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G = C32xD8order 144 = 24·32

Direct product of C32 and D8

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

Aliases: C32xD8, C24:3C6, C4.1C62, D4:(C3xC6), C8:1(C3xC6), (C3xC24):5C2, (C3xD4):4C6, (C3xC6).41D4, C6.20(C3xD4), C12.23(C2xC6), (D4xC32):7C2, C2.3(D4xC32), (C3xC12).50C22, SmallGroup(144,106)

Series: Derived Chief Lower central Upper central

C1C4 — C32xD8
C1C2C4C12C3xC12D4xC32 — C32xD8
C1C2C4 — C32xD8
C1C3xC6C3xC12 — C32xD8

Generators and relations for C32xD8
 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, C2xC6, D8, C3xC6, C3xC6, C24, C3xD4, C3xC12, C62, C3xD8, C3xC24, D4xC32, C32xD8
Quotients: C1, C2, C3, C22, C6, D4, C32, C2xC6, D8, C3xC6, C3xD4, C62, C3xD8, D4xC32, C32xD8

Smallest permutation representation of C32xD8
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)]])

C32xD8 is a maximal subgroup of   C32:7D16  C32:8SD32  C24:8D6  C24.26D6

63 conjugacy classes

class 1 2A2B2C3A···3H 4 6A···6H6I···6X8A8B12A···12H24A···24P
order12223···346···66···68812···1224···24
size11441···121···14···4222···22···2

63 irreducible representations

dim1111112222
type+++++
imageC1C2C2C3C6C6D4D8C3xD4C3xD8
kernelC32xD8C3xC24D4xC32C3xD8C24C3xD4C3xC6C32C6C3
# reps112881612816

Matrix representation of C32xD8 in GL3(F73) generated by

800
080
008
,
100
0640
0064
,
7200
01657
01616
,
7200
010
0072
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] >;

C32xD8 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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