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G = C62⋊4D4  order 288 = 25·32

1st semidirect product of C62 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial

Aliases: C62⋊4D4, C62.118C23, (S3×C6)⋊15D4, C23.26S32, D6⋊7(C3⋊D4), (S3×C23)⋊4S3, (C2×Dic3)⋊3D6, C6.170(S3×D4), C32⋊4C22≀C2, (C22×S3)⋊3D6, C3⋊5(C23⋊2D6), D6⋊Dic3⋊35C2, C62⋊5C4⋊10C2, C3⋊2(C24⋊4S3), (C22×C6).118D6, C22⋊3(D6⋊S3), (C6×Dic3)⋊15C22, (C2×C62).37C22, (C2×C3⋊D4)⋊4S3, (S3×C22×C6)⋊1C2, (S3×C2×C6)⋊2C22, (C6×C3⋊D4)⋊10C2, C2.42(S3×C3⋊D4), C6.84(C2×C3⋊D4), (C2×D6⋊S3)⋊8C2, (C2×C6)⋊14(C3⋊D4), C22.141(C2×S32), (C3×C6).164(C2×D4), C2.16(C2×D6⋊S3), (C2×C3⋊Dic3)⋊4C22, (C2×C6).137(C22×S3), SmallGroup(288,624)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — C62⋊4D4
Chief series
C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×D6⋊S3 — C62⋊4D4
Lower central
C32 — C62 — C62⋊4D4
Upper central
C1 — C22 — C23

Generators and relations for C62⋊4D4
 G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=a-1b3, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 1018 in 277 conjugacy classes, 60 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, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C2×D4, C24, C3×S3, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×S3, C22×S3, C22×C6, C22×C6, C22≀C2, C3×Dic3, C3⋊Dic3, S3×C6, S3×C6, C62, C62, C62, D6⋊C4, C6.D4, C2×C3⋊D4, C2×C3⋊D4, C6×D4, S3×C23, C23×C6, D6⋊S3, C6×Dic3, C3×C3⋊D4, C2×C3⋊Dic3, S3×C2×C6, S3×C2×C6, S3×C2×C6, C2×C62, C23⋊2D6, C24⋊4S3, D6⋊Dic3, C62⋊5C4, C2×D6⋊S3, C6×C3⋊D4, S3×C22×C6, C62⋊4D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C22≀C2, S32, S3×D4, C2×C3⋊D4, D6⋊S3, C2×S32, C23⋊2D6, C24⋊4S3, C2×D6⋊S3, S3×C3⋊D4, C62⋊4D4

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

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

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

48 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J3A3B3C4A4B4C6A···6J6K···6S6T···6AA6AB6AC12A12B
order122222222223334446···66···66···6661212
size1111226666122241236362···24···46···612121212

48 irreducible representations

dim11111122222222244444
type+++++++++++++++-+
imageC1C2C2C2C2C2S3S3D4D4D6D6D6C3⋊D4C3⋊D4S32S3×D4D6⋊S3C2×S32S3×C3⋊D4
kernelC62⋊4D4D6⋊Dic3C62⋊5C4C2×D6⋊S3C6×C3⋊D4S3×C22×C6C2×C3⋊D4S3×C23S3×C6C62C2×Dic3C22×S3C22×C6D6C2×C6C23C6C22C22C2
# reps12121111421328812214

Matrix representation of C62⋊4D4 ►in GL8(𝔽13)

120000000
012000000
00100000
000120000
00001000
000031200
000000012
00000011
,
01000000
1212000000
001200000
000120000
000012000
000001200
00000010
00000001
,
10000000
1212000000
00010000
001200000
000012500
00000100
000000120
00000011
,
10000000
1212000000
00100000
000120000
000012000
000001200
00000010
00000001

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

C62⋊4D4 in GAP, Magma, Sage, TeX 

C_6^2\rtimes_4D_4
 
% in TeX
 
G:=Group("C6^2:4D4");
 
// GroupNames label
 
G:=SmallGroup(288,624);
 
// by ID
 
G=gap.SmallGroup(288,624);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,219,1356,9414]);
 
// 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*b^3,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

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