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## G = C2×D6⋊D4order 192 = 26·3

### Direct product of C2 and D6⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C2×D6⋊D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C24 — C2×D6⋊D4
 Lower central C3 — C2×C6 — C2×D6⋊D4
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C2×D6⋊D4
G = < a,b,c,d,e | a2=b6=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b3c, ece=bc, ede=d-1 >

Subgroups: 2088 in 662 conjugacy classes, 143 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C22×C6, C22×C6, C2×C22⋊C4, C2×C22⋊C4, C22≀C2, C22×D4, C25, D6⋊C4, C3×C22⋊C4, C2×D12, C2×D12, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, S3×C23, S3×C23, S3×C23, C23×C6, C2×C22≀C2, D6⋊D4, C2×D6⋊C4, C6×C22⋊C4, C22×D12, C22×C3⋊D4, S3×C24, C2×D6⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, D12, C22×S3, C22≀C2, C22×D4, C2×D12, S3×D4, S3×C23, C2×C22≀C2, D6⋊D4, C22×D12, C2×S3×D4, C2×D6⋊D4

Smallest permutation representation of C2×D6⋊D4
On 48 points
Generators in S48
(1 40)(2 41)(3 42)(4 37)(5 38)(6 39)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(13 33)(14 34)(15 35)(16 36)(17 31)(18 32)(25 47)(26 48)(27 43)(28 44)(29 45)(30 46)
(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 17)(2 16)(3 15)(4 14)(5 13)(6 18)(7 47)(8 46)(9 45)(10 44)(11 43)(12 48)(19 25)(20 30)(21 29)(22 28)(23 27)(24 26)(31 40)(32 39)(33 38)(34 37)(35 42)(36 41)
(1 44 18 8)(2 45 13 9)(3 46 14 10)(4 47 15 11)(5 48 16 12)(6 43 17 7)(19 39 27 31)(20 40 28 32)(21 41 29 33)(22 42 30 34)(23 37 25 35)(24 38 26 36)
(1 25)(2 30)(3 29)(4 28)(5 27)(6 26)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 22)(14 21)(15 20)(16 19)(17 24)(18 23)(37 44)(38 43)(39 48)(40 47)(41 46)(42 45)

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

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

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

48 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L ··· 2S 2T 2U 3 4A 4B 4C 4D 4E 4F 6A ··· 6G 6H 6I 6J 6K 12A ··· 12H order 1 2 ··· 2 2 2 2 2 2 ··· 2 2 2 3 4 4 4 4 4 4 6 ··· 6 6 6 6 6 12 ··· 12 size 1 1 ··· 1 2 2 2 2 6 ··· 6 12 12 2 4 4 4 4 12 12 2 ··· 2 4 4 4 4 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 D12 S3×D4 kernel C2×D6⋊D4 D6⋊D4 C2×D6⋊C4 C6×C22⋊C4 C22×D12 C22×C3⋊D4 S3×C24 C2×C22⋊C4 C22×S3 C22×C6 C22⋊C4 C22×C4 C24 C23 C22 # reps 1 8 2 1 2 1 1 1 8 4 4 2 1 8 4

Matrix representation of C2×D6⋊D4 in GL5(𝔽13)

 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 12 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 3 6 0 0 0 7 10 0 0 0 0 0 0 12 0 0 0 1 0
,
 1 0 0 0 0 0 3 6 0 0 0 3 10 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,3,7,0,0,0,6,10,0,0,0,0,0,0,1,0,0,0,12,0],[1,0,0,0,0,0,3,3,0,0,0,6,10,0,0,0,0,0,0,1,0,0,0,1,0] >;

C2×D6⋊D4 in GAP, Magma, Sage, TeX

C_2\times D_6\rtimes D_4
% in TeX

G:=Group("C2xD6:D4");
// GroupNames label

G:=SmallGroup(192,1046);
// by ID

G=gap.SmallGroup(192,1046);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,675,297,80,6278]);
// Polycyclic

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

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