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## G = (C6×Q8)⋊C4order 192 = 26·3

### 1st semidirect product of C6×Q8 and C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — (C6×Q8)⋊C4
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×C12 — C6.C42 — (C6×Q8)⋊C4
 Lower central C3 — C2×C6 — C2×C12 — (C6×Q8)⋊C4
 Upper central C1 — C22 — C22×C4 — C22⋊Q8

Generators and relations for (C6×Q8)⋊C4
G = < a,b,c,d | a6=b4=d4=1, c2=b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=b-1, dbd-1=a3b, dcd-1=b-1c >

Subgroups: 224 in 80 conjugacy classes, 31 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C23, Dic3, C12, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×C6, C2.C42, C22⋊C8, C22⋊Q8, C2×C3⋊C8, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C22×Dic3, C22×C12, C6×Q8, C23.31D4, C12.55D4, C6.C42, C3×C22⋊Q8, (C6×Q8)⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, SD16, Q16, C2×Dic3, C3⋊D4, C23⋊C4, Q8⋊C4, C4≀C2, Q82S3, C3⋊Q16, C6.D4, C23.31D4, C23.7D6, Q82Dic3, Q83Dic3, (C6×Q8)⋊C4

Smallest permutation representation of (C6×Q8)⋊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 28 15 23)(2 29 16 24)(3 30 17 19)(4 25 18 20)(5 26 13 21)(6 27 14 22)(7 42 46 31)(8 37 47 32)(9 38 48 33)(10 39 43 34)(11 40 44 35)(12 41 45 36)
(1 40 15 35)(2 41 16 36)(3 42 17 31)(4 37 18 32)(5 38 13 33)(6 39 14 34)(7 19 46 30)(8 20 47 25)(9 21 48 26)(10 22 43 27)(11 23 44 28)(12 24 45 29)
(1 15)(2 14)(3 13)(4 18)(5 17)(6 16)(7 38 43 36)(8 37 44 35)(9 42 45 34)(10 41 46 33)(11 40 47 32)(12 39 48 31)(19 29)(20 28)(21 27)(22 26)(23 25)(24 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), (1,28,15,23)(2,29,16,24)(3,30,17,19)(4,25,18,20)(5,26,13,21)(6,27,14,22)(7,42,46,31)(8,37,47,32)(9,38,48,33)(10,39,43,34)(11,40,44,35)(12,41,45,36), (1,40,15,35)(2,41,16,36)(3,42,17,31)(4,37,18,32)(5,38,13,33)(6,39,14,34)(7,19,46,30)(8,20,47,25)(9,21,48,26)(10,22,43,27)(11,23,44,28)(12,24,45,29), (1,15)(2,14)(3,13)(4,18)(5,17)(6,16)(7,38,43,36)(8,37,44,35)(9,42,45,34)(10,41,46,33)(11,40,47,32)(12,39,48,31)(19,29)(20,28)(21,27)(22,26)(23,25)(24,30)>;

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

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

33 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 6A 6B 6C 6D 6E 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G 12H order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 6 6 6 6 6 8 8 8 8 12 12 12 12 12 12 12 12 size 1 1 1 1 2 2 2 2 2 4 8 8 12 12 12 12 2 2 2 4 4 12 12 12 12 4 4 4 4 8 8 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 SD16 Q16 C3⋊D4 C3⋊D4 C4≀C2 C23⋊C4 Q8⋊2S3 C3⋊Q16 C23.7D6 Q8⋊3Dic3 kernel (C6×Q8)⋊C4 C12.55D4 C6.C42 C3×C22⋊Q8 C3×C4⋊C4 C6×Q8 C22⋊Q8 C2×C12 C22×C6 C4⋊C4 C22×C4 C2×Q8 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×Q8)⋊C4 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 72 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 46 0 0 0 0 0 0 27 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 46 2 0 0 0 0 0 27
,
 0 72 0 0 0 0 1 0 0 0 0 0 0 0 30 60 0 0 0 0 13 43 0 0 0 0 0 0 10 61 0 0 0 0 51 63
,
 72 0 0 0 0 0 0 27 0 0 0 0 0 0 1 0 0 0 0 0 72 72 0 0 0 0 0 0 72 0 0 0 0 0 46 1

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[46,0,0,0,0,0,0,27,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,0,0,0,0,0,2,27],[0,1,0,0,0,0,72,0,0,0,0,0,0,0,30,13,0,0,0,0,60,43,0,0,0,0,0,0,10,51,0,0,0,0,61,63],[72,0,0,0,0,0,0,27,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,72,46,0,0,0,0,0,1] >;

(C6×Q8)⋊C4 in GAP, Magma, Sage, TeX

(C_6\times Q_8)\rtimes C_4
% in TeX

G:=Group("(C6xQ8):C4");
// GroupNames label

G:=SmallGroup(192,97);
// by ID

G=gap.SmallGroup(192,97);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,232,219,1571,570,136,6278]);
// Polycyclic

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

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