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## G = C62.(C2×C4)  order 288 = 25·32

### The non-split extension by C62 of C2×C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C62.(C2×C4)
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C32⋊2C8 — C2×C32⋊2C8 — C62.(C2×C4)
 Lower central C32 — C3×C6 — C62.(C2×C4)
 Upper central C1 — C2 — D4

Generators and relations for C62.(C2×C4)
G = < a,b,c,d | a6=b6=c2=1, d4=b3, ab=ba, cac=ab3, dad-1=a-1b4, bc=cb, dbd-1=a4b, cd=dc >

Subgroups: 456 in 102 conjugacy classes, 34 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, D4, D4, Q8, C32, Dic3, C12, D6, C2×C6, C2×C8, M4(2), C4○D4, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C8○D4, C3⋊Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, D42S3, C322C8, C322C8, C324Q8, C4×C3⋊S3, C2×C3⋊Dic3, C327D4, D4×C32, C3⋊S33C8, C32⋊M4(2), C2×C322C8, C62.C4, C12.D6, C62.(C2×C4)
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C8○D4, C32⋊C4, C2×C32⋊C4, C22×C32⋊C4, C62.(C2×C4)

Character table of C62.(C2×C4)

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 12A 12B size 1 1 2 2 18 4 4 2 9 9 18 18 4 4 8 8 8 8 9 9 9 9 18 18 18 18 18 18 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 linear of order 2 ρ3 1 1 1 -1 -1 1 1 -1 1 1 -1 1 1 1 -1 1 1 -1 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 -1 -1 1 1 -1 1 1 -1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 1 -1 -1 1 1 1 -1 -1 linear of order 2 ρ5 1 1 -1 -1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 1 linear of order 2 ρ6 1 1 -1 -1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 linear of order 2 ρ7 1 1 -1 1 -1 1 1 -1 1 1 1 -1 1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 1 -1 -1 1 -1 -1 linear of order 2 ρ8 1 1 -1 1 -1 1 1 -1 1 1 1 -1 1 1 1 -1 -1 1 1 1 1 1 -1 -1 -1 1 1 -1 -1 -1 linear of order 2 ρ9 1 1 -1 -1 -1 1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 i i -i -i -i -i i -i i i 1 1 linear of order 4 ρ10 1 1 1 1 -1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 -i -i i i i -i i -i i -i 1 1 linear of order 4 ρ11 1 1 1 -1 1 1 1 -1 -1 -1 1 -1 1 1 -1 1 1 -1 -i -i i i -i -i i i -i i -1 -1 linear of order 4 ρ12 1 1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 1 i i -i -i i -i i i -i -i -1 -1 linear of order 4 ρ13 1 1 -1 -1 -1 1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 -i -i i i i i -i i -i -i 1 1 linear of order 4 ρ14 1 1 1 1 -1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 i i -i -i -i i -i i -i i 1 1 linear of order 4 ρ15 1 1 1 -1 1 1 1 -1 -1 -1 1 -1 1 1 -1 1 1 -1 i i -i -i i i -i -i i -i -1 -1 linear of order 4 ρ16 1 1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 1 -i -i i i -i i -i -i i i -1 -1 linear of order 4 ρ17 2 -2 0 0 0 2 2 0 -2i 2i 0 0 -2 -2 0 0 0 0 2ζ87 2ζ83 2ζ8 2ζ85 0 0 0 0 0 0 0 0 complex lifted from C8○D4 ρ18 2 -2 0 0 0 2 2 0 2i -2i 0 0 -2 -2 0 0 0 0 2ζ85 2ζ8 2ζ83 2ζ87 0 0 0 0 0 0 0 0 complex lifted from C8○D4 ρ19 2 -2 0 0 0 2 2 0 -2i 2i 0 0 -2 -2 0 0 0 0 2ζ83 2ζ87 2ζ85 2ζ8 0 0 0 0 0 0 0 0 complex lifted from C8○D4 ρ20 2 -2 0 0 0 2 2 0 2i -2i 0 0 -2 -2 0 0 0 0 2ζ8 2ζ85 2ζ87 2ζ83 0 0 0 0 0 0 0 0 complex lifted from C8○D4 ρ21 4 4 4 4 0 -2 1 4 0 0 0 0 1 -2 -2 1 -2 1 0 0 0 0 0 0 0 0 0 0 1 -2 orthogonal lifted from C32⋊C4 ρ22 4 4 -4 4 0 -2 1 -4 0 0 0 0 1 -2 -2 -1 2 1 0 0 0 0 0 0 0 0 0 0 -1 2 orthogonal lifted from C2×C32⋊C4 ρ23 4 4 4 4 0 1 -2 4 0 0 0 0 -2 1 1 -2 1 -2 0 0 0 0 0 0 0 0 0 0 -2 1 orthogonal lifted from C32⋊C4 ρ24 4 4 -4 4 0 1 -2 -4 0 0 0 0 -2 1 1 2 -1 -2 0 0 0 0 0 0 0 0 0 0 2 -1 orthogonal lifted from C2×C32⋊C4 ρ25 4 4 4 -4 0 1 -2 -4 0 0 0 0 -2 1 -1 -2 1 2 0 0 0 0 0 0 0 0 0 0 2 -1 orthogonal lifted from C2×C32⋊C4 ρ26 4 4 -4 -4 0 1 -2 4 0 0 0 0 -2 1 -1 2 -1 2 0 0 0 0 0 0 0 0 0 0 -2 1 orthogonal lifted from C2×C32⋊C4 ρ27 4 4 4 -4 0 -2 1 -4 0 0 0 0 1 -2 2 1 -2 -1 0 0 0 0 0 0 0 0 0 0 -1 2 orthogonal lifted from C2×C32⋊C4 ρ28 4 4 -4 -4 0 -2 1 4 0 0 0 0 1 -2 2 -1 2 -1 0 0 0 0 0 0 0 0 0 0 1 -2 orthogonal lifted from C2×C32⋊C4 ρ29 8 -8 0 0 0 -4 2 0 0 0 0 0 -2 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ30 8 -8 0 0 0 2 -4 0 0 0 0 0 4 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

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

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

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

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

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

 0 46 0 0 0 0 27 0 0 0 0 0 0 0 0 72 0 0 0 0 1 72 0 0 0 0 0 0 0 72 0 0 0 0 1 72
,
 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 72 1 0 0 0 0 72 0
,
 0 1 0 0 0 0 1 0 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
,
 22 0 0 0 0 0 0 22 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0

G:=sub<GL(6,GF(73))| [0,27,0,0,0,0,46,0,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[22,0,0,0,0,0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;

C62.(C2×C4) in GAP, Magma, Sage, TeX

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

G:=Group("C6^2.(C2xC4)");
// GroupNames label

G:=SmallGroup(288,935);
// by ID

G=gap.SmallGroup(288,935);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,219,80,9413,362,12550,1203]);
// Polycyclic

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

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