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G = C32×M4(2)  order 144 = 24·32

Direct product of C32 and M4(2)

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

Aliases: C32×M4(2), C247C6, C12.6C12, C62.3C4, C4.6C62, C83(C3×C6), C4.(C3×C12), (C3×C24)⋊11C2, C2.3(C6×C12), (C2×C6).4C12, C22.(C3×C12), (C6×C12).14C2, C6.15(C2×C12), (C2×C12).14C6, (C3×C12).10C4, C12.29(C2×C6), (C3×C12).54C22, (C2×C4).2(C3×C6), (C3×C6).37(C2×C4), SmallGroup(144,105)

Series: Derived Chief Lower central Upper central

C1C2 — C32×M4(2)
C1C2C4C12C3×C12C3×C24 — C32×M4(2)
C1C2 — C32×M4(2)
C1C3×C12 — C32×M4(2)

Generators and relations for C32×M4(2)
 G = < a,b,c,d | a3=b3=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c5 >

Subgroups: 66 in 60 conjugacy classes, 54 normal (14 characteristic)
C1, C2, C2, C3 [×4], C4 [×2], C22, C6 [×4], C6 [×4], C8 [×2], C2×C4, C32, C12 [×8], C2×C6 [×4], M4(2), C3×C6, C3×C6, C24 [×8], C2×C12 [×4], C3×C12 [×2], C62, C3×M4(2) [×4], C3×C24 [×2], C6×C12, C32×M4(2)
Quotients: C1, C2 [×3], C3 [×4], C4 [×2], C22, C6 [×12], C2×C4, C32, C12 [×8], C2×C6 [×4], M4(2), C3×C6 [×3], C2×C12 [×4], C3×C12 [×2], C62, C3×M4(2) [×4], C6×C12, C32×M4(2)

Smallest permutation representation of C32×M4(2)
On 72 points
Generators in S72
(1 34 59)(2 35 60)(3 36 61)(4 37 62)(5 38 63)(6 39 64)(7 40 57)(8 33 58)(9 26 52)(10 27 53)(11 28 54)(12 29 55)(13 30 56)(14 31 49)(15 32 50)(16 25 51)(17 65 48)(18 66 41)(19 67 42)(20 68 43)(21 69 44)(22 70 45)(23 71 46)(24 72 47)
(1 31 19)(2 32 20)(3 25 21)(4 26 22)(5 27 23)(6 28 24)(7 29 17)(8 30 18)(9 45 62)(10 46 63)(11 47 64)(12 48 57)(13 41 58)(14 42 59)(15 43 60)(16 44 61)(33 56 66)(34 49 67)(35 50 68)(36 51 69)(37 52 70)(38 53 71)(39 54 72)(40 55 65)
(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 6)(4 8)(9 13)(11 15)(18 22)(20 24)(26 30)(28 32)(33 37)(35 39)(41 45)(43 47)(50 54)(52 56)(58 62)(60 64)(66 70)(68 72)

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

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

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

C32×M4(2) is a maximal subgroup of   C62.8Q8  C12.19D12  C12.20D12  C62.37D4  C24.47D6  C243D6  C24.5D6

90 conjugacy classes

class 1 2A2B3A···3H4A4B4C6A···6H6I···6P8A8B8C8D12A···12P12Q···12X24A···24AF
order1223···34446···66···6888812···1212···1224···24
size1121···11121···12···222221···12···22···2

90 irreducible representations

dim111111111122
type+++
imageC1C2C2C3C4C4C6C6C12C12M4(2)C3×M4(2)
kernelC32×M4(2)C3×C24C6×C12C3×M4(2)C3×C12C62C24C2×C12C12C2×C6C32C3
# reps1218221681616216

Matrix representation of C32×M4(2) in GL3(𝔽73) generated by

100
0640
0064
,
800
080
008
,
7200
001
0270
,
7200
010
0072
G:=sub<GL(3,GF(73))| [1,0,0,0,64,0,0,0,64],[8,0,0,0,8,0,0,0,8],[72,0,0,0,0,27,0,1,0],[72,0,0,0,1,0,0,0,72] >;

C32×M4(2) in GAP, Magma, Sage, TeX

C_3^2\times M_4(2)
% in TeX

G:=Group("C3^2xM4(2)");
// GroupNames label

G:=SmallGroup(144,105);
// by ID

G=gap.SmallGroup(144,105);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-2,-2,216,889,88]);
// 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^5>;
// generators/relations

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