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G = C3×C32⋊Q16order 432 = 24·33

Direct product of C3 and C32⋊Q16

direct product, non-abelian, soluble, monomial

Aliases: C3×C32⋊Q16, C331Q16, C6.23S3≀C2, C32⋊(C3×Q16), C322Q8.C6, (C32×C6).5D4, C322C8.2C6, C2.5(C3×S3≀C2), (C3×C6).5(C3×D4), C3⋊Dic3.7(C2×C6), (C3×C322C8).3C2, (C3×C322Q8).2C2, (C3×C3⋊Dic3).33C22, SmallGroup(432,578)

Series: Derived Chief Lower central Upper central

C1C32C3⋊Dic3 — C3×C32⋊Q16
C1C32C3×C6C3⋊Dic3C3×C3⋊Dic3C3×C322Q8 — C3×C32⋊Q16
C32C3×C6C3⋊Dic3 — C3×C32⋊Q16
C1C6

Generators and relations for C3×C32⋊Q16
 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=c, dcd-1=b-1, ece-1=b, ede-1=d-1 >

Subgroups: 300 in 72 conjugacy classes, 18 normal (14 characteristic)
C1, C2, C3, C3 [×4], C4 [×3], C6, C6 [×4], C8, Q8 [×2], C32, C32 [×4], Dic3 [×4], C12 [×7], Q16, C3×C6, C3×C6 [×4], C24, Dic6 [×2], C3×Q8 [×2], C33, C3×Dic3 [×8], C3⋊Dic3, C3×C12 [×2], C3×Q16, C32×C6, C322C8, C322Q8 [×2], C3×Dic6 [×2], C32×Dic3 [×2], C3×C3⋊Dic3, C32⋊Q16, C3×C322C8, C3×C322Q8 [×2], C3×C32⋊Q16
Quotients: C1, C2 [×3], C3, C22, C6 [×3], D4, C2×C6, Q16, C3×D4, C3×Q16, S3≀C2, C32⋊Q16, C3×S3≀C2, C3×C32⋊Q16

Smallest permutation representation of C3×C32⋊Q16
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 33 47)(3 48 34)(4 41 35)(5 36 42)(6 37 43)(7 44 38)(8 45 39)(9 30 19)(10 31 20)(11 21 32)(12 22 25)(13 26 23)(14 27 24)(15 17 28)(16 18 29)
(1 40 46)(2 47 33)(3 48 34)(4 35 41)(5 36 42)(6 43 37)(7 44 38)(8 39 45)(9 30 19)(10 20 31)(11 21 32)(12 25 22)(13 26 23)(14 24 27)(15 17 28)(16 29 18)
(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)(2,33,47)(3,48,34)(4,41,35)(5,36,42)(6,37,43)(7,44,38)(8,45,39)(9,30,19)(10,31,20)(11,21,32)(12,22,25)(13,26,23)(14,27,24)(15,17,28)(16,18,29), (1,40,46)(2,47,33)(3,48,34)(4,35,41)(5,36,42)(6,43,37)(7,44,38)(8,39,45)(9,30,19)(10,20,31)(11,21,32)(12,25,22)(13,26,23)(14,24,27)(15,17,28)(16,29,18), (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)(2,33,47)(3,48,34)(4,41,35)(5,36,42)(6,37,43)(7,44,38)(8,45,39)(9,30,19)(10,31,20)(11,21,32)(12,22,25)(13,26,23)(14,27,24)(15,17,28)(16,18,29), (1,40,46)(2,47,33)(3,48,34)(4,35,41)(5,36,42)(6,43,37)(7,44,38)(8,39,45)(9,30,19)(10,20,31)(11,21,32)(12,25,22)(13,26,23)(14,24,27)(15,17,28)(16,29,18), (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),(2,33,47),(3,48,34),(4,41,35),(5,36,42),(6,37,43),(7,44,38),(8,45,39),(9,30,19),(10,31,20),(11,21,32),(12,22,25),(13,26,23),(14,27,24),(15,17,28),(16,18,29)], [(1,40,46),(2,47,33),(3,48,34),(4,35,41),(5,36,42),(6,43,37),(7,44,38),(8,39,45),(9,30,19),(10,20,31),(11,21,32),(12,25,22),(13,26,23),(14,24,27),(15,17,28),(16,29,18)], [(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)])

45 conjugacy classes

class 1  2 3A3B3C···3H4A4B4C6A6B6C···6H8A8B12A···12P12Q12R24A24B24C24D
order12333···3444666···68812···12121224242424
size11114···4121218114···4181812···12181818181818

45 irreducible representations

dim11111122224444
type++++-+-
imageC1C2C2C3C6C6D4Q16C3×D4C3×Q16S3≀C2C32⋊Q16C3×S3≀C2C3×C32⋊Q16
kernelC3×C32⋊Q16C3×C322C8C3×C322Q8C32⋊Q16C322C8C322Q8C32×C6C33C3×C6C32C6C3C2C1
# reps11222412244488

Matrix representation of C3×C32⋊Q16 in GL4(𝔽73) generated by

8000
0800
0080
0008
,
8000
06400
00640
0008
,
8000
06400
0080
00064
,
00460
00027
04600
46000
,
46000
02700
00027
00270
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[8,0,0,0,0,64,0,0,0,0,64,0,0,0,0,8],[8,0,0,0,0,64,0,0,0,0,8,0,0,0,0,64],[0,0,0,46,0,0,46,0,46,0,0,0,0,27,0,0],[46,0,0,0,0,27,0,0,0,0,0,27,0,0,27,0] >;

C3×C32⋊Q16 in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes Q_{16}
% in TeX

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

G:=SmallGroup(432,578);
// by ID

G=gap.SmallGroup(432,578);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,3,168,197,176,1011,514,80,4037,3036,362,1189,1203]);
// 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=c,d*c*d^-1=b^-1,e*c*e^-1=b,e*d*e^-1=d^-1>;
// generators/relations

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