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## G = C32×C4.Dic3order 432 = 24·33

### Direct product of C32 and C4.Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C32×C4.Dic3
 Chief series C1 — C3 — C6 — C12 — C3×C12 — C32×C12 — C32×C3⋊C8 — C32×C4.Dic3
 Lower central C3 — C6 — C32×C4.Dic3
 Upper central C1 — C3×C12 — C6×C12

Generators and relations for C32×C4.Dic3
G = < a,b,c,d,e | a3=b3=c4=1, d6=c2, e2=c2d3, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d5 >

Subgroups: 296 in 184 conjugacy classes, 90 normal (26 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, C6, C6, C6, C8, C2×C4, C32, C32, C32, C12, C12, C12, C2×C6, C2×C6, C2×C6, M4(2), C3×C6, C3×C6, C3×C6, C3⋊C8, C24, C2×C12, C2×C12, C2×C12, C33, C3×C12, C3×C12, C3×C12, C62, C62, C62, C4.Dic3, C3×M4(2), C32×C6, C32×C6, C3×C3⋊C8, C3×C24, C6×C12, C6×C12, C6×C12, C32×C12, C3×C62, C3×C4.Dic3, C32×M4(2), C32×C3⋊C8, C3×C6×C12, C32×C4.Dic3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C32, Dic3, C12, D6, C2×C6, M4(2), C3×S3, C3×C6, C2×Dic3, C2×C12, C3×Dic3, C3×C12, S3×C6, C62, C4.Dic3, C3×M4(2), S3×C32, C6×Dic3, C6×C12, C32×Dic3, S3×C3×C6, C3×C4.Dic3, C32×M4(2), Dic3×C3×C6, C32×C4.Dic3

Smallest permutation representation of C32×C4.Dic3
On 72 points
Generators in S72
(1 33 23)(2 34 24)(3 35 13)(4 36 14)(5 25 15)(6 26 16)(7 27 17)(8 28 18)(9 29 19)(10 30 20)(11 31 21)(12 32 22)(37 66 51)(38 67 52)(39 68 53)(40 69 54)(41 70 55)(42 71 56)(43 72 57)(44 61 58)(45 62 59)(46 63 60)(47 64 49)(48 65 50)
(1 25 19)(2 26 20)(3 27 21)(4 28 22)(5 29 23)(6 30 24)(7 31 13)(8 32 14)(9 33 15)(10 34 16)(11 35 17)(12 36 18)(37 62 55)(38 63 56)(39 64 57)(40 65 58)(41 66 59)(42 67 60)(43 68 49)(44 69 50)(45 70 51)(46 71 52)(47 72 53)(48 61 54)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 22 19 16)(14 23 20 17)(15 24 21 18)(25 34 31 28)(26 35 32 29)(27 36 33 30)(37 40 43 46)(38 41 44 47)(39 42 45 48)(49 52 55 58)(50 53 56 59)(51 54 57 60)(61 64 67 70)(62 65 68 71)(63 66 69 72)
(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)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)
(1 48 10 45 7 42 4 39)(2 41 11 38 8 47 5 44)(3 46 12 43 9 40 6 37)(13 60 22 57 19 54 16 51)(14 53 23 50 20 59 17 56)(15 58 24 55 21 52 18 49)(25 61 34 70 31 67 28 64)(26 66 35 63 32 72 29 69)(27 71 36 68 33 65 30 62)

G:=sub<Sym(72)| (1,33,23)(2,34,24)(3,35,13)(4,36,14)(5,25,15)(6,26,16)(7,27,17)(8,28,18)(9,29,19)(10,30,20)(11,31,21)(12,32,22)(37,66,51)(38,67,52)(39,68,53)(40,69,54)(41,70,55)(42,71,56)(43,72,57)(44,61,58)(45,62,59)(46,63,60)(47,64,49)(48,65,50), (1,25,19)(2,26,20)(3,27,21)(4,28,22)(5,29,23)(6,30,24)(7,31,13)(8,32,14)(9,33,15)(10,34,16)(11,35,17)(12,36,18)(37,62,55)(38,63,56)(39,64,57)(40,65,58)(41,66,59)(42,67,60)(43,68,49)(44,69,50)(45,70,51)(46,71,52)(47,72,53)(48,61,54), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,22,19,16)(14,23,20,17)(15,24,21,18)(25,34,31,28)(26,35,32,29)(27,36,33,30)(37,40,43,46)(38,41,44,47)(39,42,45,48)(49,52,55,58)(50,53,56,59)(51,54,57,60)(61,64,67,70)(62,65,68,71)(63,66,69,72), (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,48,10,45,7,42,4,39)(2,41,11,38,8,47,5,44)(3,46,12,43,9,40,6,37)(13,60,22,57,19,54,16,51)(14,53,23,50,20,59,17,56)(15,58,24,55,21,52,18,49)(25,61,34,70,31,67,28,64)(26,66,35,63,32,72,29,69)(27,71,36,68,33,65,30,62)>;

G:=Group( (1,33,23)(2,34,24)(3,35,13)(4,36,14)(5,25,15)(6,26,16)(7,27,17)(8,28,18)(9,29,19)(10,30,20)(11,31,21)(12,32,22)(37,66,51)(38,67,52)(39,68,53)(40,69,54)(41,70,55)(42,71,56)(43,72,57)(44,61,58)(45,62,59)(46,63,60)(47,64,49)(48,65,50), (1,25,19)(2,26,20)(3,27,21)(4,28,22)(5,29,23)(6,30,24)(7,31,13)(8,32,14)(9,33,15)(10,34,16)(11,35,17)(12,36,18)(37,62,55)(38,63,56)(39,64,57)(40,65,58)(41,66,59)(42,67,60)(43,68,49)(44,69,50)(45,70,51)(46,71,52)(47,72,53)(48,61,54), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,22,19,16)(14,23,20,17)(15,24,21,18)(25,34,31,28)(26,35,32,29)(27,36,33,30)(37,40,43,46)(38,41,44,47)(39,42,45,48)(49,52,55,58)(50,53,56,59)(51,54,57,60)(61,64,67,70)(62,65,68,71)(63,66,69,72), (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,48,10,45,7,42,4,39)(2,41,11,38,8,47,5,44)(3,46,12,43,9,40,6,37)(13,60,22,57,19,54,16,51)(14,53,23,50,20,59,17,56)(15,58,24,55,21,52,18,49)(25,61,34,70,31,67,28,64)(26,66,35,63,32,72,29,69)(27,71,36,68,33,65,30,62) );

G=PermutationGroup([[(1,33,23),(2,34,24),(3,35,13),(4,36,14),(5,25,15),(6,26,16),(7,27,17),(8,28,18),(9,29,19),(10,30,20),(11,31,21),(12,32,22),(37,66,51),(38,67,52),(39,68,53),(40,69,54),(41,70,55),(42,71,56),(43,72,57),(44,61,58),(45,62,59),(46,63,60),(47,64,49),(48,65,50)], [(1,25,19),(2,26,20),(3,27,21),(4,28,22),(5,29,23),(6,30,24),(7,31,13),(8,32,14),(9,33,15),(10,34,16),(11,35,17),(12,36,18),(37,62,55),(38,63,56),(39,64,57),(40,65,58),(41,66,59),(42,67,60),(43,68,49),(44,69,50),(45,70,51),(46,71,52),(47,72,53),(48,61,54)], [(1,10,7,4),(2,11,8,5),(3,12,9,6),(13,22,19,16),(14,23,20,17),(15,24,21,18),(25,34,31,28),(26,35,32,29),(27,36,33,30),(37,40,43,46),(38,41,44,47),(39,42,45,48),(49,52,55,58),(50,53,56,59),(51,54,57,60),(61,64,67,70),(62,65,68,71),(63,66,69,72)], [(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),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72)], [(1,48,10,45,7,42,4,39),(2,41,11,38,8,47,5,44),(3,46,12,43,9,40,6,37),(13,60,22,57,19,54,16,51),(14,53,23,50,20,59,17,56),(15,58,24,55,21,52,18,49),(25,61,34,70,31,67,28,64),(26,66,35,63,32,72,29,69),(27,71,36,68,33,65,30,62)]])

162 conjugacy classes

 class 1 2A 2B 3A ··· 3H 3I ··· 3Q 4A 4B 4C 6A ··· 6H 6I ··· 6AQ 8A 8B 8C 8D 12A ··· 12P 12Q ··· 12BH 24A ··· 24AF order 1 2 2 3 ··· 3 3 ··· 3 4 4 4 6 ··· 6 6 ··· 6 8 8 8 8 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 2 1 ··· 1 2 ··· 2 1 1 2 1 ··· 1 2 ··· 2 6 6 6 6 1 ··· 1 2 ··· 2 6 ··· 6

162 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 S3 Dic3 D6 Dic3 M4(2) C3×S3 C3×Dic3 S3×C6 C3×Dic3 C4.Dic3 C3×M4(2) C3×C4.Dic3 kernel C32×C4.Dic3 C32×C3⋊C8 C3×C6×C12 C3×C4.Dic3 C32×C12 C3×C62 C3×C3⋊C8 C6×C12 C3×C12 C62 C6×C12 C3×C12 C3×C12 C62 C33 C2×C12 C12 C12 C2×C6 C32 C32 C3 # reps 1 2 1 8 2 2 16 8 16 16 1 1 1 1 2 8 8 8 8 4 16 32

Matrix representation of C32×C4.Dic3 in GL3(𝔽73) generated by

 64 0 0 0 1 0 0 0 1
,
 8 0 0 0 64 0 0 0 64
,
 1 0 0 0 46 0 0 0 27
,
 1 0 0 0 24 0 0 0 3
,
 1 0 0 0 0 1 0 46 0
G:=sub<GL(3,GF(73))| [64,0,0,0,1,0,0,0,1],[8,0,0,0,64,0,0,0,64],[1,0,0,0,46,0,0,0,27],[1,0,0,0,24,0,0,0,3],[1,0,0,0,0,46,0,1,0] >;

C32×C4.Dic3 in GAP, Magma, Sage, TeX

C_3^2\times C_4.{\rm Dic}_3
% in TeX

G:=Group("C3^2xC4.Dic3");
// GroupNames label

G:=SmallGroup(432,470);
// by ID

G=gap.SmallGroup(432,470);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,252,1037,102,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^4=1,d^6=c^2,e^2=c^2*d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^5>;
// generators/relations

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