Copied to
clipboard

G = C62⋊7D4  order 288 = 25·32

4th semidirect product of C62 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial

Aliases: C62⋊7D4, C62.122C23, C23.34S32, C6.73(S3×D4), C3⋊Dic3⋊14D4, D6⋊Dic3⋊36C2, (C22×C6).80D6, C32⋊15(C4⋊D4), (C2×Dic3).47D6, (C22×S3).28D6, C3⋊6(C23.14D6), C2.32(Dic3⋊D6), C6.70(D4⋊2S3), C22⋊2(D6⋊S3), C62.C22⋊25C2, (C2×C62).41C22, C2.18(D6.4D6), (C6×Dic3).86C22, (C2×C3⋊D4)⋊7S3, (C6×C3⋊D4)⋊12C2, (C2×C6)⋊6(C3⋊D4), C6.85(C2×C3⋊D4), C22.145(C2×S32), (C3×C6).168(C2×D4), (S3×C2×C6).50C22, (C2×D6⋊S3)⋊10C2, (C3×C6).89(C4○D4), C2.17(C2×D6⋊S3), (C22×C3⋊Dic3)⋊4C2, (C2×C6).141(C22×S3), (C2×C3⋊Dic3).148C22, SmallGroup(288,628)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — C62⋊7D4
Chief series
C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×D6⋊S3 — C62⋊7D4
Lower central
C32 — C62 — C62⋊7D4
Upper central
C1 — C22 — C23

Generators and relations for C62⋊7D4
 G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=a-1, dad=ab3, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 786 in 215 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3, C3×C6, C3×C6, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C4⋊D4, C3×Dic3, C3⋊Dic3, C3⋊Dic3, S3×C6, C62, C62, C62, Dic3⋊C4, D6⋊C4, C6.D4, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C6×D4, D6⋊S3, C6×Dic3, C3×C3⋊D4, C2×C3⋊Dic3, C2×C3⋊Dic3, S3×C2×C6, C2×C62, C23.14D6, D6⋊Dic3, C62.C22, C2×D6⋊S3, C6×C3⋊D4, C22×C3⋊Dic3, C62⋊7D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, S32, S3×D4, D4⋊2S3, C2×C3⋊D4, D6⋊S3, C2×S32, C23.14D6, D6.4D6, C2×D6⋊S3, Dic3⋊D6, C62⋊7D4

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C4D4E4F6A···6F6G···6Q6R6S6T6U12A12B12C12D
order122222223334444446···66···6666612121212
size11112212122241212181818182···24···41212121212121212

42 irreducible representations

dim111111222222224444444
type++++++++++++++--+-+
imageC1C2C2C2C2C2S3D4D4D6D6D6C4○D4C3⋊D4S32S3×D4D4⋊2S3D6⋊S3C2×S32D6.4D6Dic3⋊D6
kernelC62⋊7D4D6⋊Dic3C62.C22C2×D6⋊S3C6×C3⋊D4C22×C3⋊Dic3C2×C3⋊D4C3⋊Dic3C62C2×Dic3C22×S3C22×C6C3×C6C2×C6C23C6C6C22C22C2C2
# reps121121222222281222122

Matrix representation of C62⋊7D4 ►in GL8(𝔽13)

120000000
01000000
000120000
00110000
000012000
000001200
00000010
00000001
,
120000000
012000000
00100000
00010000
00001000
00000100
000000121
000000120
,
120000000
012000000
00100000
0012120000
00003700
000061000
00000001
00000010
,
012000000
120000000
00100000
00010000
00007300
000010600
00000001
00000010

G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,3,6,0,0,0,0,0,0,7,10,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,7,10,0,0,0,0,0,0,3,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
 

C62⋊7D4 in GAP, Magma, Sage, TeX 

C_6^2\rtimes_7D_4
 
% in TeX
 
G:=Group("C6^2:7D4");
 
// GroupNames label
 
G:=SmallGroup(288,628);
 
// by ID
 
G=gap.SmallGroup(288,628);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,422,219,1356,9414]);
 
// 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,d*a*d=a*b^3,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁