Copied to
clipboard

G = C24.52D6order 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, D63D440C2, C123D429C2, (C2×D4).230D6, C12.252(C2×D4), (C22×D4)⋊14S3, C244S313C2, (C2×D12)⋊57C22, (C2×C6).300C24, C4⋊Dic378C22, C6.147(C22×D4), (C22×C4).288D6, C23.12D628C2, C2.93(D46D6), (C2×C12).545C23, C36(C22.29C24), (C4×Dic3)⋊42C22, (C2×Dic6)⋊68C22, (C6×D4).271C22, (C23×C6).79C22, C6.D439C22, C23.26D633C2, C23.146(C22×S3), C22.313(S3×C23), (C22×C6).234C23, (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, C22.36(C2×C3⋊D4), C2.20(C22×C3⋊D4), (C2×C4).628(C22×S3), SmallGroup(192,1364)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C24.52D6
C1C3C6C2×C6C22×S3S3×C2×C4C2×C4○D12 — C24.52D6
C3C2×C6 — C24.52D6
C1C22C22×D4

Generators and relations for C24.52D6
 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 >

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], C41D4 [×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], D63D4 [×4], C123D4 [×2], C244S3 [×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, D46D6 [×2], C22×C3⋊D4, C24.52D6

Smallest permutation representation of C24.52D6
On 48 points
Generators in S48
(2 8)(4 10)(6 12)(13 29)(14 36)(15 31)(16 26)(17 33)(18 28)(19 35)(20 30)(21 25)(22 32)(23 27)(24 34)(37 43)(39 45)(41 47)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 37)(9 38)(10 39)(11 40)(12 41)(13 35)(14 36)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 33)(24 34)
(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 33 7 27)(2 26 8 32)(3 31 9 25)(4 36 10 30)(5 29 11 35)(6 34 12 28)(13 46 19 40)(14 39 20 45)(15 44 21 38)(16 37 22 43)(17 42 23 48)(18 47 24 41)

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

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

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

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+4D46D6
kernelC24.52D6C23.26D6C23.12D6D63D4C123D4C244S3C2×C4○D12D4×C2×C6C22×D4C2×C12C22×C4C2×D4C24C2×C4C6C2
# reps1124241114142824

Matrix representation of C24.52D6 in 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] >;

C24.52D6 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

׿
×
𝔽