Copied to
clipboard

G = C23.53D12order 192 = 26·3

19th non-split extension by C23 of D12 acting via D12/D6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.53D12, (C2×C8)⋊21D6, (C2×D12)⋊14C4, C4.13(D6⋊C4), (C2×C24)⋊36C22, D12.24(C2×C4), C2.D2439C2, (C2×C12).173D4, (C2×C4).153D12, C12.417(C2×D4), C2.4(C8⋊D6), C6.20(C8⋊C22), (C6×M4(2))⋊19C2, (C2×M4(2))⋊11S3, C4⋊Dic348C22, (C22×C4).155D6, C22.57(C2×D12), (C22×C6).101D4, C12.27(C22⋊C4), (C2×C12).773C23, C12.115(C22×C4), C22.28(D6⋊C4), (C22×D12).15C2, C33(C23.37D4), (C2×D12).200C22, C23.26D616C2, (C22×C12).188C22, C4.73(S3×C2×C4), (C2×C4).53(C4×S3), C2.30(C2×D6⋊C4), (C2×C6).163(C2×D4), C4.110(C2×C3⋊D4), C6.58(C2×C22⋊C4), (C2×C12).108(C2×C4), (C2×C4).76(C3⋊D4), (C2×C6).21(C22⋊C4), (C2×C4).722(C22×S3), SmallGroup(192,690)

Series: Derived Chief Lower central Upper central

C1C12 — C23.53D12
C1C3C6C12C2×C12C2×D12C22×D12 — C23.53D12
C3C6C12 — C23.53D12
C1C22C22×C4C2×M4(2)

Generators and relations for C23.53D12
 G = < a,b,c,d,e | a2=b2=c2=1, d12=c, e2=cb=bc, ab=ba, dad-1=eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd11 >

Subgroups: 696 in 190 conjugacy classes, 63 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×18], S3 [×4], C6, C6 [×2], C6 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×2], D4 [×10], C23, C23 [×10], Dic3 [×2], C12 [×2], C12 [×2], D6 [×16], C2×C6, C2×C6 [×2], C2×C6 [×2], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], M4(2) [×2], C22×C4, C2×D4 [×9], C24, C24 [×2], D12 [×4], D12 [×6], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×4], C22×S3 [×10], C22×C6, D4⋊C4 [×4], C42⋊C2, C2×M4(2), C22×D4, C4×Dic3, C4⋊Dic3 [×2], C6.D4, C2×C24 [×2], C3×M4(2) [×2], C2×D12 [×6], C2×D12 [×3], C22×C12, S3×C23, C23.37D4, C2.D24 [×4], C23.26D6, C6×M4(2), C22×D12, C23.53D12
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C22×S3, C2×C22⋊C4, C8⋊C22 [×2], D6⋊C4 [×4], S3×C2×C4, C2×D12, C2×C3⋊D4, C23.37D4, C8⋊D6 [×2], C2×D6⋊C4, C23.53D12

Smallest permutation representation of C23.53D12
On 48 points
Generators in S48
(2 14)(4 16)(6 18)(8 20)(10 22)(12 24)(26 38)(28 40)(30 42)(32 44)(34 46)(36 48)
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 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 44 45 24)(2 23 46 43)(3 42 47 22)(4 21 48 41)(5 40 25 20)(6 19 26 39)(7 38 27 18)(8 17 28 37)(9 36 29 16)(10 15 30 35)(11 34 31 14)(12 13 32 33)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E4F4G4H6A6B6C6D6E8A8B8C8D12A12B12C12D12E12F24A···24H
order122222222234444444466666888812121212121224···24
size1111221212121222222121212122224444442222444···4

42 irreducible representations

dim11111122222222244
type++++++++++++++
imageC1C2C2C2C2C4S3D4D4D6D6C4×S3D12C3⋊D4D12C8⋊C22C8⋊D6
kernelC23.53D12C2.D24C23.26D6C6×M4(2)C22×D12C2×D12C2×M4(2)C2×C12C22×C6C2×C8C22×C4C2×C4C2×C4C2×C4C23C6C2
# reps14111813121424224

Matrix representation of C23.53D12 in GL6(𝔽73)

7200000
0720000
001000
000100
004726720
002147072
,
7200000
0720000
0072000
0007200
0000720
0000072
,
100000
010000
0072000
0007200
0000720
0000072
,
52700000
25210000
0047472523
002602525
001972026
0020194726
,
2130000
23520000
0026264850
000475048
0064552647
005564047

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,47,21,0,0,0,1,26,47,0,0,0,0,72,0,0,0,0,0,0,72],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[52,25,0,0,0,0,70,21,0,0,0,0,0,0,47,26,19,20,0,0,47,0,72,19,0,0,25,25,0,47,0,0,23,25,26,26],[21,23,0,0,0,0,3,52,0,0,0,0,0,0,26,0,64,55,0,0,26,47,55,64,0,0,48,50,26,0,0,0,50,48,47,47] >;

C23.53D12 in GAP, Magma, Sage, TeX

C_2^3._{53}D_{12}
% in TeX

G:=Group("C2^3.53D12");
// GroupNames label

G:=SmallGroup(192,690);
// by ID

G=gap.SmallGroup(192,690);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,387,142,1123,136,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^12=c,e^2=c*b=b*c,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^11>;
// generators/relations

׿
×
𝔽