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G = C3xC32:2Q16order 432 = 24·33

Direct product of C3 and C32:2Q16

direct product, metabelian, supersoluble, monomial

Aliases: C3xC32:2Q16, C33:4Q16, C12.98S32, C12.30(S3xC6), C32:4(C3xQ16), (C3xC12).111D6, C32:4C8.2C6, (C3xDic6).9S3, Dic6.2(C3xS3), (C3xDic6).5C6, (C32xC6).23D4, C6.29(D6:S3), C32:10(C3:Q16), (C32xC12).6C22, (C32xDic6).1C2, C4.10(C3xS32), C3:2(C3xC3:Q16), C6.9(C3xC3:D4), (C3xC6).22(C3xD4), (C3xC12).40(C2xC6), C2.5(C3xD6:S3), (C3xC6).84(C3:D4), (C3xC32:4C8).4C2, SmallGroup(432,423)

Series: Derived Chief Lower central Upper central

C1C3xC12 — C3xC32:2Q16
C1C3C32C3xC6C3xC12C32xC12C32xDic6 — C3xC32:2Q16
C32C3xC6C3xC12 — C3xC32:2Q16
C1C6C12

Generators and relations for C3xC32:2Q16
 G = < a,b,c,d,e | a3=b3=c3=d8=1, e2=d4, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe-1=b-1, dcd-1=c-1, ce=ec, ede-1=d-1 >

Subgroups: 304 in 110 conjugacy classes, 36 normal (16 characteristic)
C1, C2, C3, C3, C3, C4, C4, C6, C6, C6, C8, Q8, C32, C32, C32, Dic3, C12, C12, C12, Q16, C3xC6, C3xC6, C3xC6, C3:C8, C24, Dic6, C3xQ8, C33, C3xDic3, C3xC12, C3xC12, C3xC12, C3:Q16, C3xQ16, C32xC6, C3xC3:C8, C32:4C8, C3xDic6, C3xDic6, Q8xC32, C32xDic3, C32xC12, C32:2Q16, C3xC3:Q16, C3xC32:4C8, C32xDic6, C3xC32:2Q16
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2xC6, Q16, C3xS3, C3:D4, C3xD4, S32, S3xC6, C3:Q16, C3xQ16, D6:S3, C3xC3:D4, C3xS32, C32:2Q16, C3xC3:Q16, C3xD6:S3, C3xC32:2Q16

Smallest permutation representation of C3xC32:2Q16
On 48 points
Generators in S48
(1 40 46)(2 33 47)(3 34 48)(4 35 41)(5 36 42)(6 37 43)(7 38 44)(8 39 45)(9 30 19)(10 31 20)(11 32 21)(12 25 22)(13 26 23)(14 27 24)(15 28 17)(16 29 18)
(1 40 46)(2 47 33)(3 34 48)(4 41 35)(5 36 42)(6 43 37)(7 38 44)(8 45 39)(9 19 30)(10 31 20)(11 21 32)(12 25 22)(13 23 26)(14 27 24)(15 17 28)(16 29 18)
(1 40 46)(2 47 33)(3 34 48)(4 41 35)(5 36 42)(6 43 37)(7 38 44)(8 45 39)(9 30 19)(10 20 31)(11 32 21)(12 22 25)(13 26 23)(14 24 27)(15 28 17)(16 18 29)
(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 23 5 19)(2 22 6 18)(3 21 7 17)(4 20 8 24)(9 40 13 36)(10 39 14 35)(11 38 15 34)(12 37 16 33)(25 43 29 47)(26 42 30 46)(27 41 31 45)(28 48 32 44)

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

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

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

63 conjugacy classes

class 1  2 3A3B3C···3H3I3J3K4A4B4C6A6B6C···6H6I6J6K8A8B12A12B12C···12N12O···12AD24A24B24C24D
order12333···3333444666···666688121212···1212···1224242424
size11112···244421212112···24441818224···412···1218181818

63 irreducible representations

dim111111222222222244444444
type++++++-+--
imageC1C2C2C3C6C6S3D4D6Q16C3xS3C3:D4C3xD4S3xC6C3xQ16C3xC3:D4S32C3:Q16D6:S3C3xS32C32:2Q16C3xC3:Q16C3xD6:S3C3xC32:2Q16
kernelC3xC32:2Q16C3xC32:4C8C32xDic6C32:2Q16C32:4C8C3xDic6C3xDic6C32xC6C3xC12C33Dic6C3xC6C3xC6C12C32C6C12C32C6C4C3C3C2C1
# reps112224212244244812122424

Matrix representation of C3xC32:2Q16 in GL6(F73)

100000
010000
0064000
0006400
000010
000001
,
100000
010000
001000
000100
0000072
0000172
,
100000
010000
0072100
0072000
000010
000001
,
2200000
0100000
0007200
0072000
000001
000010
,
010000
7200000
0072000
0007200
000001
000010

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[22,0,0,0,0,0,0,10,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C3xC32:2Q16 in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes_2Q_{16}
% in TeX

G:=Group("C3xC3^2:2Q16");
// GroupNames label

G:=SmallGroup(432,423);
// by ID

G=gap.SmallGroup(432,423);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,197,176,1011,514,80,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=1,e^2=d^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=e*b*e^-1=b^-1,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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