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G = C6xC32:2Q8order 432 = 24·33

Direct product of C6 and C32:2Q8

direct product, metabelian, supersoluble, monomial

Aliases: C6xC32:2Q8, C62.112D6, C6:1(C3xDic6), (C3xC6):7Dic6, C3:2(C6xDic6), (C32xC6):3Q8, C32:6(C6xQ8), C33:13(C2xQ8), C62.28(C2xC6), Dic3.7(S3xC6), (C6xDic3).8C6, (C3xDic3).48D6, (C6xDic3).19S3, C32:14(C2xDic6), (C32xC6).35C23, (C3xC62).22C22, (C32xDic3).25C22, C2.16(S32xC6), (C2xC6).75S32, C6.16(S3xC2xC6), (C3xC6):3(C3xQ8), C6.119(C2xS32), (C2xC6).30(S3xC6), C22.12(C3xS32), (Dic3xC3xC6).9C2, (C2xC3:Dic3).12C6, C3:Dic3.20(C2xC6), (C6xC3:Dic3).18C2, (C3xC6).26(C22xC6), (C3xDic3).7(C2xC6), (C2xDic3).3(C3xS3), (C3xC6).140(C22xS3), (C3xC3:Dic3).57C22, SmallGroup(432,657)

Series: Derived Chief Lower central Upper central

C1C3xC6 — C6xC32:2Q8
C1C3C32C3xC6C32xC6C32xDic3C3xC32:2Q8 — C6xC32:2Q8
C32C3xC6 — C6xC32:2Q8
C1C2xC6

Generators and relations for C6xC32:2Q8
 G = < a,b,c,d,e | a6=b3=c3=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 576 in 210 conjugacy classes, 80 normal (16 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, C6, C6, C6, C2xC4, Q8, C32, C32, C32, Dic3, Dic3, C12, C2xC6, C2xC6, C2xC6, C2xQ8, C3xC6, C3xC6, C3xC6, Dic6, C2xDic3, C2xDic3, C2xC12, C3xQ8, C33, C3xDic3, C3xDic3, C3:Dic3, C3xC12, C62, C62, C62, C2xDic6, C6xQ8, C32xC6, C32xC6, C32:2Q8, C3xDic6, C6xDic3, C6xDic3, C2xC3:Dic3, C6xC12, C32xDic3, C3xC3:Dic3, C3xC62, C2xC32:2Q8, C6xDic6, C3xC32:2Q8, Dic3xC3xC6, C6xC3:Dic3, C6xC32:2Q8
Quotients: C1, C2, C3, C22, S3, C6, Q8, C23, D6, C2xC6, C2xQ8, C3xS3, Dic6, C3xQ8, C22xS3, C22xC6, S32, S3xC6, C2xDic6, C6xQ8, C32:2Q8, C3xDic6, C2xS32, S3xC2xC6, C3xS32, C2xC32:2Q8, C6xDic6, C3xC32:2Q8, S32xC6, C6xC32:2Q8

Smallest permutation representation of C6xC32:2Q8
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 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 47 45)(44 48 46)
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 48)
(1 28 16 24)(2 29 17 19)(3 30 18 20)(4 25 13 21)(5 26 14 22)(6 27 15 23)(7 35 47 39)(8 36 48 40)(9 31 43 41)(10 32 44 42)(11 33 45 37)(12 34 46 38)
(1 37 16 33)(2 38 17 34)(3 39 18 35)(4 40 13 36)(5 41 14 31)(6 42 15 32)(7 30 47 20)(8 25 48 21)(9 26 43 22)(10 27 44 23)(11 28 45 24)(12 29 46 19)

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

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

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

90 conjugacy classes

class 1 2A2B2C3A3B3C···3H3I3J3K4A4B4C4D4E4F6A···6F6G···6X6Y···6AG12A···12AF12AG12AH12AI12AJ
order1222333···33334444446···66···66···612···1212121212
size1111112···2444666618181···12···24···46···618181818

90 irreducible representations

dim111111112222222222444444
type+++++-++-+-+
imageC1C2C2C2C3C6C6C6S3Q8D6D6C3xS3Dic6C3xQ8S3xC6S3xC6C3xDic6S32C32:2Q8C2xS32C3xS32C3xC32:2Q8S32xC6
kernelC6xC32:2Q8C3xC32:2Q8Dic3xC3xC6C6xC3:Dic3C2xC32:2Q8C32:2Q8C6xDic3C2xC3:Dic3C6xDic3C32xC6C3xDic3C62C2xDic3C3xC6C3xC6Dic3C2xC6C6C2xC6C6C6C22C2C2
# reps1421284222424848416121242

Matrix representation of C6xC32:2Q8 in GL6(F13)

1200000
0120000
003000
000300
000010
000001
,
100000
010000
001000
000100
0000012
0000112
,
12120000
100000
001000
000100
000010
000001
,
100000
010000
000100
0012000
000001
000010
,
100000
12120000
0010900
009300
000010
000001

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,12,0,0,0,0,0,12,0,0,0,0,0,0,10,9,0,0,0,0,9,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C6xC32:2Q8 in GAP, Magma, Sage, TeX

C_6\times C_3^2\rtimes_2Q_8
% in TeX

G:=Group("C6xC3^2:2Q8");
// GroupNames label

G:=SmallGroup(432,657);
// by ID

G=gap.SmallGroup(432,657);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,168,365,176,2028,14118]);
// Polycyclic

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

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