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## G = M4(2)×He3order 432 = 24·33

### Direct product of M4(2) and He3

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — M4(2)×He3
 Chief series C1 — C2 — C6 — C12 — C3×C12 — C4×He3 — C8×He3 — M4(2)×He3
 Lower central C1 — C6 — M4(2)×He3
 Upper central C1 — C12 — M4(2)×He3

Generators and relations for M4(2)×He3
G = < a,b,c,d,e | a8=b2=c3=d3=e3=1, bab=a5, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, de=ed >

Subgroups: 209 in 110 conjugacy classes, 63 normal (21 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C8, C2×C4, C32, C12, C12, C2×C6, C2×C6, M4(2), C3×C6, C3×C6, C24, C24, C2×C12, C2×C12, He3, C3×C12, C62, C3×M4(2), C3×M4(2), C2×He3, C2×He3, C3×C24, C6×C12, C4×He3, C22×He3, C32×M4(2), C8×He3, C2×C4×He3, M4(2)×He3
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C32, C12, C2×C6, M4(2), C3×C6, C2×C12, He3, C3×C12, C62, C3×M4(2), C2×He3, C6×C12, C4×He3, C22×He3, C32×M4(2), C2×C4×He3, M4(2)×He3

Smallest permutation representation of M4(2)×He3
On 72 points
Generators in S72
(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)(17 21)(19 23)(25 29)(27 31)(34 38)(36 40)(41 45)(43 47)(49 53)(51 55)(58 62)(60 64)(66 70)(68 72)
(1 50 67)(2 51 68)(3 52 69)(4 53 70)(5 54 71)(6 55 72)(7 56 65)(8 49 66)(9 61 35)(10 62 36)(11 63 37)(12 64 38)(13 57 39)(14 58 40)(15 59 33)(16 60 34)(17 27 41)(18 28 42)(19 29 43)(20 30 44)(21 31 45)(22 32 46)(23 25 47)(24 26 48)
(1 32 61)(2 25 62)(3 26 63)(4 27 64)(5 28 57)(6 29 58)(7 30 59)(8 31 60)(9 67 22)(10 68 23)(11 69 24)(12 70 17)(13 71 18)(14 72 19)(15 65 20)(16 66 21)(33 56 44)(34 49 45)(35 50 46)(36 51 47)(37 52 48)(38 53 41)(39 54 42)(40 55 43)
(1 67 35)(2 68 36)(3 69 37)(4 70 38)(5 71 39)(6 72 40)(7 65 33)(8 66 34)(9 46 61)(10 47 62)(11 48 63)(12 41 64)(13 42 57)(14 43 58)(15 44 59)(16 45 60)(17 53 27)(18 54 28)(19 55 29)(20 56 30)(21 49 31)(22 50 32)(23 51 25)(24 52 26)

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

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

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

110 conjugacy classes

 class 1 2A 2B 3A 3B 3C ··· 3J 4A 4B 4C 6A 6B 6C 6D 6E ··· 6L 6M ··· 6T 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G ··· 12V 12W ··· 12AD 24A ··· 24H 24I ··· 24AN order 1 2 2 3 3 3 ··· 3 4 4 4 6 6 6 6 6 ··· 6 6 ··· 6 8 8 8 8 12 12 12 12 12 12 12 ··· 12 12 ··· 12 24 ··· 24 24 ··· 24 size 1 1 2 1 1 3 ··· 3 1 1 2 1 1 2 2 3 ··· 3 6 ··· 6 2 2 2 2 1 1 1 1 2 2 3 ··· 3 6 ··· 6 2 ··· 2 6 ··· 6

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 3 3 3 3 3 6 type + + + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 M4(2) C3×M4(2) He3 C2×He3 C2×He3 C4×He3 C4×He3 M4(2)×He3 kernel M4(2)×He3 C8×He3 C2×C4×He3 C32×M4(2) C4×He3 C22×He3 C3×C24 C6×C12 C3×C12 C62 He3 C32 M4(2) C8 C2×C4 C4 C22 C1 # reps 1 2 1 8 2 2 16 8 16 16 2 16 2 4 2 4 4 4

Matrix representation of M4(2)×He3 in GL5(𝔽73)

 18 54 0 0 0 30 55 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 72
,
 1 0 0 0 0 48 72 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 40 9 65
,
 1 0 0 0 0 0 1 0 0 0 0 0 64 0 0 0 0 0 64 0 0 0 0 0 64
,
 8 0 0 0 0 0 8 0 0 0 0 0 64 16 18 0 0 40 9 65 0 0 0 8 0

G:=sub<GL(5,GF(73))| [18,30,0,0,0,54,55,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[1,48,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,8,0,40,0,0,0,0,9,0,0,0,1,65],[1,0,0,0,0,0,1,0,0,0,0,0,64,0,0,0,0,0,64,0,0,0,0,0,64],[8,0,0,0,0,0,8,0,0,0,0,0,64,40,0,0,0,16,9,8,0,0,18,65,0] >;

M4(2)×He3 in GAP, Magma, Sage, TeX

M_4(2)\times {\rm He}_3
% in TeX

G:=Group("M4(2)xHe3");
// GroupNames label

G:=SmallGroup(432,213);
// by ID

G=gap.SmallGroup(432,213);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-3,-2,252,3053,605,242]);
// Polycyclic

G:=Group<a,b,c,d,e|a^8=b^2=c^3=d^3=e^3=1,b*a*b=a^5,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c*d^-1,d*e=e*d>;
// generators/relations

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