He3:2D8 - GroupNames

G = He₃⋊₂D₈ order 432 = 2⁴·3³

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

Aliases: He₃⋊₂D₈, C₁₂.₈₂S₃², (C₃×C₁₂).₂D₆, C₁₂⋊S₃⋊₁S₃, He₃⋊₄C₈⋊₁C₂, He₃⋊₄D₄⋊₁C₂, (C₂×He₃).₇D₄, C₃²⋊₁(D₄⋊S₃), C₄.₉(C₃²⋊D₆), C₆.₂(D₆⋊S₃), C₂.₄(He₃⋊₂D₄), C₃.₁(C₃²⋊₂D₈), (C₄×He₃).₂C₂², (C₃×C₆).₂(C₃⋊D₄), SmallGroup(432,79)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃ — C₄×He₃ — He₃⋊₂D₈

Chief series

C₁ — C₃ — C₃² — He₃ — C₂×He₃ — C₄×He₃ — He₃⋊₄D₄ — He₃⋊₂D₈

Lower central

He₃ — C₂×He₃ — C₄×He₃ — He₃⋊₂D₈

Upper central

C₁ — C₂ — C₄

Generators and relations for He₃⋊₂D₈
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, He₃⋊₄C₈, He₃⋊₄D₄, He₃⋊₂D₈
Quotients: C₁, C₂, C₂², S₃, D₄, D₆, D₈, C₃⋊D₄, S₃², D₄⋊S₃, D₆⋊S₃, C₃²⋊D₆, C₃²⋊₂D₈, He₃⋊₂D₄, He₃⋊₂D₈

Character table of He₃⋊₂D₈

class	1	2A	2B	2C	3A	3B	3C	3D	4	6A	6B	6C	6D	6E	6F	6G	6H	8A	8B	12A	12B	12C	12D	12E	12F	24A	24B	24C	24D
size	1	1	36	36	2	6	6	12	2	2	6	6	12	36	36	36	36	18	18	2	2	12	12	12	12	18	18	18	18
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	-1	-1	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	1	1	1	1	1	1	1	1	1	1	1	1	linear of order 2
ρ₃	1	1	-1	1	1	1	1	1	1	1	1	1	1	-1	1	1	-1	-1	-1	1	1	1	1	1	1	-1	-1	-1	-1	linear of order 2
ρ₄	1	1	1	-1	1	1	1	1	1	1	1	1	1	1	-1	-1	1	-1	-1	1	1	1	1	1	1	-1	-1	-1	-1	linear of order 2
ρ₅	2	2	0	-2	2	2	-1	-1	2	2	2	-1	-1	0	1	1	0	0	0	2	2	2	-1	-1	-1	0	0	0	0	orthogonal lifted from D₆
ρ₆	2	2	2	0	2	-1	2	-1	2	2	-1	2	-1	-1	0	0	-1	0	0	2	2	-1	-1	2	-1	0	0	0	0	orthogonal lifted from S₃
ρ₇	2	2	0	2	2	2	-1	-1	2	2	2	-1	-1	0	-1	-1	0	0	0	2	2	2	-1	-1	-1	0	0	0	0	orthogonal lifted from S₃
ρ₈	2	2	0	0	2	2	2	2	-2	2	2	2	2	0	0	0	0	0	0	-2	-2	-2	-2	-2	-2	0	0	0	0	orthogonal lifted from D₄
ρ₉	2	2	-2	0	2	-1	2	-1	2	2	-1	2	-1	1	0	0	1	0	0	2	2	-1	-1	2	-1	0	0	0	0	orthogonal lifted from D₆
ρ₁₀	2	-2	0	0	2	2	2	2	0	-2	-2	-2	-2	0	0	0	0	√2	-√2	0	0	0	0	0	0	-√2	-√2	√2	√2	orthogonal lifted from D₈
ρ₁₁	2	-2	0	0	2	2	2	2	0	-2	-2	-2	-2	0	0	0	0	-√2	√2	0	0	0	0	0	0	√2	√2	-√2	-√2	orthogonal lifted from D₈
ρ₁₂	2	2	0	0	2	2	-1	-1	-2	2	2	-1	-1	0	-√-3	√-3	0	0	0	-2	-2	-2	1	1	1	0	0	0	0	complex lifted from C₃⋊D₄
ρ₁₃	2	2	0	0	2	-1	2	-1	-2	2	-1	2	-1	-√-3	0	0	√-3	0	0	-2	-2	1	1	-2	1	0	0	0	0	complex lifted from C₃⋊D₄
ρ₁₄	2	2	0	0	2	-1	2	-1	-2	2	-1	2	-1	√-3	0	0	-√-3	0	0	-2	-2	1	1	-2	1	0	0	0	0	complex lifted from C₃⋊D₄
ρ₁₅	2	2	0	0	2	2	-1	-1	-2	2	2	-1	-1	0	√-3	-√-3	0	0	0	-2	-2	-2	1	1	1	0	0	0	0	complex lifted from C₃⋊D₄
ρ₁₆	4	4	0	0	4	-2	-2	1	4	4	-2	-2	1	0	0	0	0	0	0	4	4	-2	1	-2	1	0	0	0	0	orthogonal lifted from S₃²
ρ₁₇	4	-4	0	0	4	-2	4	-2	0	-4	2	-4	2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄⋊S₃, Schur index 2
ρ₁₈	4	-4	0	0	4	4	-2	-2	0	-4	-4	2	2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄⋊S₃, Schur index 2
ρ₁₉	4	4	0	0	4	-2	-2	1	-4	4	-2	-2	1	0	0	0	0	0	0	-4	-4	2	-1	2	-1	0	0	0	0	symplectic lifted from D₆⋊S₃, Schur index 2
ρ₂₀	4	-4	0	0	4	-2	-2	1	0	-4	2	2	-1	0	0	0	0	0	0	0	0	0	3i	0	-3i	0	0	0	0	complex lifted from C₃²⋊₂D₈
ρ₂₁	4	-4	0	0	4	-2	-2	1	0	-4	2	2	-1	0	0	0	0	0	0	0	0	0	-3i	0	3i	0	0	0	0	complex lifted from C₃²⋊₂D₈
ρ₂₂	6	6	0	0	-3	0	0	0	6	-3	0	0	0	0	0	0	0	2	2	-3	-3	0	0	0	0	-1	-1	-1	-1	orthogonal lifted from C₃²⋊D₆
ρ₂₃	6	6	0	0	-3	0	0	0	6	-3	0	0	0	0	0	0	0	-2	-2	-3	-3	0	0	0	0	1	1	1	1	orthogonal lifted from C₃²⋊D₆
ρ₂₄	6	6	0	0	-3	0	0	0	-6	-3	0	0	0	0	0	0	0	0	0	3	3	0	0	0	0	-√3	√3	√3	-√3	orthogonal lifted from He₃⋊₂D₄
ρ₂₅	6	6	0	0	-3	0	0	0	-6	-3	0	0	0	0	0	0	0	0	0	3	3	0	0	0	0	√3	-√3	-√3	√3	orthogonal lifted from He₃⋊₂D₄
ρ₂₆	6	-6	0	0	-3	0	0	0	0	3	0	0	0	0	0	0	0	-√2	√2	3√3	-3√3	0	0	0	0	ζ₈³ζ₃+ζ₈³+ζ₈ζ₃	ζ₈⁷ζ₃+ζ₈⁵ζ₃+ζ₈⁵	ζ₈⁷ζ₃²+ζ₈⁷+ζ₈⁵ζ₃²	ζ₈³ζ₃²+ζ₈ζ₃²+ζ₈	orthogonal faithful
ρ₂₇	6	-6	0	0	-3	0	0	0	0	3	0	0	0	0	0	0	0	√2	-√2	-3√3	3√3	0	0	0	0	ζ₈⁷ζ₃²+ζ₈⁷+ζ₈⁵ζ₃²	ζ₈³ζ₃²+ζ₈ζ₃²+ζ₈	ζ₈³ζ₃+ζ₈³+ζ₈ζ₃	ζ₈⁷ζ₃+ζ₈⁵ζ₃+ζ₈⁵	orthogonal faithful
ρ₂₈	6	-6	0	0	-3	0	0	0	0	3	0	0	0	0	0	0	0	-√2	√2	-3√3	3√3	0	0	0	0	ζ₈⁷ζ₃+ζ₈⁵ζ₃+ζ₈⁵	ζ₈³ζ₃+ζ₈³+ζ₈ζ₃	ζ₈³ζ₃²+ζ₈ζ₃²+ζ₈	ζ₈⁷ζ₃²+ζ₈⁷+ζ₈⁵ζ₃²	orthogonal faithful
ρ₂₉	6	-6	0	0	-3	0	0	0	0	3	0	0	0	0	0	0	0	√2	-√2	3√3	-3√3	0	0	0	0	ζ₈³ζ₃²+ζ₈ζ₃²+ζ₈	ζ₈⁷ζ₃²+ζ₈⁷+ζ₈⁵ζ₃²	ζ₈⁷ζ₃+ζ₈⁵ζ₃+ζ₈⁵	ζ₈³ζ₃+ζ₈³+ζ₈ζ₃	orthogonal faithful

Smallest permutation representation of He₃⋊₂D₈
►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 He₃⋊₂D₈ ►in GL₆(𝔽₇₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	72	1	0	0
0	0	72	0	0	0
0	0	0	0	0	72
0	0	0	0	1	72

72	1	0	0	0	0
72	0	0	0	0	0
0	0	72	1	0	0
0	0	72	0	0	0
0	0	0	0	72	1
0	0	0	0	72	0

0	0	1	0	0	0
0	0	0	1	0	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

50	5	0	0	0	0
68	55	0	0	0	0
0	0	0	0	50	5
0	0	0	0	68	55
0	0	50	5	0	0
0	0	68	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

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] >;

He₃⋊₂D₈ 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 He₃⋊₂D₈ in TeX

G = He3⋊2D8 order 432 = 24·33

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

G = He₃⋊₂D₈ order 432 = 2⁴·3³

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