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## G = C2×C62.C4order 288 = 25·32

### Direct product of C2 and C62.C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×C62.C4
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C32⋊2C8 — C2×C32⋊2C8 — C2×C62.C4
 Lower central C32 — C3×C6 — C2×C62.C4
 Upper central C1 — C22 — C23

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, C2, C3, C4, C22, C22, C22, C6, C8, C2×C4, C23, C32, Dic3, C2×C6, C2×C8, M4(2), C22×C4, C3×C6, C3×C6, C3×C6, C2×Dic3, C22×C6, C2×M4(2), C3⋊Dic3, C3⋊Dic3, C62, C62, C62, C22×Dic3, C322C8, C2×C3⋊Dic3, C2×C3⋊Dic3, C2×C62, C2×C322C8, C62.C4, C22×C3⋊Dic3, C2×C62.C4
Quotients: C1, C2, C4, C22, C2×C4, C23, M4(2), C22×C4, C2×M4(2), C32⋊C4, C2×C32⋊C4, C62.C4, 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 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 33)(25 45)(26 46)(27 47)(28 48)(29 41)(30 42)(31 43)(32 44)
(1 46 11)(2 16 47 6 12 43)(3 13 48)(4 45 14 8 41 10)(5 42 15)(7 9 44)(17 33 27 21 37 31)(18 38 28)(19 25 39 23 29 35)(20 30 40)(22 34 32)(24 26 36)
(1 5)(2 43 12 6 47 16)(3 7)(4 10 41 8 14 45)(9 13)(11 15)(17 31 37 21 27 33)(18 22)(19 35 29 23 39 25)(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)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,33)(25,45)(26,46)(27,47)(28,48)(29,41)(30,42)(31,43)(32,44), (1,46,11)(2,16,47,6,12,43)(3,13,48)(4,45,14,8,41,10)(5,42,15)(7,9,44)(17,33,27,21,37,31)(18,38,28)(19,25,39,23,29,35)(20,30,40)(22,34,32)(24,26,36), (1,5)(2,43,12,6,47,16)(3,7)(4,10,41,8,14,45)(9,13)(11,15)(17,31,37,21,27,33)(18,22)(19,35,29,23,39,25)(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)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,33)(25,45)(26,46)(27,47)(28,48)(29,41)(30,42)(31,43)(32,44), (1,46,11)(2,16,47,6,12,43)(3,13,48)(4,45,14,8,41,10)(5,42,15)(7,9,44)(17,33,27,21,37,31)(18,38,28)(19,25,39,23,29,35)(20,30,40)(22,34,32)(24,26,36), (1,5)(2,43,12,6,47,16)(3,7)(4,10,41,8,14,45)(9,13)(11,15)(17,31,37,21,27,33)(18,22)(19,35,29,23,39,25)(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),(9,34),(10,35),(11,36),(12,37),(13,38),(14,39),(15,40),(16,33),(25,45),(26,46),(27,47),(28,48),(29,41),(30,42),(31,43),(32,44)], [(1,46,11),(2,16,47,6,12,43),(3,13,48),(4,45,14,8,41,10),(5,42,15),(7,9,44),(17,33,27,21,37,31),(18,38,28),(19,25,39,23,29,35),(20,30,40),(22,34,32),(24,26,36)], [(1,5),(2,43,12,6,47,16),(3,7),(4,10,41,8,14,45),(9,13),(11,15),(17,31,37,21,27,33),(18,22),(19,35,29,23,39,25),(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 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E 4F 6A ··· 6N 8A ··· 8H order 1 2 2 2 2 2 3 3 4 4 4 4 4 4 6 ··· 6 8 ··· 8 size 1 1 1 1 2 2 4 4 9 9 9 9 18 18 4 ··· 4 18 ··· 18

36 irreducible representations

 dim 1 1 1 1 1 1 2 4 4 4 type + + + + + + - image C1 C2 C2 C2 C4 C4 M4(2) C32⋊C4 C2×C32⋊C4 C62.C4 kernel C2×C62.C4 C2×C32⋊2C8 C62.C4 C22×C3⋊Dic3 C2×C3⋊Dic3 C2×C62 C3×C6 C23 C22 C2 # reps 1 2 4 1 6 2 4 2 6 8

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

 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 0 0 0 0 0 68 72 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 1 72 0 0 0 0 1 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 1 0 0 0 0 72 1
,
 47 48 0 0 0 0 61 26 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 19 68 0 0 0 0 14 54 0 0

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

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