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G = C62.57D4  order 288 = 25·32

41st non-split extension by C62 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.57D4, C62.104C23, C23.31S32, (C2×C6).15D12, C6.82(C2×D12), D6⋊Dic3⋊15C2, C6.D4⋊6S3, (C22×C6).68D6, (C2×Dic3).41D6, (C22×S3).25D6, Dic3⋊Dic3⋊35C2, C6.67(D4⋊2S3), (C2×C62).23C22, C2.15(D6.4D6), C3⋊5(C23.21D6), C3⋊2(C23.23D6), (C6×Dic3).24C22, C22.11(C3⋊D12), C32⋊11(C22.D4), (C6×C3⋊D4).7C2, (C2×C3⋊D4).4S3, C6.19(C2×C3⋊D4), C22.134(C2×S32), (C3×C6).150(C2×D4), (S3×C2×C6).42C22, (C3×C6).79(C4○D4), C2.22(C2×C3⋊D12), (C3×C6.D4)⋊7C2, (C2×C6).23(C3⋊D4), (C22×C3⋊Dic3)⋊2C2, (C2×C6).123(C22×S3), (C2×C3⋊Dic3).144C22, SmallGroup(288,610)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — C62.57D4
Chief series
C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — D6⋊Dic3 — C62.57D4
Lower central
C32 — C62 — C62.57D4
Upper central
C1 — C22 — C23

Generators and relations for C62.57D4
 G = < a,b,c,d | a6=b6=c4=1, d2=b3, ab=ba, cac-1=a-1b3, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=b3c-1 >

Subgroups: 626 in 183 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, C4⋊Dic3, D6⋊C4, C6.D4, C6.D4, C3×C22⋊C4, C22×Dic3, C2×C3⋊D4, C6×D4, C6×Dic3, C6×Dic3, C3×C3⋊D4, C2×C3⋊Dic3, C2×C3⋊Dic3, S3×C2×C6, C2×C62, C23.21D6, C23.23D6, D6⋊Dic3, Dic3⋊Dic3, C3×C6.D4, C6×C3⋊D4, C22×C3⋊Dic3, C62.57D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C3⋊D4, C22×S3, C22.D4, S32, C2×D12, D4⋊2S3, C2×C3⋊D4, C3⋊D12, C2×S32, C23.21D6, C23.23D6, D6.4D6, C2×C3⋊D12, C62.57D4

Smallest permutation representation of C62.57D4
►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 13 3 15 5 17)(2 14 4 16 6 18)(7 48 9 44 11 46)(8 43 10 45 12 47)(19 25 23 29 21 27)(20 26 24 30 22 28)(31 41 35 39 33 37)(32 42 36 40 34 38)

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F3A3B3C4A4B4C4D4E4F4G6A···6F6G···6Q6R6S12A···12F
order122222233344444446···66···66612···12
size11112212224121212181818182···24···4121212···12

42 irreducible representations

dim11111122222222244444
type++++++++++++++-++-
imageC1C2C2C2C2C2S3S3D4D6D6D6C4○D4D12C3⋊D4S32D4⋊2S3C3⋊D12C2×S32D6.4D6
kernelC62.57D4D6⋊Dic3Dic3⋊Dic3C3×C6.D4C6×C3⋊D4C22×C3⋊Dic3C6.D4C2×C3⋊D4C62C2×Dic3C22×S3C22×C6C3×C6C2×C6C2×C6C23C6C22C22C2
# reps12211111231244414214

Matrix representation of C62.57D4 ►in GL8(𝔽13)

10000000
1012000000
001120000
00100000
00001000
00000100
00000010
00000001
,
120000000
012000000
00100000
00010000
00001000
00000100
000000121
000000120
,
128000000
31000000
00010000
00100000
00001200
0000121200
00000010
00000001
,
50000000
05000000
00010000
00100000
000012000
00001100
00000001
00000010

G:=sub<GL(8,GF(13))| [1,10,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[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],[12,3,0,0,0,0,0,0,8,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,2,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
 

C62.57D4 in GAP, Magma, Sage, TeX 

C_6^2._{57}D_4
 
% in TeX
 
G:=Group("C6^2.57D4");
 
// GroupNames label
 
G:=SmallGroup(288,610);
 
// by ID
 
G=gap.SmallGroup(288,610);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,176,422,219,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^6=c^4=1,d^2=b^3,a*b=b*a,c*a*c^-1=a^-1*b^3,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^3*c^-1>;
 
// generators/relations
 

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