroup(288,940);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,422,80,9413,362,12550,1203]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^6=1,d^4=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c,d*c*d^-1=b^4*c>;
// generators/relations

׿
×
𝔽