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G = He34C8order 216 = 23·33

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

non-abelian, supersoluble, monomial

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

C1C3He3 — He34C8
C1C3C32He3C2×He3C4×He3 — He34C8
He3 — He34C8
C1C12

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 >

3C3
3C3
3C3
3C3
3C6
3C6
3C6
3C6
9C8
3C12
3C12
3C12
3C12
3C3⋊C8
3C3⋊C8
3C3⋊C8
3C3⋊C8
9C24
3C3×C3⋊C8
3C3×C3⋊C8
3C3×C3⋊C8
3C3×C3⋊C8

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 3A3B3C3D3E3F4A4B6A6B6C6D6E6F8A8B8C8D12A12B12C12D12E···12L24A···24H
order123333334466666688881212121212···1224···24
size1111666611116666999911116···69···9

40 irreducible representations

dim1111222333
type+++-
imageC1C2C4C8S3Dic3C3⋊C8He3⋊C2He33C4He34C8
kernelHe34C8C4×He3C2×He3He3C3×C12C3×C6C32C4C2C1
# reps1124448448

Matrix representation of He34C8 in GL3(𝔽73) generated by

010
001
100
,
800
080
008
,
0640
008
100
,
0063
0630
6300
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

Subgroup lattice of He34C8 in TeX

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