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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C62.6(C2×C4)
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C2×C3⋊Dic3 — C2×C32⋊2C8 — C62.6(C2×C4)
 Lower central C32 — C3×C6 — C62.6(C2×C4)
 Upper central C1 — C22 — C2×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 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E 4F 6A ··· 6F 8A ··· 8H 12A ··· 12H order 1 2 2 2 2 2 3 3 4 4 4 4 4 4 6 ··· 6 8 ··· 8 12 ··· 12 size 1 1 1 1 18 18 4 4 2 2 9 9 9 9 4 ··· 4 18 ··· 18 4 ··· 4

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 4 4 4 4 4 type + + + + + + + image C1 C2 C2 C4 C4 C8 D4 M4(2) C32⋊C4 C2×C32⋊C4 C3⋊S3⋊3C8 C32⋊M4(2) C62⋊C4 kernel C62.6(C2×C4) C2×C32⋊2C8 C2×C4×C3⋊S3 C6×C12 C22×C3⋊S3 C2×C3⋊S3 C3⋊Dic3 C3×C6 C2×C4 C22 C2 C2 C2 # reps 1 2 1 2 2 8 2 2 2 2 4 4 4

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

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 1 0 0 0 0 72 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 72 0 0 0 0 1 0 0 0 0 0 0 0 72 72 0 0 0 0 1 0
,
 46 0 0 0 0 0 7 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 48 0 0 0 0 0 10 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 72 72 0 0

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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