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G = He3:7D8order 432 = 24·33

2nd semidirect product of He3 and D8 acting via D8/D4=C2

non-abelian, supersoluble, monomial

Aliases: He3:7D8, (D4xHe3):2C2, D4:(He3:C2), He3:4C8:3C2, He3:5D4:2C2, (C3xC12).16D6, (D4xC32):2S3, C32:5(D4:S3), (C2xHe3).34D4, C2.4(He3:7D4), C3.2(C32:7D8), C6.40(C32:7D4), (C4xHe3).12C22, C12.44(C2xC3:S3), (C3xD4).6(C3:S3), C4.1(C2xHe3:C2), (C3xC6).35(C3:D4), SmallGroup(432,192)

Series: Derived Chief Lower central Upper central

C1C3C4xHe3 — He3:7D8
C1C3C32He3C2xHe3C4xHe3He3:5D4 — He3:7D8
He3C2xHe3C4xHe3 — He3:7D8
C1C6C12C3xD4

Generators and relations for He3:7D8
 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, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 577 in 121 conjugacy classes, 31 normal (15 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, D4, C32, C12, C12, D6, C2xC6, D8, C3xS3, C3xC6, C3xC6, C3:C8, C24, D12, C3xD4, C3xD4, He3, C3xC12, S3xC6, C62, D4:S3, C3xD8, He3:C2, C2xHe3, C2xHe3, C3xC3:C8, C3xD12, D4xC32, C4xHe3, C2xHe3:C2, C22xHe3, C3xD4:S3, He3:4C8, He3:5D4, D4xHe3, He3:7D8
Quotients: C1, C2, C22, S3, D4, D6, D8, C3:S3, C3:D4, C2xC3:S3, D4:S3, He3:C2, C32:7D4, C2xHe3:C2, C32:7D8, He3:7D4, He3:7D8

Smallest permutation representation of He3:7D8
On 72 points
Generators in S72
(1 43 59)(2 60 44)(3 45 61)(4 62 46)(5 47 63)(6 64 48)(7 41 57)(8 58 42)(9 26 37)(10 38 27)(11 28 39)(12 40 29)(13 30 33)(14 34 31)(15 32 35)(16 36 25)(17 54 67)(18 68 55)(19 56 69)(20 70 49)(21 50 71)(22 72 51)(23 52 65)(24 66 53)
(1 31 20)(2 32 21)(3 25 22)(4 26 23)(5 27 24)(6 28 17)(7 29 18)(8 30 19)(9 65 46)(10 66 47)(11 67 48)(12 68 41)(13 69 42)(14 70 43)(15 71 44)(16 72 45)(33 56 58)(34 49 59)(35 50 60)(36 51 61)(37 52 62)(38 53 63)(39 54 64)(40 55 57)
(1 70 49)(2 50 71)(3 72 51)(4 52 65)(5 66 53)(6 54 67)(7 68 55)(8 56 69)(9 23 37)(10 38 24)(11 17 39)(12 40 18)(13 19 33)(14 34 20)(15 21 35)(16 36 22)(25 45 61)(26 62 46)(27 47 63)(28 64 48)(29 41 57)(30 58 42)(31 43 59)(32 60 44)
(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 39)(10 38)(11 37)(12 36)(13 35)(14 34)(15 33)(16 40)(17 23)(18 22)(19 21)(25 29)(26 28)(30 32)(41 61)(42 60)(43 59)(44 58)(45 57)(46 64)(47 63)(48 62)(49 70)(50 69)(51 68)(52 67)(53 66)(54 65)(55 72)(56 71)

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

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

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

41 conjugacy classes

class 1 2A2B2C3A3B3C3D3E3F 4 6A6B6C6D6E6F6G6H6I···6P6Q6R8A8B12A12B12C12D12E12F24A24B24C24D
order12223333334666666666···6668812121212121224242424
size1143611666621144666612···1236361818221212121218181818

41 irreducible representations

dim11112222233466
type+++++++++
imageC1C2C2C2S3D4D6D8C3:D4He3:C2C2xHe3:C2D4:S3He3:7D4He3:7D8
kernelHe3:7D8He3:4C8He3:5D4D4xHe3D4xC32C2xHe3C3xC12He3C3xC6D4C4C32C2C1
# reps11114142844424

Matrix representation of He3:7D8 in GL5(F73)

10000
01000
00010
00001
00100
,
10000
01000
00800
00080
00008
,
10000
01000
00080
00001
006400
,
014000
2641000
00100
00001
00010
,
10000
2972000
00100
00001
00010

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[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],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,64,0,0,8,0,0,0,0,0,1,0],[0,26,0,0,0,14,41,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0],[1,29,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

He3:7D8 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_7D_8
% in TeX

G:=Group("He3:7D8");
// GroupNames label

G:=SmallGroup(432,192);
// by ID

G=gap.SmallGroup(432,192);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,254,135,58,1124,4037,537]);
// 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=e*a*e=a^-1,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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