Copied to
clipboard

G = C624Q8order 288 = 25·32

2nd semidirect product of C62 and Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial

Aliases: C624Q8, C62.124C23, C23.36S32, (C2×C6)⋊4Dic6, C6.75(S3×D4), C3⋊Dic3.62D4, C6.27(C2×Dic6), (C22×C6).82D6, (C2×Dic3).48D6, C62.C226C2, C6.D4.4S3, Dic3⋊Dic311C2, C2.34(Dic3⋊D6), C6.71(D42S3), C3215(C22⋊Q8), C223(C322Q8), (C2×C62).43C22, C34(Dic3.D4), C2.19(D6.4D6), (C6×Dic3).28C22, (C3×C6).43(C2×Q8), C22.147(C2×S32), (C3×C6).170(C2×D4), (C2×C322Q8)⋊8C2, C2.9(C2×C322Q8), (C3×C6).90(C4○D4), (C2×C6).143(C22×S3), (C3×C6.D4).3C2, (C22×C3⋊Dic3).8C2, (C2×C3⋊Dic3).149C22, SmallGroup(288,630)

Series: Derived Chief Lower central Upper central

C1C62 — C624Q8
C1C3C32C3×C6C62C6×Dic3C2×C322Q8 — C624Q8
C32C62 — C624Q8
C1C22C23

Generators and relations for C624Q8
 G = < a,b,c,d | a6=b6=c4=1, d2=c2, ab=ba, cac-1=a-1b3, dad-1=ab3, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 594 in 175 conjugacy classes, 54 normal (18 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C3, C4 [×7], C22, C22 [×2], C22 [×2], C6 [×6], C6 [×11], C2×C4 [×8], Q8 [×2], C23, C32, Dic3 [×14], C12 [×4], C2×C6 [×6], C2×C6 [×11], C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, C3×C6 [×3], C3×C6 [×2], Dic6 [×4], C2×Dic3 [×4], C2×Dic3 [×14], C2×C12 [×4], C22×C6 [×2], C22×C6, C22⋊Q8, C3×Dic3 [×4], C3⋊Dic3 [×2], C3⋊Dic3, C62, C62 [×2], C62 [×2], Dic3⋊C4 [×4], C4⋊Dic3 [×2], C6.D4 [×2], C3×C22⋊C4 [×2], C2×Dic6 [×2], C22×Dic3 [×3], C322Q8 [×2], C6×Dic3 [×4], C2×C3⋊Dic3 [×2], C2×C3⋊Dic3 [×2], C2×C62, Dic3.D4 [×2], Dic3⋊Dic3 [×2], C62.C22, C3×C6.D4 [×2], C2×C322Q8, C22×C3⋊Dic3, C624Q8
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], Q8 [×2], C23, D6 [×6], C2×D4, C2×Q8, C4○D4, Dic6 [×4], C22×S3 [×2], C22⋊Q8, S32, C2×Dic6 [×2], S3×D4 [×2], D42S3 [×2], C322Q8 [×2], C2×S32, Dic3.D4 [×2], D6.4D6, C2×C322Q8, Dic3⋊D6, C624Q8

Smallest permutation representation of C624Q8
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 46 9 48 11 44)(8 47 10 43 12 45)(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 23 4 30)(2 26 5 19)(3 21 6 28)(7 39 10 34)(8 36 11 41)(9 37 12 32)(13 29 16 22)(14 24 17 25)(15 27 18 20)(31 45 42 48)(33 43 38 46)(35 47 40 44)
(1 31 4 42)(2 40 5 35)(3 33 6 38)(7 20 10 27)(8 25 11 24)(9 22 12 29)(13 37 16 32)(14 36 17 41)(15 39 18 34)(19 44 26 47)(21 46 28 43)(23 48 30 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), (1,13,3,15,5,17)(2,14,4,16,6,18)(7,46,9,48,11,44)(8,47,10,43,12,45)(19,25,23,29,21,27)(20,26,24,30,22,28)(31,41,35,39,33,37)(32,42,36,40,34,38), (1,23,4,30)(2,26,5,19)(3,21,6,28)(7,39,10,34)(8,36,11,41)(9,37,12,32)(13,29,16,22)(14,24,17,25)(15,27,18,20)(31,45,42,48)(33,43,38,46)(35,47,40,44), (1,31,4,42)(2,40,5,35)(3,33,6,38)(7,20,10,27)(8,25,11,24)(9,22,12,29)(13,37,16,32)(14,36,17,41)(15,39,18,34)(19,44,26,47)(21,46,28,43)(23,48,30,45)>;

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

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

42 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D4E4F4G4H6A···6F6G···6Q12A···12H
order122222333444444446···66···612···12
size11112222412121212181818182···24···412···12

42 irreducible representations

dim11111122222224444444
type++++++++-++-++--+-+
imageC1C2C2C2C2C2S3D4Q8D6D6C4○D4Dic6S32S3×D4D42S3C322Q8C2×S32D6.4D6Dic3⋊D6
kernelC624Q8Dic3⋊Dic3C62.C22C3×C6.D4C2×C322Q8C22×C3⋊Dic3C6.D4C3⋊Dic3C62C2×Dic3C22×C6C3×C6C2×C6C23C6C6C22C22C2C2
# reps12121122242281222122

Matrix representation of C624Q8 in GL8(𝔽13)

120000000
01000000
001120000
00100000
00001000
00000100
00000010
00000001
,
120000000
012000000
00100000
00010000
00001000
00000100
00000001
0000001212
,
01000000
120000000
00010000
00100000
000001200
00001000
00000010
00000001
,
05000000
50000000
00100000
00010000
00003900
000091000
00000010
0000001212

G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12],[0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,9,0,0,0,0,0,0,9,10,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12] >;

C624Q8 in GAP, Magma, Sage, TeX

C_6^2\rtimes_4Q_8
% in TeX

G:=Group("C6^2:4Q8");
// GroupNames label

G:=SmallGroup(288,630);
// by ID

G=gap.SmallGroup(288,630);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,141,64,422,219,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=c^4=1,d^2=c^2,a*b=b*a,c*a*c^-1=a^-1*b^3,d*a*d^-1=a*b^3,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽