Copied to
clipboard

G = C62.56D4  order 288 = 25·32

40th non-split extension by C62 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.56D4, C62.103C23, C23.21S32, D6⋊Dic3⋊6C2, C62⋊5C4⋊5C2, (C22×C6).67D6, C6.63(C4○D12), (C2×Dic3).40D6, (C22×Dic3)⋊6S3, (C22×S3).24D6, C6.50(D4⋊2S3), C62.C22⋊24C2, (C2×C62).22C22, C2.24(D6.3D6), C22.6(D6⋊S3), C3⋊4(C23.28D6), C3⋊4(C23.23D6), (C6×Dic3).117C22, C32⋊10(C22.D4), (Dic3×C2×C6)⋊2C2, (C2×C3⋊D4).3S3, (C6×C3⋊D4).6C2, C6.82(C2×C3⋊D4), C22.133(C2×S32), (C3×C6).149(C2×D4), (S3×C2×C6).41C22, (C3×C6).78(C4○D4), C2.15(C2×D6⋊S3), (C2×C6).22(C3⋊D4), (C2×C6).122(C22×S3), (C2×C3⋊Dic3).64C22, SmallGroup(288,609)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — C62.56D4
Chief series
C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — D6⋊Dic3 — C62.56D4
Lower central
C32 — C62 — C62.56D4
Upper central
C1 — C22 — C23

Generators and relations for C62.56D4
 G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=a-1b3, dad=ab3, cbc-1=dbd=b-1, dcd=b3c-1 >

Subgroups: 570 in 173 conjugacy classes, 52 normal (26 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C22.D4, C3×Dic3, C3⋊Dic3, S3×C6, C62, C62, C62, Dic3⋊C4, D6⋊C4, C6.D4, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, C6×Dic3, C6×Dic3, C6×Dic3, C3×C3⋊D4, C2×C3⋊Dic3, S3×C2×C6, C2×C62, C23.28D6, C23.23D6, D6⋊Dic3, C62.C22, C62⋊5C4, Dic3×C2×C6, C6×C3⋊D4, C62.56D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C22.D4, S32, C4○D12, D4⋊2S3, C2×C3⋊D4, D6⋊S3, C2×S32, C23.28D6, C23.23D6, D6.3D6, C2×D6⋊S3, C62.56D4

Smallest permutation representation of C62.56D4
►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 5 2 6 3 4)(7 11 8 12 9 10)(13 17 15 16 14 18)(19 23 21 22 20 24)(25 26 27 28 29 30)(31 32 33 34 35 36)(37 42 41 40 39 38)(43 48 47 46 45 44)

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

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

48 conjugacy classes

class 1 2A2B2C2D2E2F3A3B3C4A4B4C4D4E4F4G6A···6J6K···6S6T6U12A···12H12I12J
order122222233344444446···66···66612···121212
size1111221222466661236362···24···412126···61212

48 irreducible representations

dim11111122222222244444
type+++++++++++++--+
imageC1C2C2C2C2C2S3S3D4D6D6D6C4○D4C3⋊D4C4○D12S32D4⋊2S3D6⋊S3C2×S32D6.3D6
kernelC62.56D4D6⋊Dic3C62.C22C62⋊5C4Dic3×C2×C6C6×C3⋊D4C22×Dic3C2×C3⋊D4C62C2×Dic3C22×S3C22×C6C3×C6C2×C6C6C23C6C22C22C2
# reps12211111231248812214

Matrix representation of C62.56D4 ►in GL8(𝔽13)

10000000
012000000
001200000
000120000
00001000
00000100
000000112
00000010
,
120000000
012000000
001210000
001200000
00001000
00000100
00000010
00000001
,
05000000
80000000
00010000
00100000
00003100
000031000
00000001
00000010
,
01000000
10000000
00010000
00100000
000010700
000010300
000000120
000000012

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,1,10,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,10,10,0,0,0,0,0,0,7,3,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12] >;
 

C62.56D4 in GAP, Magma, Sage, TeX 

C_6^2._{56}D_4
 
% in TeX
 
G:=Group("C6^2.56D4");
 
// GroupNames label
 
G:=SmallGroup(288,609);
 
// by ID
 
G=gap.SmallGroup(288,609);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,120,422,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^6=c^4=d^2=1,a*b=b*a,c*a*c^-1=a^-1*b^3,d*a*d=a*b^3,c*b*c^-1=d*b*d=b^-1,d*c*d=b^3*c^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁