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G = C62.60D4  order 288 = 25·32

44th non-split extension by C62 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.60D4, C62.108C23, C23.23S32, C6.85(C2×D12), (C2×C6).67D12, C6.D4⋊7S3, (C22×C6).70D6, C6.65(C4○D12), C6.D12⋊5C2, (C2×Dic3).43D6, (C22×Dic3)⋊8S3, Dic3⋊Dic3⋊10C2, C6.52(D4⋊2S3), (C2×C62).27C22, C2.26(D6.3D6), C22.6(C3⋊D12), C3⋊3(C23.28D6), C3⋊4(C23.21D6), (C6×Dic3).25C22, C32⋊12(C22.D4), (Dic3×C2×C6)⋊4C2, C6.22(C2×C3⋊D4), C22.136(C2×S32), (C3×C6).154(C2×D4), (C3×C6).81(C4○D4), C2.23(C2×C3⋊D12), (C3×C6.D4)⋊8C2, (C2×C6).24(C3⋊D4), (C2×C32⋊7D4).7C2, (C2×C6).127(C22×S3), (C22×C3⋊S3).31C22, (C2×C3⋊Dic3).66C22, SmallGroup(288,614)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — C62.60D4
Chief series
C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — Dic3⋊Dic3 — C62.60D4
Lower central
C32 — C62 — C62.60D4
Upper central
C1 — C22 — C23

Generators and relations for C62.60D4
 G = < a,b,c,d | a6=b6=c4=1, d2=b3, ab=ba, cac-1=ab3, dad-1=a-1b3, cbc-1=dbd-1=b-1, dcd-1=c-1 >

Subgroups: 738 in 183 conjugacy classes, 52 normal (24 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, C3⋊D4, C2×C12, C22×S3, C22×C6, C22×C6, C22.D4, C3×Dic3, C3⋊Dic3, C2×C3⋊S3, C62, C62, C62, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C22×Dic3, C2×C3⋊D4, C22×C12, C6×Dic3, C6×Dic3, C2×C3⋊Dic3, C32⋊7D4, C22×C3⋊S3, C2×C62, C23.21D6, C23.28D6, C6.D12, Dic3⋊Dic3, C3×C6.D4, Dic3×C2×C6, C2×C32⋊7D4, C62.60D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C3⋊D4, C22×S3, C22.D4, S32, C2×D12, C4○D12, D4⋊2S3, C2×C3⋊D4, C3⋊D12, C2×S32, C23.21D6, C23.28D6, D6.3D6, C2×C3⋊D12, C62.60D4

Smallest permutation representation of C62.60D4
►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 11 3 7 5 9)(2 12 4 8 6 10)(13 43 17 47 15 45)(14 44 18 48 16 46)(19 28 21 30 23 26)(20 29 22 25 24 27)(31 37 35 41 33 39)(32 38 36 42 34 40)

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

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

48 conjugacy classes

class 1 2A2B2C2D2E2F3A3B3C4A4B4C4D4E4F4G6A···6J6K···6S12A···12H12I12J12K12L
order122222233344444446···66···612···1212121212
size1111223622466661212362···24···46···612121212

48 irreducible representations

dim11111122222222244444
type+++++++++++++-++
imageC1C2C2C2C2C2S3S3D4D6D6C4○D4D12C3⋊D4C4○D12S32D4⋊2S3C3⋊D12C2×S32D6.3D6
kernelC62.60D4C6.D12Dic3⋊Dic3C3×C6.D4Dic3×C2×C6C2×C32⋊7D4C6.D4C22×Dic3C62C2×Dic3C22×C6C3×C6C2×C6C2×C6C6C23C6C22C22C2
# reps12211111242444812214

Matrix representation of C62.60D4 ►in GL8(𝔽13)

120000000
012000000
00010000
00100000
00001000
00000100
00000001
0000001212
,
10000000
01000000
001200000
000120000
00000100
0000121200
00000010
00000001
,
01000000
120000000
00080000
00500000
000012000
00001100
00000010
00000001
,
120000000
01000000
000120000
00100000
00001000
0000121200
00000010
0000001212

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

C62.60D4 in GAP, Magma, Sage, TeX 

C_6^2._{60}D_4
 
% in TeX
 
G:=Group("C6^2.60D4");
 
// GroupNames label
 
G:=SmallGroup(288,614);
 
// by ID
 
G=gap.SmallGroup(288,614);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,253,64,254,1356,9414]);
 
// 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*b^3,d*a*d^-1=a^-1*b^3,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
 
// generators/relations
 

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