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G = (C6×D4)⋊C4  order 192 = 26·3

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C6×D4)⋊1C4, (C2×C6).3D8, C6.16C4≀C2, C4⋊C4⋊1Dic3, (C2×D4)⋊1Dic3, C4⋊D4.1S3, (C2×C12).229D4, (C2×C6).10SD16, (C22×C6).44D4, (C22×C4).75D6, C6.20(C23⋊C4), C22.2(D4⋊S3), C3⋊3(C22.SD16), C6.23(D4⋊C4), C12.55D4⋊28C2, C6.C42⋊41C2, C2.3(D4⋊Dic3), C23.55(C3⋊D4), C22.2(D4.S3), C2.4(Q8⋊3Dic3), C2.5(C23.7D6), (C22×C12).371C22, C22.37(C6.D4), (C3×C4⋊C4)⋊1C4, (C2×C4).7(C2×Dic3), (C2×C12).167(C2×C4), (C3×C4⋊D4).10C2, (C2×C4).163(C3⋊D4), (C2×C6).95(C22⋊C4), SmallGroup(192,96)

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, Q8⋊3Dic3, (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 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G4H6A6B6C6D6E6F6G8A8B8C8D12A12B12C12D12E12F
order122222234444444466666668888121212121212
size11112282224812121212222448812121212444488

33 irreducible representations

dim1111112222222222244444
type+++++++-+-+++-
imageC1C2C2C2C4C4S3D4D4Dic3D6Dic3D8SD16C3⋊D4C3⋊D4C4≀C2C23⋊C4D4⋊S3D4.S3C23.7D6Q8⋊3Dic3
kernel(C6×D4)⋊C4C12.55D4C6.C42C3×C4⋊D4C3×C4⋊C4C6×D4C4⋊D4C2×C12C22×C6C4⋊C4C22×C4C2×D4C2×C6C2×C6C2×C4C23C6C6C22C22C2C2
# reps1111221111112222411122

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

7210000
7200000
0072000
0007200
000010
000001
,
7200000
0720000
0027300
0004600
0000460
0000027
,
43600000
13300000
00222500
00425100
0000046
0000270
,
010000
100000
001000
00557200
000010
0000027

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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