Copied to
clipboard

## G = He3⋊4C8order 216 = 23·33

### 2nd semidirect product of He3 and C8 acting via C8/C4=C2

Aliases: He34C8, C323(C3⋊C8), (C3×C12).7S3, C2.(He33C4), (C4×He3).4C2, (C2×He3).3C4, C12.10(C3⋊S3), (C3×C6).3Dic3, C6.4(C3⋊Dic3), C4.2(He3⋊C2), C3.2(C324C8), SmallGroup(216,17)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — He3⋊4C8
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C4×He3 — He3⋊4C8
 Lower central He3 — He3⋊4C8
 Upper central C1 — C12

Generators and relations for He34C8
G = < a,b,c,d | a3=b3=c3=d8=1, ab=ba, cac-1=ab-1, dad-1=a-1, bc=cb, bd=db, dcd-1=c-1 >

Smallest permutation representation of He34C8
On 72 points
Generators in S72
(1 68 56)(2 49 69)(3 70 50)(4 51 71)(5 72 52)(6 53 65)(7 66 54)(8 55 67)(9 57 33)(10 34 58)(11 59 35)(12 36 60)(13 61 37)(14 38 62)(15 63 39)(16 40 64)(17 41 27)(18 28 42)(19 43 29)(20 30 44)(21 45 31)(22 32 46)(23 47 25)(24 26 48)
(1 15 41)(2 16 42)(3 9 43)(4 10 44)(5 11 45)(6 12 46)(7 13 47)(8 14 48)(17 56 39)(18 49 40)(19 50 33)(20 51 34)(21 52 35)(22 53 36)(23 54 37)(24 55 38)(25 66 61)(26 67 62)(27 68 63)(28 69 64)(29 70 57)(30 71 58)(31 72 59)(32 65 60)
(1 68 39)(2 40 69)(3 70 33)(4 34 71)(5 72 35)(6 36 65)(7 66 37)(8 38 67)(9 57 19)(10 20 58)(11 59 21)(12 22 60)(13 61 23)(14 24 62)(15 63 17)(16 18 64)(25 54 47)(26 48 55)(27 56 41)(28 42 49)(29 50 43)(30 44 51)(31 52 45)(32 46 53)
(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)

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

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

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

He34C8 is a maximal subgroup of
He32C16  C32⋊C6⋊C8  He3⋊M4(2)  He33SD16  He32D8  He32Q16  C8×He3⋊C2  He36M4(2)  He38M4(2)  He37D8  He39SD16  He311SD16  He37Q16
He34C8 is a maximal quotient of
He34C16

40 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 4A 4B 6A 6B 6C 6D 6E 6F 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12L 24A ··· 24H order 1 2 3 3 3 3 3 3 4 4 6 6 6 6 6 6 8 8 8 8 12 12 12 12 12 ··· 12 24 ··· 24 size 1 1 1 1 6 6 6 6 1 1 1 1 6 6 6 6 9 9 9 9 1 1 1 1 6 ··· 6 9 ··· 9

40 irreducible representations

 dim 1 1 1 1 2 2 2 3 3 3 type + + + - image C1 C2 C4 C8 S3 Dic3 C3⋊C8 He3⋊C2 He3⋊3C4 He3⋊4C8 kernel He3⋊4C8 C4×He3 C2×He3 He3 C3×C12 C3×C6 C32 C4 C2 C1 # reps 1 1 2 4 4 4 8 4 4 8

Matrix representation of He34C8 in GL3(𝔽73) generated by

 0 1 0 0 0 1 1 0 0
,
 8 0 0 0 8 0 0 0 8
,
 0 64 0 0 0 8 1 0 0
,
 0 0 63 0 63 0 63 0 0
G:=sub<GL(3,GF(73))| [0,0,1,1,0,0,0,1,0],[8,0,0,0,8,0,0,0,8],[0,0,1,64,0,0,0,8,0],[0,0,63,0,63,0,63,0,0] >;

He34C8 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_4C_8
% in TeX

G:=Group("He3:4C8");
// GroupNames label

G:=SmallGroup(216,17);
// by ID

G=gap.SmallGroup(216,17);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,12,31,387,1444,382]);
// Polycyclic

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

Export

׿
×
𝔽