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## G = C32×D8order 144 = 24·32

### Direct product of C32 and D8

direct product, metacyclic, nilpotent (class 3), monomial

Aliases: C32×D8, C243C6, C4.1C62, D4⋊(C3×C6), C81(C3×C6), (C3×C24)⋊5C2, (C3×D4)⋊4C6, (C3×C6).41D4, C6.20(C3×D4), C12.23(C2×C6), (D4×C32)⋊7C2, C2.3(D4×C32), (C3×C12).50C22, SmallGroup(144,106)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C32×D8
 Chief series C1 — C2 — C4 — C12 — C3×C12 — D4×C32 — C32×D8
 Lower central C1 — C2 — C4 — C32×D8
 Upper central C1 — C3×C6 — C3×C12 — C32×D8

Generators and relations for C32×D8
G = < a,b,c,d | a3=b3=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 114 in 66 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C8, D4, C32, C12, C2×C6, D8, C3×C6, C3×C6, C24, C3×D4, C3×C12, C62, C3×D8, C3×C24, D4×C32, C32×D8
Quotients: C1, C2, C3, C22, C6, D4, C32, C2×C6, D8, C3×C6, C3×D4, C62, C3×D8, D4×C32, C32×D8

Smallest permutation representation of C32×D8
On 72 points
Generators in S72
(1 34 61)(2 35 62)(3 36 63)(4 37 64)(5 38 57)(6 39 58)(7 40 59)(8 33 60)(9 28 54)(10 29 55)(11 30 56)(12 31 49)(13 32 50)(14 25 51)(15 26 52)(16 27 53)(17 69 42)(18 70 43)(19 71 44)(20 72 45)(21 65 46)(22 66 47)(23 67 48)(24 68 41)
(1 31 17)(2 32 18)(3 25 19)(4 26 20)(5 27 21)(6 28 22)(7 29 23)(8 30 24)(9 47 58)(10 48 59)(11 41 60)(12 42 61)(13 43 62)(14 44 63)(15 45 64)(16 46 57)(33 56 68)(34 49 69)(35 50 70)(36 51 71)(37 52 72)(38 53 65)(39 54 66)(40 55 67)
(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)
(2 8)(3 7)(4 6)(9 15)(10 14)(11 13)(18 24)(19 23)(20 22)(25 29)(26 28)(30 32)(33 35)(36 40)(37 39)(41 43)(44 48)(45 47)(50 56)(51 55)(52 54)(58 64)(59 63)(60 62)(66 72)(67 71)(68 70)

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

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

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

C32×D8 is a maximal subgroup of   C327D16  C328SD32  C248D6  C24.26D6

63 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 4 6A ··· 6H 6I ··· 6X 8A 8B 12A ··· 12H 24A ··· 24P order 1 2 2 2 3 ··· 3 4 6 ··· 6 6 ··· 6 8 8 12 ··· 12 24 ··· 24 size 1 1 4 4 1 ··· 1 2 1 ··· 1 4 ··· 4 2 2 2 ··· 2 2 ··· 2

63 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C3 C6 C6 D4 D8 C3×D4 C3×D8 kernel C32×D8 C3×C24 D4×C32 C3×D8 C24 C3×D4 C3×C6 C32 C6 C3 # reps 1 1 2 8 8 16 1 2 8 16

Matrix representation of C32×D8 in GL3(𝔽73) generated by

 8 0 0 0 8 0 0 0 8
,
 1 0 0 0 64 0 0 0 64
,
 72 0 0 0 16 57 0 16 16
,
 72 0 0 0 1 0 0 0 72
G:=sub<GL(3,GF(73))| [8,0,0,0,8,0,0,0,8],[1,0,0,0,64,0,0,0,64],[72,0,0,0,16,16,0,57,16],[72,0,0,0,1,0,0,0,72] >;

C32×D8 in GAP, Magma, Sage, TeX

C_3^2\times D_8
% in TeX

G:=Group("C3^2xD8");
// GroupNames label

G:=SmallGroup(144,106);
// by ID

G=gap.SmallGroup(144,106);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-2,-2,457,3244,1630,88]);
// Polycyclic

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

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