Copied to
clipboard

?

G = C24.52D6  order 192 = 26·3

41st non-split extension by C24 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.52D6, C6.902+ (1+4), (C2×C12)⋊14D4, D6⋊3D4⋊40C2, C12⋊3D4⋊29C2, (C2×D4).230D6, C12.252(C2×D4), (C22×D4)⋊14S3, C24⋊4S3⋊13C2, (C2×D12)⋊57C22, (C2×C6).300C24, C4⋊Dic3⋊78C22, (C22×C4).288D6, C6.147(C22×D4), C2.93(D4⋊6D6), C23.12D6⋊28C2, (C2×C12).545C23, C3⋊6(C22.29C24), (C4×Dic3)⋊42C22, (C2×Dic6)⋊68C22, (C6×D4).271C22, (C23×C6).79C22, C23.26D6⋊33C2, C6.D4⋊39C22, (C22×C6).234C23, C22.313(S3×C23), C23.146(C22×S3), (C22×S3).131C23, (C22×C12).277C22, (C2×Dic3).155C23, (D4×C2×C6)⋊7C2, (C2×C4)⋊6(C3⋊D4), (S3×C2×C4)⋊31C22, C4.97(C2×C3⋊D4), (C2×C4○D12)⋊29C2, (C2×C6).583(C2×D4), (C2×C3⋊D4)⋊28C22, C2.20(C22×C3⋊D4), C22.36(C2×C3⋊D4), (C2×C4).628(C22×S3), SmallGroup(192,1364)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C24.52D6
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C2×C4○D12 — C24.52D6
Lower central
C3 — C2×C6 — C24.52D6

Subgroups: 904 in 334 conjugacy classes, 111 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×4], C4 [×6], C22, C22 [×2], C22 [×28], S3 [×2], C6, C6 [×2], C6 [×6], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×22], Q8 [×2], C23, C23 [×4], C23 [×10], Dic3 [×6], C12 [×4], D6 [×6], C2×C6, C2×C6 [×2], C2×C6 [×22], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C22×C4, C22×C4 [×2], C2×D4 [×4], C2×D4 [×15], C2×Q8, C4○D4 [×4], C24 [×2], Dic6 [×2], C4×S3 [×4], D12 [×2], C2×Dic3 [×6], C3⋊D4 [×12], C2×C12 [×2], C2×C12 [×4], C3×D4 [×8], C22×S3 [×2], C22×C6, C22×C6 [×4], C22×C6 [×8], C42⋊C2, C22≀C2 [×4], C4⋊D4 [×4], C4.4D4 [×2], C4⋊1D4 [×2], C22×D4, C2×C4○D4, C4×Dic3 [×2], C4⋊Dic3 [×2], C6.D4 [×10], C2×Dic6, S3×C2×C4 [×2], C2×D12, C4○D12 [×4], C2×C3⋊D4 [×10], C22×C12, C6×D4 [×4], C6×D4 [×4], C23×C6 [×2], C22.29C24, C23.26D6, C23.12D6 [×2], D6⋊3D4 [×4], C12⋊3D4 [×2], C24⋊4S3 [×4], C2×C4○D12, D4×C2×C6, C24.52D6

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, 2+ (1+4) [×2], C2×C3⋊D4 [×6], S3×C23, C22.29C24, D4⋊6D6 [×2], C22×C3⋊D4, C24.52D6

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

Smallest permutation representation
►On 48 points
Generators in S48
(2 8)(4 10)(6 12)(13 38)(14 45)(15 40)(16 47)(17 42)(18 37)(19 44)(20 39)(21 46)(22 41)(23 48)(24 43)(25 31)(27 33)(29 35)

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

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

Matrix representation ►G ⊆ GL6(𝔽13)

1200000
010000
001000
0001200
0000120
000001
,
1200000
0120000
001000
000100
0000120
0000012
,
1200000
0120000
0012000
0001200
0000120
0000012
,
100000
010000
0012000
0001200
0000120
0000012
,
1000000
040000
0001200
001000
0000012
000010
,
040000
1000000
0000012
000010
0001200
001000

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1],[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],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[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],[10,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0],[0,10,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,1,0,0,0,0,12,0,0,0] >;
 

42 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K 3 4A4B4C4D4E···4J6A···6G6H···6O12A12B12C12D
order122222222222344444···46···66···612121212
size111122444412122222212···122···24···44444

42 irreducible representations

dim1111111122222244
type++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D6D6D6C3⋊D42+ (1+4)D4⋊6D6
kernelC24.52D6C23.26D6C23.12D6D6⋊3D4C12⋊3D4C24⋊4S3C2×C4○D12D4×C2×C6C22×D4C2×C12C22×C4C2×D4C24C2×C4C6C2
# reps1124241114142824

In GAP, Magma, Sage, TeX 

C_2^4._{52}D_6
 
% in TeX
 
G:=Group("C2^4.52D6");
 
// GroupNames label
 
G:=SmallGroup(192,1364);
 
// by ID
 
G=gap.SmallGroup(192,1364);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,675,570,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=f^2=d,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*c*d,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,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
○
≀