Copied to
clipboard

?

G = C24.41D6  order 192 = 26·3

30th non-split extension by C24 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.41D6, C6.52+ (1+4), Dic3⋊D4⋊2C2, C12⋊7D4⋊4C2, D6⋊C4⋊2C22, C24⋊4S3⋊3C2, C23.9D6⋊2C2, (C2×D12)⋊4C22, (C2×C6).38C24, C4⋊Dic3⋊6C22, C22⋊C4.87D6, (C22×C4).61D6, C12.48D4⋊4C2, C2.9(D4⋊6D6), Dic3⋊C4⋊2C22, (C2×Dic6)⋊3C22, C23.8D6⋊1C2, (C2×C12).131C23, C23.11D6⋊2C2, C3⋊1(C22.32C24), (C4×Dic3)⋊48C22, C22.77(S3×C23), C23.92(C22×S3), (C23×C6).64C22, C22.23(C4○D12), (C22×S3).10C23, (C22×C6).128C23, (C2×Dic3).11C23, C6.D4.2C22, (C22×C12).355C22, (C4×C3⋊D4)⋊34C2, (S3×C2×C4)⋊41C22, C6.16(C2×C4○D4), (C6×C22⋊C4)⋊20C2, (C2×C22⋊C4)⋊17S3, C2.18(C2×C4○D12), (C2×C3⋊D4).7C22, (C2×C6).104(C4○D4), (C2×C4).261(C22×S3), (C3×C22⋊C4).109C22, SmallGroup(192,1053)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C24.41D6
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — C2×C3⋊D4 — C4×C3⋊D4 — C24.41D6
Lower central
C3 — C2×C6 — C24.41D6

Subgroups: 680 in 250 conjugacy classes, 95 normal (31 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×10], C22, C22 [×2], C22 [×18], S3 [×2], C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×2], C2×C4 [×10], D4 [×9], Q8, C23, C23 [×2], C23 [×6], Dic3 [×6], C12 [×4], D6 [×6], C2×C6, C2×C6 [×2], C2×C6 [×12], C42 [×2], C22⋊C4 [×4], C22⋊C4 [×10], C4⋊C4 [×6], C22×C4 [×2], C22×C4 [×2], C2×D4 [×7], C2×Q8, C24, Dic6, C4×S3 [×2], D12, C2×Dic3 [×6], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×2], C2×C12 [×2], C22×S3 [×2], C22×C6, C22×C6 [×2], C22×C6 [×4], C2×C22⋊C4, C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4 [×2], C42⋊2C2 [×2], C4×Dic3 [×2], Dic3⋊C4 [×4], C4⋊Dic3 [×2], D6⋊C4 [×4], C6.D4 [×6], C3×C22⋊C4 [×4], C2×Dic6, S3×C2×C4 [×2], C2×D12, C2×C3⋊D4 [×6], C22×C12 [×2], C23×C6, C22.32C24, C23.8D6 [×2], C23.9D6 [×2], Dic3⋊D4 [×2], C23.11D6 [×2], C12.48D4, C4×C3⋊D4 [×2], C12⋊7D4, C24⋊4S3 [×2], C6×C22⋊C4, C24.41D6

Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×2], C24, C22×S3 [×7], C2×C4○D4, 2+ (1+4) [×2], C4○D12 [×2], S3×C23, C22.32C24, C2×C4○D12, D4⋊6D6 [×2], C24.41D6

Generators and relations
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=f2=c, ab=ba, ac=ca, faf-1=ad=da, ae=ea, bc=cb, ebe-1=bd=db, fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >

Smallest permutation representation
►On 48 points
Generators in S48
(1 34)(2 35)(3 36)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 33)

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

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

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

Matrix representation ►G ⊆ GL6(𝔽13)

1200000
0120000
0012000
0001200
000010
000001
,
1200000
010000
001000
0001200
000010
0000012
,
1200000
0120000
001000
000100
000010
000001
,
100000
010000
0012000
0001200
0000120
0000012
,
700000
0110000
000100
001000
000001
000010
,
020000
600000
000001
000010
000100
001000

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[7,0,0,0,0,0,0,11,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,6,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0] >;
 

42 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E4F4G···4L6A···6G6H6I6J6K12A···12H
order122222222234444444···46···6666612···12
size111122441212222224412···122···244444···4

42 irreducible representations

dim111111111122222244
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2S3D6D6D6C4○D4C4○D122+ (1+4)D4⋊6D6
kernelC24.41D6C23.8D6C23.9D6Dic3⋊D4C23.11D6C12.48D4C4×C3⋊D4C12⋊7D4C24⋊4S3C6×C22⋊C4C2×C22⋊C4C22⋊C4C22×C4C24C2×C6C22C6C2
# reps122221212114214824

In GAP, Magma, Sage, TeX 

C_2^4._{41}D_6
 
% in TeX
 
G:=Group("C2^4.41D6");
 
// GroupNames label
 
G:=SmallGroup(192,1053);
 
// by ID
 
G=gap.SmallGroup(192,1053);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,100,675,297,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=f^2=c,a*b=b*a,a*c=c*a,f*a*f^-1=a*d=d*a,a*e=e*a,b*c=c*b,e*b*e^-1=b*d=d*b,f*b*f^-1=b*c*d,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^5>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀