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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, (C22xC6).67D6, C6.63(C4oD12), (C2xDic3).40D6, (C22xDic3):6S3, (C22xS3).24D6, C6.50(D4:2S3), C62.C22:24C2, (C2xC62).22C22, C2.24(D6.3D6), C22.6(D6:S3), C3:4(C23.28D6), C3:4(C23.23D6), (C6xDic3).117C22, C32:10(C22.D4), (Dic3xC2xC6):2C2, (C2xC3:D4).3S3, (C6xC3:D4).6C2, C6.82(C2xC3:D4), C22.133(C2xS32), (C3xC6).149(C2xD4), (S3xC2xC6).41C22, (C3xC6).78(C4oD4), C2.15(C2xD6:S3), (C2xC6).22(C3:D4), (C2xC6).122(C22xS3), (C2xC3:Dic3).64C22, SmallGroup(288,609)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — C62.56D4
Chief series
C1 — C3 — C32 — C3xC6 — C62 — S3xC2xC6 — 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, C2xC4, D4, C23, C23, C32, Dic3, C12, D6, C2xC6, C2xC6, C2xC6, C22:C4, C4:C4, C22xC4, C2xD4, C3xS3, C3xC6, C3xC6, C3xC6, C2xDic3, C2xDic3, C2xDic3, C3:D4, C2xC12, C3xD4, C22xS3, C22xC6, C22xC6, C22.D4, C3xDic3, C3:Dic3, S3xC6, C62, C62, C62, Dic3:C4, D6:C4, C6.D4, C22xDic3, C2xC3:D4, C22xC12, C6xD4, C6xDic3, C6xDic3, C6xDic3, C3xC3:D4, C2xC3:Dic3, S3xC2xC6, C2xC62, C23.28D6, C23.23D6, D6:Dic3, C62.C22, C62:5C4, Dic3xC2xC6, C6xC3:D4, C62.56D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, C3:D4, C22xS3, C22.D4, S32, C4oD12, D4:2S3, C2xC3:D4, D6:S3, C2xS32, C23.28D6, C23.23D6, D6.3D6, C2xD6: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+++++++++++++--+
imageC1C2C2C2C2C2S3S3D4D6D6D6C4oD4C3:D4C4oD12S32D4:2S3D6:S3C2xS32D6.3D6
kernelC62.56D4D6:Dic3C62.C22C62:5C4Dic3xC2xC6C6xC3:D4C22xDic3C2xC3:D4C62C2xDic3C22xS3C22xC6C3xC6C2xC6C6C23C6C22C22C2
# 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
 

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