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G = Q8×C32order 72 = 23·32

Direct product of C32 and Q8

direct product, metacyclic, nilpotent (class 2), monomial

Aliases: Q8×C32, C12.5C6, C2.2C62, C4.(C3×C6), C6.9(C2×C6), (C3×C12).5C2, (C3×C6).17C22, SmallGroup(72,38)

Series: Derived Chief Lower central Upper central

C1C2 — Q8×C32
C1C2C6C3×C6C3×C12 — Q8×C32
C1C2 — Q8×C32
C1C3×C6 — Q8×C32

Generators and relations for Q8×C32
 G = < a,b,c,d | a3=b3=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >


Smallest permutation representation of Q8×C32
Regular action on 72 points
Generators in S72
(1 40 31)(2 37 32)(3 38 29)(4 39 30)(5 35 27)(6 36 28)(7 33 25)(8 34 26)(9 24 14)(10 21 15)(11 22 16)(12 23 13)(17 54 50)(18 55 51)(19 56 52)(20 53 49)(41 71 45)(42 72 46)(43 69 47)(44 70 48)(57 68 61)(58 65 62)(59 66 63)(60 67 64)
(1 13 5)(2 14 6)(3 15 7)(4 16 8)(9 36 37)(10 33 38)(11 34 39)(12 35 40)(17 62 69)(18 63 70)(19 64 71)(20 61 72)(21 25 29)(22 26 30)(23 27 31)(24 28 32)(41 52 67)(42 49 68)(43 50 65)(44 51 66)(45 56 60)(46 53 57)(47 54 58)(48 55 59)
(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 52 3 50)(2 51 4 49)(5 41 7 43)(6 44 8 42)(9 63 11 61)(10 62 12 64)(13 67 15 65)(14 66 16 68)(17 40 19 38)(18 39 20 37)(21 58 23 60)(22 57 24 59)(25 47 27 45)(26 46 28 48)(29 54 31 56)(30 53 32 55)(33 69 35 71)(34 72 36 70)

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

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

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

Q8×C32 is a maximal subgroup of   C3211SD16  C327Q16  C12.26D6  Q8⋊3- 1+2  Q8⋊He3

45 conjugacy classes

class 1  2 3A···3H4A4B4C6A···6H12A···12X
order123···34446···612···12
size111···12221···12···2

45 irreducible representations

dim111122
type++-
imageC1C2C3C6Q8C3×Q8
kernelQ8×C32C3×C12C3×Q8C12C32C3
# reps1382418

Matrix representation of Q8×C32 in GL3(𝔽13) generated by

100
030
003
,
900
030
003
,
1200
001
0120
,
100
0910
0104
G:=sub<GL(3,GF(13))| [1,0,0,0,3,0,0,0,3],[9,0,0,0,3,0,0,0,3],[12,0,0,0,0,12,0,1,0],[1,0,0,0,9,10,0,10,4] >;

Q8×C32 in GAP, Magma, Sage, TeX

Q_8\times C_3^2
% in TeX

G:=Group("Q8xC3^2");
// GroupNames label

G:=SmallGroup(72,38);
// by ID

G=gap.SmallGroup(72,38);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-2,180,381,186]);
// Polycyclic

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

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Subgroup lattice of Q8×C32 in TeX

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