Copied to
clipboard

## G = He3⋊8M4(2)  order 432 = 24·33

### 2nd semidirect product of He3 and M4(2) acting via M4(2)/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×He3 — He3⋊8M4(2)
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C4×He3 — He3⋊4C8 — He3⋊8M4(2)
 Lower central He3 — C2×He3 — He3⋊8M4(2)
 Upper central C1 — C12 — C2×C12

Generators and relations for He38M4(2)
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, cac-1=ab-1, dad-1=a-1, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ce=ec, ede=d5 >

Subgroups: 281 in 110 conjugacy classes, 45 normal (19 characteristic)
C1, C2, C2, C3, C3 [×4], C4 [×2], C22, C6, C6 [×9], C8 [×2], C2×C4, C32 [×4], C12 [×2], C12 [×8], C2×C6, C2×C6 [×4], M4(2), C3×C6 [×4], C3×C6 [×4], C3⋊C8 [×8], C24 [×2], C2×C12, C2×C12 [×4], He3, C3×C12 [×8], C62 [×4], C4.Dic3 [×4], C3×M4(2), C2×He3, C2×He3, C3×C3⋊C8 [×8], C6×C12 [×4], C4×He3 [×2], C22×He3, C3×C4.Dic3 [×4], He34C8 [×2], C2×C4×He3, He38M4(2)
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, Dic3 [×8], D6 [×4], M4(2), C3⋊S3, C2×Dic3 [×4], C3⋊Dic3 [×2], C2×C3⋊S3, C4.Dic3 [×4], He3⋊C2, C2×C3⋊Dic3, He33C4 [×2], C2×He3⋊C2, C12.58D6, C2×He33C4, He38M4(2)

Smallest permutation representation of He38M4(2)
On 72 points
Generators in S72
(1 38 29)(2 30 39)(3 40 31)(4 32 33)(5 34 25)(6 26 35)(7 36 27)(8 28 37)(9 43 52)(10 53 44)(11 45 54)(12 55 46)(13 47 56)(14 49 48)(15 41 50)(16 51 42)(17 57 67)(18 68 58)(19 59 69)(20 70 60)(21 61 71)(22 72 62)(23 63 65)(24 66 64)
(1 41 18)(2 42 19)(3 43 20)(4 44 21)(5 45 22)(6 46 23)(7 47 24)(8 48 17)(9 60 31)(10 61 32)(11 62 25)(12 63 26)(13 64 27)(14 57 28)(15 58 29)(16 59 30)(33 53 71)(34 54 72)(35 55 65)(36 56 66)(37 49 67)(38 50 68)(39 51 69)(40 52 70)
(1 38 15)(2 16 39)(3 40 9)(4 10 33)(5 34 11)(6 12 35)(7 36 13)(8 14 37)(17 28 67)(18 68 29)(19 30 69)(20 70 31)(21 32 71)(22 72 25)(23 26 65)(24 66 27)(41 50 58)(42 59 51)(43 52 60)(44 61 53)(45 54 62)(46 63 55)(47 56 64)(48 57 49)
(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)(26 30)(28 32)(33 37)(35 39)(42 46)(44 48)(49 53)(51 55)(57 61)(59 63)(65 69)(67 71)

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

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

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

62 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 3E 3F 4A 4B 4C 6A 6B 6C 6D 6E ··· 6P 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G ··· 12V 24A ··· 24H order 1 2 2 3 3 3 3 3 3 4 4 4 6 6 6 6 6 ··· 6 8 8 8 8 12 12 12 12 12 12 12 ··· 12 24 ··· 24 size 1 1 2 1 1 6 6 6 6 1 1 2 1 1 2 2 6 ··· 6 18 18 18 18 1 1 1 1 2 2 6 ··· 6 18 ··· 18

62 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 6 type + + + + - + - image C1 C2 C2 C4 C4 S3 Dic3 D6 Dic3 M4(2) C4.Dic3 He3⋊C2 He3⋊3C4 C2×He3⋊C2 He3⋊3C4 He3⋊8M4(2) kernel He3⋊8M4(2) He3⋊4C8 C2×C4×He3 C4×He3 C22×He3 C6×C12 C3×C12 C3×C12 C62 He3 C32 C2×C4 C4 C4 C22 C1 # reps 1 2 1 2 2 4 4 4 4 2 16 4 4 4 4 4

Matrix representation of He38M4(2) in GL5(𝔽73)

 8 0 0 0 0 49 64 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 8 0 0 0 0 0 8 0 0 0 0 0 8
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 64 0 0 0 0 0 8 0 0 1 0 0
,
 3 66 0 0 0 60 70 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0
,
 1 0 0 0 0 53 72 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 72

G:=sub<GL(5,GF(73))| [8,49,0,0,0,0,64,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,8],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,64,0,0,0,0,0,8,0],[3,60,0,0,0,66,70,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0],[1,53,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72] >;

He38M4(2) in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_8M_4(2)
% in TeX

G:=Group("He3:8M4(2)");
// GroupNames label

G:=SmallGroup(432,185);
// by ID

G=gap.SmallGroup(432,185);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,141,58,1124,4037,537]);
// Polycyclic

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

׿
×
𝔽