Copied to
clipboard

?

G = D12.39C23  order 192 = 26·3

20th non-split extension by D12 of C23 acting via C23/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C6.18C25, C12.53C24, D6.10C24, D12.39C23, 2- (1+4)⋊7S3, Dic6.40C23, Dic3.13C24, C4○D4⋊16D6, (C2×Q8)⋊28D6, D4○D12⋊12C2, (C2×C6).9C24, (S3×D4)⋊15C22, C4.50(S3×C23), C2.19(S3×C24), (S3×Q8)⋊18C22, (C6×Q8)⋊25C22, C3⋊D4.5C23, C4○D12⋊15C22, (C2×D12)⋊41C22, (C4×S3).22C23, C3⋊3(C2.C25), Q8.15D6⋊8C2, D4.33(C22×S3), (C3×D4).33C23, C22.6(S3×C23), (C3×Q8).34C23, Q8.44(C22×S3), D4⋊2S3⋊19C22, (C2×C12).124C23, Q8⋊3S3⋊17C22, (C3×2- (1+4))⋊5C2, (C22×S3).144C23, (C2×Dic3).300C23, (S3×C4○D4)⋊10C2, (S3×C2×C4)⋊38C22, (C2×Q8⋊3S3)⋊22C2, (C3×C4○D4)⋊13C22, (C2×C4).108(C22×S3), SmallGroup(192,1527)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — D12.39C23
Chief series
C1 — C3 — C6 — D6 — C22×S3 — S3×C2×C4 — S3×C4○D4 — D12.39C23
Lower central
C3 — C6 — D12.39C23

Subgroups: 1704 in 810 conjugacy classes, 443 normal (8 characteristic)
C1, C2, C2 [×15], C3, C4 [×10], C4 [×6], C22 [×5], C22 [×25], S3 [×10], C6, C6 [×5], C2×C4 [×15], C2×C4 [×45], D4 [×10], D4 [×50], Q8 [×10], Q8 [×10], C23 [×15], Dic3, Dic3 [×5], C12 [×10], D6 [×10], D6 [×15], C2×C6 [×5], C22×C4 [×15], C2×D4 [×45], C2×Q8 [×5], C2×Q8 [×10], C4○D4 [×10], C4○D4 [×70], Dic6 [×10], C4×S3 [×40], D12 [×30], C2×Dic3 [×5], C3⋊D4 [×20], C2×C12 [×15], C3×D4 [×10], C3×Q8 [×10], C22×S3 [×15], C2×C4○D4 [×15], 2+ (1+4) [×10], 2- (1+4), 2- (1+4) [×5], S3×C2×C4 [×15], C2×D12 [×15], C4○D12 [×30], S3×D4 [×30], D4⋊2S3 [×10], S3×Q8 [×10], Q8⋊3S3 [×30], C6×Q8 [×5], C3×C4○D4 [×10], C2.C25, C2×Q8⋊3S3 [×5], Q8.15D6 [×5], S3×C4○D4 [×10], D4○D12 [×10], C3×2- (1+4), D12.39C23

Quotients:
C1, C2 [×31], C22 [×155], S3, C23 [×155], D6 [×15], C24 [×31], C22×S3 [×35], C25, S3×C23 [×15], C2.C25, S3×C24, D12.39C23

Generators and relations
 G = < a,b,c,d,e | a12=b2=c2=d2=e2=1, bab=a-1, ac=ca, ad=da, eae=a5, cbc=a6b, bd=db, ebe=a10b, dcd=ece=a6c, de=ed >

Smallest permutation representation
►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 9)(2 8)(3 7)(4 6)(10 12)(14 24)(15 23)(16 22)(17 21)(18 20)(25 29)(26 28)(30 36)(31 35)(32 34)(37 41)(38 40)(42 48)(43 47)(44 46)

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

(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)

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

Matrix representation ►G ⊆ GL6(𝔽13)

12120000
100000
001500
00101200
000015
00001012
,
12120000
010000
001500
0001200
0000128
000001
,
100000
010000
000010
000001
001000
000100
,
1200000
0120000
001000
000100
0000120
0000012
,
100000
12120000
008100
002500
0000512
0000118

G:=sub<GL(6,GF(13))| [12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,10,0,0,0,0,5,12,0,0,0,0,0,0,1,10,0,0,0,0,5,12],[12,0,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,5,12,0,0,0,0,0,0,12,0,0,0,0,0,8,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,12,0,0,0,0,0,12,0,0,0,0,0,0,8,2,0,0,0,0,1,5,0,0,0,0,0,0,5,11,0,0,0,0,12,8] >;
 

51 conjugacy classes

class 1 2A2B···2F2G···2P 3 4A···4J4K4L4M···4Q6A6B···6F12A···12J
order122···22···234···4444···466···612···12
size112···26···622···2336···624···44···4

51 irreducible representations

dim11111122248
type++++++++++
imageC1C2C2C2C2C2S3D6D6C2.C25D12.39C23
kernelD12.39C23C2×Q8⋊3S3Q8.15D6S3×C4○D4D4○D12C3×2- (1+4)2- (1+4)C2×Q8C4○D4C3C1
# reps15510101151021

In GAP, Magma, Sage, TeX 

D_{12}._{39}C_2^3
 
% in TeX
 
G:=Group("D12.39C2^3");
 
// GroupNames label
 
G:=SmallGroup(192,1527);
 
// by ID
 
G=gap.SmallGroup(192,1527);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,570,1684,438,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^12=b^2=c^2=d^2=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e=a^5,c*b*c=a^6*b,b*d=d*b,e*b*e=a^10*b,d*c*d=e*c*e=a^6*c,d*e=e*d>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀