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G = (C6×D4)⋊C4order 192 = 26·3

1st semidirect product of C6×D4 and C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — (C6×D4)⋊C4
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×C12 — C6.C42 — (C6×D4)⋊C4
 Lower central C3 — C2×C6 — C2×C12 — (C6×D4)⋊C4
 Upper central C1 — C22 — C22×C4 — C4⋊D4

Generators and relations for (C6×D4)⋊C4
G = < a,b,c,d | a6=b4=c2=d4=1, ab=ba, ac=ca, dad-1=a-1, cbc=b-1, dbd-1=a3b, dcd-1=bc >

Subgroups: 272 in 90 conjugacy classes, 31 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C2.C42, C22⋊C8, C4⋊D4, C2×C3⋊C8, C3×C22⋊C4, C3×C4⋊C4, C22×Dic3, C22×C12, C6×D4, C6×D4, C22.SD16, C12.55D4, C6.C42, C3×C4⋊D4, (C6×D4)⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, D8, SD16, C2×Dic3, C3⋊D4, C23⋊C4, D4⋊C4, C4≀C2, D4⋊S3, D4.S3, C6.D4, C22.SD16, D4⋊Dic3, C23.7D6, Q83Dic3, (C6×D4)⋊C4

Smallest permutation representation of (C6×D4)⋊C4
On 48 points
Generators in S48
(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)
(1 20 15 25)(2 21 16 26)(3 22 17 27)(4 23 18 28)(5 24 13 29)(6 19 14 30)(7 34 46 39)(8 35 47 40)(9 36 48 41)(10 31 43 42)(11 32 44 37)(12 33 45 38)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 7)(13 45)(14 46)(15 47)(16 48)(17 43)(18 44)(19 39)(20 40)(21 41)(22 42)(23 37)(24 38)(25 35)(26 36)(27 31)(28 32)(29 33)(30 34)
(2 6)(3 5)(7 33 43 41)(8 32 44 40)(9 31 45 39)(10 36 46 38)(11 35 47 37)(12 34 48 42)(13 17)(14 16)(19 24)(20 23)(21 22)(25 28)(26 27)(29 30)

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

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

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

33 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F order 1 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 6 6 6 6 6 6 6 8 8 8 8 12 12 12 12 12 12 size 1 1 1 1 2 2 8 2 2 2 4 8 12 12 12 12 2 2 2 4 4 8 8 12 12 12 12 4 4 4 4 8 8

33 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + - + - + + + - image C1 C2 C2 C2 C4 C4 S3 D4 D4 Dic3 D6 Dic3 D8 SD16 C3⋊D4 C3⋊D4 C4≀C2 C23⋊C4 D4⋊S3 D4.S3 C23.7D6 Q8⋊3Dic3 kernel (C6×D4)⋊C4 C12.55D4 C6.C42 C3×C4⋊D4 C3×C4⋊C4 C6×D4 C4⋊D4 C2×C12 C22×C6 C4⋊C4 C22×C4 C2×D4 C2×C6 C2×C6 C2×C4 C23 C6 C6 C22 C22 C2 C2 # reps 1 1 1 1 2 2 1 1 1 1 1 1 2 2 2 2 4 1 1 1 2 2

Matrix representation of (C6×D4)⋊C4 in GL6(𝔽73)

 72 1 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 27 3 0 0 0 0 0 46 0 0 0 0 0 0 46 0 0 0 0 0 0 27
,
 43 60 0 0 0 0 13 30 0 0 0 0 0 0 22 25 0 0 0 0 42 51 0 0 0 0 0 0 0 46 0 0 0 0 27 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 55 72 0 0 0 0 0 0 1 0 0 0 0 0 0 27

G:=sub<GL(6,GF(73))| [72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,27,0,0,0,0,0,3,46,0,0,0,0,0,0,46,0,0,0,0,0,0,27],[43,13,0,0,0,0,60,30,0,0,0,0,0,0,22,42,0,0,0,0,25,51,0,0,0,0,0,0,0,27,0,0,0,0,46,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,55,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,27] >;

(C6×D4)⋊C4 in GAP, Magma, Sage, TeX

(C_6\times D_4)\rtimes C_4
% in TeX

G:=Group("(C6xD4):C4");
// GroupNames label

G:=SmallGroup(192,96);
// by ID

G=gap.SmallGroup(192,96);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,219,1571,570,136,6278]);
// Polycyclic

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

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