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G = C627D4order 288 = 25·32

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

metabelian, supersoluble, monomial

Aliases: C627D4, C62.122C23, C23.34S32, C6.73(S3×D4), C3⋊Dic314D4, D6⋊Dic336C2, (C22×C6).80D6, C3215(C4⋊D4), (C2×Dic3).47D6, (C22×S3).28D6, C36(C23.14D6), C2.32(Dic3⋊D6), C6.70(D42S3), C222(D6⋊S3), C62.C2225C2, (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

C1C62 — C627D4
C1C3C32C3×C6C62S3×C2×C6C2×D6⋊S3 — C627D4
C32C62 — C627D4
C1C22C23

Generators and relations for C627D4
 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 [×3], C2 [×4], C3 [×2], C3, C4 [×5], C22, C22 [×2], C22 [×8], S3 [×2], C6 [×6], C6 [×13], C2×C4 [×6], D4 [×6], C23, C23 [×2], C32, Dic3 [×12], C12 [×2], D6 [×6], C2×C6 [×6], C2×C6 [×17], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], C3×S3 [×2], C3×C6 [×3], C3×C6 [×2], C2×Dic3 [×2], C2×Dic3 [×14], C3⋊D4 [×8], C2×C12 [×2], C3×D4 [×4], C22×S3 [×2], C22×C6 [×2], C22×C6 [×3], C4⋊D4, C3×Dic3 [×2], C3⋊Dic3 [×2], C3⋊Dic3, S3×C6 [×6], C62, C62 [×2], C62 [×2], Dic3⋊C4 [×2], D6⋊C4 [×2], C6.D4 [×2], C22×Dic3 [×3], C2×C3⋊D4 [×2], C2×C3⋊D4 [×2], C6×D4 [×2], D6⋊S3 [×2], C6×Dic3 [×2], C3×C3⋊D4 [×4], C2×C3⋊Dic3 [×2], C2×C3⋊Dic3 [×2], S3×C2×C6 [×2], C2×C62, C23.14D6 [×2], D6⋊Dic3 [×2], C62.C22, C2×D6⋊S3, C6×C3⋊D4 [×2], C22×C3⋊Dic3, C627D4
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×4], C23, D6 [×6], C2×D4 [×2], C4○D4, C3⋊D4 [×4], C22×S3 [×2], C4⋊D4, S32, S3×D4 [×2], D42S3 [×2], C2×C3⋊D4 [×2], D6⋊S3 [×2], C2×S32, C23.14D6 [×2], D6.4D6, C2×D6⋊S3, Dic3⋊D6, C627D4

Smallest permutation representation of C627D4
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 22 2 23 3 24)(4 16 5 17 6 18)(7 12 8 10 9 11)(13 21 14 19 15 20)(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 17 19 7)(2 16 20 9)(3 18 21 8)(4 13 10 23)(5 15 11 22)(6 14 12 24)(25 41 46 31)(26 40 47 36)(27 39 48 35)(28 38 43 34)(29 37 44 33)(30 42 45 32)
(1 30)(2 28)(3 26)(4 35)(5 33)(6 31)(7 42)(8 40)(9 38)(10 39)(11 37)(12 41)(13 48)(14 46)(15 44)(16 34)(17 32)(18 36)(19 45)(20 43)(21 47)(22 29)(23 27)(24 25)

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

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

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

42 conjugacy classes

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

42 irreducible representations

dim111111222222224444444
type++++++++++++++--+-+
imageC1C2C2C2C2C2S3D4D4D6D6D6C4○D4C3⋊D4S32S3×D4D42S3D6⋊S3C2×S32D6.4D6Dic3⋊D6
kernelC627D4D6⋊Dic3C62.C22C2×D6⋊S3C6×C3⋊D4C22×C3⋊Dic3C2×C3⋊D4C3⋊Dic3C62C2×Dic3C22×S3C22×C6C3×C6C2×C6C23C6C6C22C22C2C2
# reps121121222222281222122

Matrix representation of C627D4 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] >;

C627D4 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

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