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G = He32D8order 432 = 24·33

1st semidirect product of He3 and D8 acting via D8/C4=C22

non-abelian, supersoluble, monomial

Aliases: He32D8, C12.82S32, (C3×C12).2D6, C12⋊S31S3, He34C81C2, He34D41C2, (C2×He3).7D4, C321(D4⋊S3), C4.9(C32⋊D6), C6.2(D6⋊S3), C2.4(He32D4), C3.1(C322D8), (C4×He3).2C22, (C3×C6).2(C3⋊D4), SmallGroup(432,79)

Series: Derived Chief Lower central Upper central

C1C3C4×He3 — He32D8
C1C3C32He3C2×He3C4×He3He34D4 — He32D8
He3C2×He3C4×He3 — He32D8
C1C2C4

Generators and relations for He32D8
 G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, cac-1=ab-1, dad-1=eae=a-1, bc=cb, bd=db, ebe=b-1, dcd-1=c-1, ce=ec, ede=d-1 >

Subgroups: 651 in 85 conjugacy classes, 21 normal (11 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, C32, C32, C12, C12, D6, C2×C6, D8, C3×S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, D12, C3×D4, He3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, D24, D4⋊S3, C32⋊C6, C2×He3, C3×C3⋊C8, C3×D12, C12⋊S3, C4×He3, C2×C32⋊C6, C3⋊D24, He34C8, He34D4, He32D8
Quotients: C1, C2, C22, S3, D4, D6, D8, C3⋊D4, S32, D4⋊S3, D6⋊S3, C32⋊D6, C322D8, He32D4, He32D8

Character table of He32D8

 class 12A2B2C3A3B3C3D46A6B6C6D6E6F6G6H8A8B12A12B12C12D12E12F24A24B24C24D
 size 11363626612226612363636361818221212121218181818
ρ111111111111111111111111111111    trivial
ρ211-1-1111111111-1-1-1-1111111111111    linear of order 2
ρ311-11111111111-111-1-1-1111111-1-1-1-1    linear of order 2
ρ4111-11111111111-1-11-1-1111111-1-1-1-1    linear of order 2
ρ5220-222-1-1222-1-1011000222-1-1-10000    orthogonal lifted from D6
ρ622202-12-122-12-1-100-10022-1-12-10000    orthogonal lifted from S3
ρ7220222-1-1222-1-10-1-1000222-1-1-10000    orthogonal lifted from S3
ρ822002222-22222000000-2-2-2-2-2-20000    orthogonal lifted from D4
ρ922-202-12-122-12-110010022-1-12-10000    orthogonal lifted from D6
ρ102-20022220-2-2-2-200002-2000000-2-222    orthogonal lifted from D8
ρ112-20022220-2-2-2-20000-2200000022-2-2    orthogonal lifted from D8
ρ12220022-1-1-222-1-10--3-3000-2-2-21110000    complex lifted from C3⋊D4
ρ1322002-12-1-22-12-1--300-300-2-211-210000    complex lifted from C3⋊D4
ρ1422002-12-1-22-12-1-300--300-2-211-210000    complex lifted from C3⋊D4
ρ15220022-1-1-222-1-10-3--3000-2-2-21110000    complex lifted from C3⋊D4
ρ1644004-2-2144-2-2100000044-21-210000    orthogonal lifted from S32
ρ174-4004-24-20-42-420000000000000000    orthogonal lifted from D4⋊S3, Schur index 2
ρ184-40044-2-20-4-4220000000000000000    orthogonal lifted from D4⋊S3, Schur index 2
ρ1944004-2-21-44-2-21000000-4-42-12-10000    symplectic lifted from D6⋊S3, Schur index 2
ρ204-4004-2-210-422-10000000003i0-3i0000    complex lifted from C322D8
ρ214-4004-2-210-422-1000000000-3i03i0000    complex lifted from C322D8
ρ226600-30006-3000000022-3-30000-1-1-1-1    orthogonal lifted from C32⋊D6
ρ236600-30006-30000000-2-2-3-300001111    orthogonal lifted from C32⋊D6
ρ246600-3000-6-3000000000330000-333-3    orthogonal lifted from He32D4
ρ256600-3000-6-30000000003300003-3-33    orthogonal lifted from He32D4
ρ266-600-3000030000000-2233-330000ζ83ζ3838ζ3ζ87ζ385ζ385ζ87ζ328785ζ32ζ83ζ328ζ328    orthogonal faithful
ρ276-600-30000300000002-2-33330000ζ87ζ328785ζ32ζ83ζ328ζ328ζ83ζ3838ζ3ζ87ζ385ζ385    orthogonal faithful
ρ286-600-3000030000000-22-33330000ζ87ζ385ζ385ζ83ζ3838ζ3ζ83ζ328ζ328ζ87ζ328785ζ32    orthogonal faithful
ρ296-600-30000300000002-233-330000ζ83ζ328ζ328ζ87ζ328785ζ32ζ87ζ385ζ385ζ83ζ3838ζ3    orthogonal faithful

Smallest permutation representation of He32D8
On 72 points
Generators in S72
(25 51 46)(26 47 52)(27 53 48)(28 41 54)(29 55 42)(30 43 56)(31 49 44)(32 45 50)(33 60 70)(34 71 61)(35 62 72)(36 65 63)(37 64 66)(38 67 57)(39 58 68)(40 69 59)
(1 20 13)(2 21 14)(3 22 15)(4 23 16)(5 24 9)(6 17 10)(7 18 11)(8 19 12)(25 51 46)(26 52 47)(27 53 48)(28 54 41)(29 55 42)(30 56 43)(31 49 44)(32 50 45)(33 60 70)(34 61 71)(35 62 72)(36 63 65)(37 64 66)(38 57 67)(39 58 68)(40 59 69)
(1 61 27)(2 28 62)(3 63 29)(4 30 64)(5 57 31)(6 32 58)(7 59 25)(8 26 60)(9 38 44)(10 45 39)(11 40 46)(12 47 33)(13 34 48)(14 41 35)(15 36 42)(16 43 37)(17 50 68)(18 69 51)(19 52 70)(20 71 53)(21 54 72)(22 65 55)(23 56 66)(24 67 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 8)(3 7)(4 6)(9 24)(10 23)(11 22)(12 21)(13 20)(14 19)(15 18)(16 17)(25 29)(26 28)(30 32)(33 72)(34 71)(35 70)(36 69)(37 68)(38 67)(39 66)(40 65)(41 52)(42 51)(43 50)(44 49)(45 56)(46 55)(47 54)(48 53)(58 64)(59 63)(60 62)

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

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

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

Matrix representation of He32D8 in GL6(𝔽73)

100000
010000
0072100
0072000
0000072
0000172
,
7210000
7200000
0072100
0072000
0000721
0000720
,
001000
000100
000010
000001
100000
010000
,
5050000
68550000
0000505
00006855
0050500
00685500
,
010000
100000
000100
001000
000001
000010

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[50,68,0,0,0,0,5,55,0,0,0,0,0,0,0,0,50,68,0,0,0,0,5,55,0,0,50,68,0,0,0,0,5,55,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

He32D8 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_2D_8
% in TeX

G:=Group("He3:2D8");
// GroupNames label

G:=SmallGroup(432,79);
// by ID

G=gap.SmallGroup(432,79);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,254,135,58,571,4037,537,14118,7069]);
// 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=e*a*e=a^-1,b*c=c*b,b*d=d*b,e*b*e=b^-1,d*c*d^-1=c^-1,c*e=e*c,e*d*e=d^-1>;
// generators/relations

Export

Character table of He32D8 in TeX

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