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G = C62.6(C2×C4)  order 288 = 25·32

5th non-split extension by C62 of C2×C4 acting via C2×C4/C2=C4

metabelian, soluble, monomial

Aliases: (C6×C12).7C4, C62.6(C2×C4), C323(C22⋊C8), C3⋊Dic3.51D4, (C3×C6).2M4(2), C2.1(C62⋊C4), C2.3(C32⋊M4(2)), (C2×C3⋊S3)⋊5C8, (C3×C6).10(C2×C8), (C2×C322C8)⋊2C2, (C22×C3⋊S3).8C4, (C2×C4).3(C32⋊C4), C2.4(C3⋊S33C8), C22.11(C2×C32⋊C4), (C3×C6).13(C22⋊C4), (C2×C3⋊Dic3).110C22, (C2×C4×C3⋊S3).13C2, SmallGroup(288,426)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C62.6(C2×C4)
C1C32C3×C6C3⋊Dic3C2×C3⋊Dic3C2×C322C8 — C62.6(C2×C4)
C32C3×C6 — C62.6(C2×C4)
C1C22C2×C4

Generators and relations for C62.6(C2×C4)
 G = < a,b,c,d | a6=b6=1, c2=d4=b3, ab=ba, ac=ca, dad-1=a-1b4, bc=cb, dbd-1=a4b, dcd-1=a3c >

Subgroups: 536 in 104 conjugacy classes, 22 normal (16 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C4 [×3], C22, C22 [×4], S3 [×8], C6 [×6], C8 [×2], C2×C4, C2×C4 [×3], C23, C32, Dic3 [×4], C12 [×4], D6 [×12], C2×C6 [×2], C2×C8 [×2], C22×C4, C3⋊S3 [×2], C3×C6 [×3], C4×S3 [×8], C2×Dic3 [×2], C2×C12 [×2], C22×S3 [×2], C22⋊C8, C3⋊Dic3 [×2], C3×C12, C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, S3×C2×C4 [×2], C322C8 [×2], C4×C3⋊S3 [×2], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C2×C322C8 [×2], C2×C4×C3⋊S3, C62.6(C2×C4)
Quotients: C1, C2 [×3], C4 [×2], C22, C8 [×2], C2×C4, D4 [×2], C22⋊C4, C2×C8, M4(2), C22⋊C8, C32⋊C4, C2×C32⋊C4, C3⋊S33C8, C32⋊M4(2), C62⋊C4, C62.6(C2×C4)

Smallest permutation representation of C62.6(C2×C4)
On 48 points
Generators in S48
(1 27 9 42 17 39)(2 43)(3 33 19 44 11 29)(4 45)(5 31 13 46 21 35)(6 47)(7 37 23 48 15 25)(8 41)(10 40)(12 34)(14 36)(16 38)(18 28)(20 30)(22 32)(24 26)
(1 21 9 5 17 13)(2 14 18 6 10 22)(3 15 19 7 11 23)(4 24 12 8 20 16)(25 44 37 29 48 33)(26 34 41 30 38 45)(27 35 42 31 39 46)(28 47 40 32 43 36)
(1 7 5 3)(2 41 6 45)(4 43 8 47)(9 23 13 19)(10 26 14 30)(11 17 15 21)(12 28 16 32)(18 38 22 34)(20 40 24 36)(25 35 29 39)(27 37 31 33)(42 48 46 44)
(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)| (1,27,9,42,17,39)(2,43)(3,33,19,44,11,29)(4,45)(5,31,13,46,21,35)(6,47)(7,37,23,48,15,25)(8,41)(10,40)(12,34)(14,36)(16,38)(18,28)(20,30)(22,32)(24,26), (1,21,9,5,17,13)(2,14,18,6,10,22)(3,15,19,7,11,23)(4,24,12,8,20,16)(25,44,37,29,48,33)(26,34,41,30,38,45)(27,35,42,31,39,46)(28,47,40,32,43,36), (1,7,5,3)(2,41,6,45)(4,43,8,47)(9,23,13,19)(10,26,14,30)(11,17,15,21)(12,28,16,32)(18,38,22,34)(20,40,24,36)(25,35,29,39)(27,37,31,33)(42,48,46,44), (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( (1,27,9,42,17,39)(2,43)(3,33,19,44,11,29)(4,45)(5,31,13,46,21,35)(6,47)(7,37,23,48,15,25)(8,41)(10,40)(12,34)(14,36)(16,38)(18,28)(20,30)(22,32)(24,26), (1,21,9,5,17,13)(2,14,18,6,10,22)(3,15,19,7,11,23)(4,24,12,8,20,16)(25,44,37,29,48,33)(26,34,41,30,38,45)(27,35,42,31,39,46)(28,47,40,32,43,36), (1,7,5,3)(2,41,6,45)(4,43,8,47)(9,23,13,19)(10,26,14,30)(11,17,15,21)(12,28,16,32)(18,38,22,34)(20,40,24,36)(25,35,29,39)(27,37,31,33)(42,48,46,44), (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([(1,27,9,42,17,39),(2,43),(3,33,19,44,11,29),(4,45),(5,31,13,46,21,35),(6,47),(7,37,23,48,15,25),(8,41),(10,40),(12,34),(14,36),(16,38),(18,28),(20,30),(22,32),(24,26)], [(1,21,9,5,17,13),(2,14,18,6,10,22),(3,15,19,7,11,23),(4,24,12,8,20,16),(25,44,37,29,48,33),(26,34,41,30,38,45),(27,35,42,31,39,46),(28,47,40,32,43,36)], [(1,7,5,3),(2,41,6,45),(4,43,8,47),(9,23,13,19),(10,26,14,30),(11,17,15,21),(12,28,16,32),(18,38,22,34),(20,40,24,36),(25,35,29,39),(27,37,31,33),(42,48,46,44)], [(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···6F8A···8H12A···12H
order122222334444446···68···812···12
size11111818442299994···418···184···4

36 irreducible representations

dim1111112244444
type+++++++
imageC1C2C2C4C4C8D4M4(2)C32⋊C4C2×C32⋊C4C3⋊S33C8C32⋊M4(2)C62⋊C4
kernelC62.6(C2×C4)C2×C322C8C2×C4×C3⋊S3C6×C12C22×C3⋊S3C2×C3⋊S3C3⋊Dic3C3×C6C2×C4C22C2C2C2
# reps1212282222444

Matrix representation of C62.6(C2×C4) in GL6(𝔽73)

7200000
0720000
001100
0072000
0000720
0000072
,
7200000
0720000
00727200
001000
00007272
000010
,
4600000
7270000
001000
000100
0000720
0000072
,
63480000
0100000
000010
000001
001000
00727200

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[46,7,0,0,0,0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[63,0,0,0,0,0,48,10,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,1,0,0] >;

C62.6(C2×C4) in GAP, Magma, Sage, TeX

C_6^2._6(C_2\times C_4)
% in TeX

G:=Group("C6^2.6(C2xC4)");
// GroupNames label

G:=SmallGroup(288,426);
// by ID

G=gap.SmallGroup(288,426);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,120,100,9413,691,12550,2372]);
// Polycyclic

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

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