Copied to
clipboard

?

G = D12.32C23  order 192 = 26·3

13rd non-split extension by D12 of C23 acting via C23/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12.37C24, D12.32C23, 2+ (1+4)⋊8S3, Dic6.32C23, C4○D4⋊8D6, C3⋊5(D4○D8), D4○D12⋊9C2, (C2×D4)⋊16D6, (C3×D4).36D4, C3⋊C8.16C23, (C3×Q8).36D4, D4⋊D6⋊11C2, D4⋊S3⋊20C22, Q8.13D6⋊8C2, C12.269(C2×D4), (C6×D4)⋊24C22, C4.37(S3×C23), D12⋊6C22⋊11C2, C4○D12⋊10C22, D4.18(C3⋊D4), (C2×D12)⋊39C22, D4.Dic3⋊10C2, D4.S3⋊19C22, Q8.25(C3⋊D4), (C3×D4).25C23, C3⋊Q16⋊21C22, D4.25(C22×S3), C6.171(C22×D4), (C3×Q8).25C23, Q8.35(C22×S3), (C2×C12).118C23, Q8⋊2S3⋊19C22, C4.Dic3⋊16C22, (C3×2+ (1+4))⋊2C2, (C2×D4⋊S3)⋊32C2, (C2×C3⋊C8)⋊24C22, (C2×C6).85(C2×D4), C4.75(C2×C3⋊D4), (C3×C4○D4)⋊8C22, C22.6(C2×C3⋊D4), C2.44(C22×C3⋊D4), (C2×C4).102(C22×S3), SmallGroup(192,1394)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C12 — D12.32C23
Chief series
C1 — C3 — C6 — C12 — D12 — C2×D12 — D4○D12 — D12.32C23
Lower central
C3 — C6 — C12 — D12.32C23

Subgroups: 728 in 268 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2 [×9], C3, C4, C4 [×3], C4 [×2], C22 [×3], C22 [×12], S3 [×3], C6, C6 [×6], C8 [×4], C2×C4 [×3], C2×C4 [×6], D4 [×6], D4 [×15], Q8 [×2], Q8, C23 [×6], Dic3, C12, C12 [×3], C12, D6 [×6], C2×C6 [×3], C2×C6 [×6], C2×C8 [×3], M4(2) [×3], D8 [×9], SD16 [×6], Q16, C2×D4 [×3], C2×D4 [×9], C4○D4, C4○D4 [×3], C4○D4 [×5], C3⋊C8, C3⋊C8 [×3], Dic6, C4×S3 [×3], D12 [×3], D12 [×3], C3⋊D4 [×3], C2×C12 [×3], C2×C12 [×3], C3×D4 [×6], C3×D4 [×6], C3×Q8 [×2], C22×S3 [×3], C22×C6 [×3], C8○D4, C2×D8 [×3], C4○D8 [×3], C8⋊C22 [×6], 2+ (1+4), 2+ (1+4), C2×C3⋊C8 [×3], C4.Dic3 [×3], D4⋊S3 [×9], D4.S3 [×3], Q8⋊2S3 [×3], C3⋊Q16, C2×D12 [×3], C4○D12 [×3], S3×D4 [×3], Q8⋊3S3, C6×D4 [×3], C6×D4 [×3], C3×C4○D4, C3×C4○D4 [×3], C3×C4○D4, D4○D8, C2×D4⋊S3 [×3], D12⋊6C22 [×3], D4.Dic3, D4⋊D6 [×3], Q8.13D6 [×3], D4○D12, C3×2+ (1+4), D12.32C23

Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C3⋊D4 [×4], C22×S3 [×7], C22×D4, C2×C3⋊D4 [×6], S3×C23, D4○D8, C22×C3⋊D4, D12.32C23

Generators and relations
 G = < a,b,c,d,e | a12=b2=c2=d2=e2=1, bab=dad=a-1, ac=ca, eae=a7, cbc=a6b, dbd=a10b, ebe=a3b, cd=dc, ce=ec, ede=a9d >

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

(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)

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

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

Matrix representation ►G ⊆ GL6(𝔽73)

72720000
100000
0007200
001000
00727212
00107272
,
72720000
010000
000010
00727212
001000
0072011
,
7200000
0720000
001000
000100
0000720
0011072
,
7200000
110000
000100
001000
00117271
000001
,
7200000
0720000
00575700
00571600
005757032
00570160

G:=sub<GL(6,GF(73))| [72,1,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,1,0,0,72,0,72,0,0,0,0,0,1,72,0,0,0,0,2,72],[72,0,0,0,0,0,72,1,0,0,0,0,0,0,0,72,1,72,0,0,0,72,0,0,0,0,1,1,0,1,0,0,0,2,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,1,0,0,0,1,0,1,0,0,0,0,72,0,0,0,0,0,0,72],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,1,0,1,0,0,0,0,0,72,0,0,0,0,0,71,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,57,57,57,57,0,0,57,16,57,0,0,0,0,0,0,16,0,0,0,0,32,0] >;
 

39 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J 3 4A4B4C4D4E4F6A6B···6J8A8B8C8D8E12A···12F
order12222222222344444466···68888812···12
size112224441212122222241224···4661212124···4

39 irreducible representations

dim11111111222222248
type+++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D6D6C3⋊D4C3⋊D4D4○D8D12.32C23
kernelD12.32C23C2×D4⋊S3D12⋊6C22D4.Dic3D4⋊D6Q8.13D6D4○D12C3×2+ (1+4)2+ (1+4)C3×D4C3×Q8C2×D4C4○D4D4Q8C3C1
# reps13313311131346221

In GAP, Magma, Sage, TeX 

D_{12}._{32}C_2^3
 
% in TeX
 
G:=Group("D12.32C2^3");
 
// GroupNames label
 
G:=SmallGroup(192,1394);
 
// by ID
 
G=gap.SmallGroup(192,1394);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,675,1684,235,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^12=b^2=c^2=d^2=e^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,e*a*e=a^7,c*b*c=a^6*b,d*b*d=a^10*b,e*b*e=a^3*b,c*d=d*c,c*e=e*c,e*d*e=a^9*d>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀