Copied to
clipboard

## G = C2×D12⋊6C22order 192 = 26·3

### Direct product of C2 and D12⋊6C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C2×D12⋊6C22
 Chief series C1 — C3 — C6 — C12 — D12 — C2×D12 — C2×C4○D12 — C2×D12⋊6C22
 Lower central C3 — C6 — C12 — C2×D12⋊6C22
 Upper central C1 — C22 — C22×C4 — C22×D4

Generators and relations for C2×D126C22
G = < a,b,c,d,e | a2=b12=c2=d2=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe=b7, dcd=b6c, ece=b3c, de=ed >

Subgroups: 744 in 298 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×22], S3 [×2], C6, C6 [×2], C6 [×6], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×5], D4 [×4], D4 [×13], Q8 [×3], C23, C23 [×11], Dic3 [×2], C12 [×2], C12 [×2], D6 [×4], C2×C6, C2×C6 [×2], C2×C6 [×18], C2×C8 [×2], M4(2) [×4], D8 [×8], SD16 [×8], C22×C4, C22×C4, C2×D4 [×6], C2×D4 [×5], C2×Q8, C4○D4 [×6], C24, C3⋊C8 [×4], Dic6 [×2], Dic6, C4×S3 [×4], D12 [×2], D12, C2×Dic3, C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×4], C3×D4 [×4], C3×D4 [×6], C22×S3, C22×C6, C22×C6 [×10], C2×M4(2), C2×D8 [×2], C2×SD16 [×2], C8⋊C22 [×8], C22×D4, C2×C4○D4, C2×C3⋊C8 [×2], C4.Dic3 [×4], D4⋊S3 [×8], D4.S3 [×8], C2×Dic6, S3×C2×C4, C2×D12, C4○D12 [×4], C4○D12 [×2], C2×C3⋊D4, C22×C12, C6×D4 [×6], C6×D4 [×3], C23×C6, C2×C8⋊C22, C2×C4.Dic3, C2×D4⋊S3 [×2], D126C22 [×8], C2×D4.S3 [×2], C2×C4○D12, D4×C2×C6, C2×D126C22
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C3⋊D4 [×4], C22×S3 [×7], C8⋊C22 [×2], C22×D4, C2×C3⋊D4 [×6], S3×C23, C2×C8⋊C22, D126C22 [×2], C22×C3⋊D4, C2×D126C22

Smallest permutation representation of C2×D126C22
On 48 points
Generators in S48
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 13)(12 14)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 37)(35 38)(36 39)
(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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 47)(14 46)(15 45)(16 44)(17 43)(18 42)(19 41)(20 40)(21 39)(22 38)(23 37)(24 48)
(1 21)(2 22)(3 23)(4 24)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 37)(35 38)(36 39)
(1 15)(2 22)(3 17)(4 24)(5 19)(6 14)(7 21)(8 16)(9 23)(10 18)(11 13)(12 20)(25 43)(26 38)(27 45)(28 40)(29 47)(30 42)(31 37)(32 44)(33 39)(34 46)(35 41)(36 48)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 4E 4F 6A ··· 6G 6H ··· 6O 8A 8B 8C 8D 12A 12B 12C 12D order 1 2 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 6 ··· 6 6 ··· 6 8 8 8 8 12 12 12 12 size 1 1 1 1 2 2 4 4 4 4 12 12 2 2 2 2 2 12 12 2 ··· 2 4 ··· 4 12 12 12 12 4 4 4 4

42 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 C3⋊D4 C3⋊D4 C8⋊C22 D12⋊6C22 kernel C2×D12⋊6C22 C2×C4.Dic3 C2×D4⋊S3 D12⋊6C22 C2×D4.S3 C2×C4○D12 D4×C2×C6 C22×D4 C2×C12 C22×C6 C22×C4 C2×D4 C2×C4 C23 C6 C2 # reps 1 1 2 8 2 1 1 1 3 1 1 6 6 2 2 4

Matrix representation of C2×D126C22 in GL6(𝔽73)

 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 72 0 0 0 0 0 0 72
,
 8 0 0 0 0 0 29 64 0 0 0 0 0 0 8 60 0 0 0 0 66 65 0 0 0 0 0 0 9 31 0 0 0 0 56 64
,
 52 40 0 0 0 0 62 21 0 0 0 0 0 0 0 0 9 31 0 0 0 0 56 64 0 0 8 60 0 0 0 0 66 65 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 29 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 10 1

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[8,29,0,0,0,0,0,64,0,0,0,0,0,0,8,66,0,0,0,0,60,65,0,0,0,0,0,0,9,56,0,0,0,0,31,64],[52,62,0,0,0,0,40,21,0,0,0,0,0,0,0,0,8,66,0,0,0,0,60,65,0,0,9,56,0,0,0,0,31,64,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,29,1,0,0,0,0,0,0,72,10,0,0,0,0,0,1] >;

C2×D126C22 in GAP, Magma, Sage, TeX

C_2\times D_{12}\rtimes_6C_2^2
% in TeX

G:=Group("C2xD12:6C2^2");
// GroupNames label

G:=SmallGroup(192,1352);
// by ID

G=gap.SmallGroup(192,1352);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,675,297,1684,235,102,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^12=c^2=d^2=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e=b^7,d*c*d=b^6*c,e*c*e=b^3*c,d*e=e*d>;
// generators/relations

׿
×
𝔽