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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C32×M4(2)
 Chief series C1 — C2 — C4 — C12 — C3×C12 — C3×C24 — C32×M4(2)
 Lower central C1 — C2 — C32×M4(2)
 Upper central C1 — C3×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, C4, C22, C6, C6, C8, C2×C4, C32, C12, C2×C6, M4(2), C3×C6, C3×C6, C24, C2×C12, C3×C12, C62, C3×M4(2), C3×C24, C6×C12, C32×M4(2)
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C32, C12, C2×C6, M4(2), C3×C6, C2×C12, C3×C12, C62, C3×M4(2), 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 65 48)(10 66 41)(11 67 42)(12 68 43)(13 69 44)(14 70 45)(15 71 46)(16 72 47)(17 29 53)(18 30 54)(19 31 55)(20 32 56)(21 25 49)(22 26 50)(23 27 51)(24 28 52)
(1 31 11)(2 32 12)(3 25 13)(4 26 14)(5 27 15)(6 28 16)(7 29 9)(8 30 10)(17 48 57)(18 41 58)(19 42 59)(20 43 60)(21 44 61)(22 45 62)(23 46 63)(24 47 64)(33 54 66)(34 55 67)(35 56 68)(36 49 69)(37 50 70)(38 51 71)(39 52 72)(40 53 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)(10 14)(12 16)(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)(9,65,48)(10,66,41)(11,67,42)(12,68,43)(13,69,44)(14,70,45)(15,71,46)(16,72,47)(17,29,53)(18,30,54)(19,31,55)(20,32,56)(21,25,49)(22,26,50)(23,27,51)(24,28,52), (1,31,11)(2,32,12)(3,25,13)(4,26,14)(5,27,15)(6,28,16)(7,29,9)(8,30,10)(17,48,57)(18,41,58)(19,42,59)(20,43,60)(21,44,61)(22,45,62)(23,46,63)(24,47,64)(33,54,66)(34,55,67)(35,56,68)(36,49,69)(37,50,70)(38,51,71)(39,52,72)(40,53,65), (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)(10,14)(12,16)(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)(9,65,48)(10,66,41)(11,67,42)(12,68,43)(13,69,44)(14,70,45)(15,71,46)(16,72,47)(17,29,53)(18,30,54)(19,31,55)(20,32,56)(21,25,49)(22,26,50)(23,27,51)(24,28,52), (1,31,11)(2,32,12)(3,25,13)(4,26,14)(5,27,15)(6,28,16)(7,29,9)(8,30,10)(17,48,57)(18,41,58)(19,42,59)(20,43,60)(21,44,61)(22,45,62)(23,46,63)(24,47,64)(33,54,66)(34,55,67)(35,56,68)(36,49,69)(37,50,70)(38,51,71)(39,52,72)(40,53,65), (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)(10,14)(12,16)(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),(9,65,48),(10,66,41),(11,67,42),(12,68,43),(13,69,44),(14,70,45),(15,71,46),(16,72,47),(17,29,53),(18,30,54),(19,31,55),(20,32,56),(21,25,49),(22,26,50),(23,27,51),(24,28,52)], [(1,31,11),(2,32,12),(3,25,13),(4,26,14),(5,27,15),(6,28,16),(7,29,9),(8,30,10),(17,48,57),(18,41,58),(19,42,59),(20,43,60),(21,44,61),(22,45,62),(23,46,63),(24,47,64),(33,54,66),(34,55,67),(35,56,68),(36,49,69),(37,50,70),(38,51,71),(39,52,72),(40,53,65)], [(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),(10,14),(12,16),(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 2A 2B 3A ··· 3H 4A 4B 4C 6A ··· 6H 6I ··· 6P 8A 8B 8C 8D 12A ··· 12P 12Q ··· 12X 24A ··· 24AF order 1 2 2 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 1 1 2 1 ··· 1 2 ··· 2 2 2 2 2 1 ··· 1 2 ··· 2 2 ··· 2

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 type + + + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 M4(2) C3×M4(2) kernel C32×M4(2) C3×C24 C6×C12 C3×M4(2) C3×C12 C62 C24 C2×C12 C12 C2×C6 C32 C3 # reps 1 2 1 8 2 2 16 8 16 16 2 16

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

 1 0 0 0 64 0 0 0 64
,
 8 0 0 0 8 0 0 0 8
,
 72 0 0 0 0 1 0 27 0
,
 72 0 0 0 1 0 0 0 72
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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