He3:2SD16 - GroupNames

G = He₃⋊₂SD₁₆ order 432 = 2⁴·3³

The semidirect product of He₃ and SD₁₆ acting via SD₁₆/C₂=D₄

Aliases: He₃⋊₂SD₁₆, C₆.₁S₃≀C₂, He₃⋊₂D₄.C₂, He₃⋊₂C₈⋊₂C₂, He₃⋊₂Q₈⋊₁C₂, C₂.₃(He₃⋊D₄), (C₂×He₃).₁D₄, C₃.(C₃²⋊₂SD₁₆), He₃⋊₃C₄.₁C₂², SmallGroup(432,234)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃ — He₃ — He₃⋊₃C₄ — He₃⋊₂SD₁₆

Chief series

C₁ — C₃ — He₃ — C₂×He₃ — He₃⋊₃C₄ — He₃⋊₂D₄ — He₃⋊₂SD₁₆

Lower central

He₃ — C₂×He₃ — He₃⋊₃C₄ — He₃⋊₂SD₁₆

Upper central

C₁ — C₂

Generators and relations for He₃⋊₂SD₁₆
G = < a,b,c,d,e | a³=b³=c³=d⁸=e²=1, eae=ab=ba, cac^-1=ab^-1, dad^-1=cb=bc, bd=db, ebe=b^-1, dcd^-1=a^-1b, ece=c^-1, ede=d³ >

Subgroups: 539 in 58 conjugacy classes, 11 normal (all characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², S₃, C₆, C₆, C₈, D₄, Q₈, C₃², Dic₃, C₁₂, D₆, C₂×C₆, SD₁₆, C₃×S₃, C₃⋊S₃, C₃×C₆, C₂₄, Dic₆, D₁₂, C₃⋊D₄, He₃, C₃×Dic₃, C₃⋊Dic₃, S₃×C₆, C₂×C₃⋊S₃, C₂₄⋊C₂, C₃²⋊C₆, C₂×He₃, C₃⋊D₁₂, C₃²⋊₂Q₈, C₃²⋊C₁₂, He₃⋊₃C₄, C₂×C₃²⋊C₆, He₃⋊₂C₈, He₃⋊₂Q₈, He₃⋊₂D₄, He₃⋊₂SD₁₆
Quotients: C₁, C₂, C₂², D₄, SD₁₆, S₃≀C₂, C₃²⋊₂SD₁₆, He₃⋊D₄, He₃⋊₂SD₁₆

Character table of He₃⋊₂SD₁₆

class	1	2A	2B	3A	3B	3C	4A	4B	6A	6B	6C	6D	6E	8A	8B	12A	12B	12C	12D	24A	24B	24C	24D
size	1	1	36	2	12	12	18	36	2	12	12	36	36	18	18	18	18	36	36	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	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	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	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	linear of order 2
ρ₅	2	2	0	2	2	2	-2	0	2	2	2	0	0	0	0	-2	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₆	2	-2	0	2	2	2	0	0	-2	-2	-2	0	0	√-2	-√-2	0	0	0	0	√-2	-√-2	√-2	-√-2	complex lifted from SD₁₆
ρ₇	2	-2	0	2	2	2	0	0	-2	-2	-2	0	0	-√-2	√-2	0	0	0	0	-√-2	√-2	-√-2	√-2	complex lifted from SD₁₆
ρ₈	4	4	0	4	1	-2	0	-2	4	-2	1	0	0	0	0	0	0	1	1	0	0	0	0	orthogonal lifted from S₃≀C₂
ρ₉	4	4	0	4	1	-2	0	2	4	-2	1	0	0	0	0	0	0	-1	-1	0	0	0	0	orthogonal lifted from S₃≀C₂
ρ₁₀	4	4	2	4	-2	1	0	0	4	1	-2	-1	-1	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from S₃≀C₂
ρ₁₁	4	4	-2	4	-2	1	0	0	4	1	-2	1	1	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from S₃≀C₂
ρ₁₂	4	-4	0	4	1	-2	0	0	-4	2	-1	0	0	0	0	0	0	√3	-√3	0	0	0	0	symplectic lifted from C₃²⋊₂SD₁₆, Schur index 2
ρ₁₃	4	-4	0	4	1	-2	0	0	-4	2	-1	0	0	0	0	0	0	-√3	√3	0	0	0	0	symplectic lifted from C₃²⋊₂SD₁₆, Schur index 2
ρ₁₄	4	-4	0	4	-2	1	0	0	-4	-1	2	-√-3	√-3	0	0	0	0	0	0	0	0	0	0	complex lifted from C₃²⋊₂SD₁₆
ρ₁₅	4	-4	0	4	-2	1	0	0	-4	-1	2	√-3	-√-3	0	0	0	0	0	0	0	0	0	0	complex lifted from C₃²⋊₂SD₁₆
ρ₁₆	6	6	0	-3	0	0	-2	0	-3	0	0	0	0	2	2	1	1	0	0	-1	-1	-1	-1	orthogonal lifted from He₃⋊D₄
ρ₁₇	6	6	0	-3	0	0	-2	0	-3	0	0	0	0	-2	-2	1	1	0	0	1	1	1	1	orthogonal lifted from He₃⋊D₄
ρ₁₈	6	6	0	-3	0	0	2	0	-3	0	0	0	0	0	0	-1	-1	0	0	√3	-√3	-√3	√3	orthogonal lifted from He₃⋊D₄
ρ₁₉	6	6	0	-3	0	0	2	0	-3	0	0	0	0	0	0	-1	-1	0	0	-√3	√3	√3	-√3	orthogonal lifted from He₃⋊D₄
ρ₂₀	6	-6	0	-3	0	0	0	0	3	0	0	0	0	-√-2	√-2	√3	-√3	0	0	ζ₈³ζ₃+ζ₈³-ζ₈ζ₃	ζ₈⁷ζ₃²+ζ₈⁷-ζ₈⁵ζ₃²	ζ₈³ζ₃²+ζ₈³-ζ₈ζ₃²	ζ₈⁷ζ₃+ζ₈⁷-ζ₈⁵ζ₃	complex faithful
ρ₂₁	6	-6	0	-3	0	0	0	0	3	0	0	0	0	-√-2	√-2	-√3	√3	0	0	ζ₈³ζ₃²+ζ₈³-ζ₈ζ₃²	ζ₈⁷ζ₃+ζ₈⁷-ζ₈⁵ζ₃	ζ₈³ζ₃+ζ₈³-ζ₈ζ₃	ζ₈⁷ζ₃²+ζ₈⁷-ζ₈⁵ζ₃²	complex faithful
ρ₂₂	6	-6	0	-3	0	0	0	0	3	0	0	0	0	√-2	-√-2	-√3	√3	0	0	ζ₈⁷ζ₃²+ζ₈⁷-ζ₈⁵ζ₃²	ζ₈³ζ₃+ζ₈³-ζ₈ζ₃	ζ₈⁷ζ₃+ζ₈⁷-ζ₈⁵ζ₃	ζ₈³ζ₃²+ζ₈³-ζ₈ζ₃²	complex faithful
ρ₂₃	6	-6	0	-3	0	0	0	0	3	0	0	0	0	√-2	-√-2	√3	-√3	0	0	ζ₈⁷ζ₃+ζ₈⁷-ζ₈⁵ζ₃	ζ₈³ζ₃²+ζ₈³-ζ₈ζ₃²	ζ₈⁷ζ₃²+ζ₈⁷-ζ₈⁵ζ₃²	ζ₈³ζ₃+ζ₈³-ζ₈ζ₃	complex faithful

Smallest permutation representation of He₃⋊₂SD₁₆
►On 72 points

Generators in S₇₂

(1 41 15)(2 32 64)(3 43 9)(4 21 33)(5 45 11)(6 28 60)(7 47 13)(8 17 37)(10 66 26)(12 53 23)(14 70 30)(16 49 19)(18 71 63)(22 67 59)(25 50 40)(29 54 36)(35 68 46)(39 72 42)(44 58 51)(48 62 55)

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

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

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

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

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

Matrix representation of He₃⋊₂SD₁₆ ►in GL₆(𝔽₇₃)

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

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

40	4	40	4	40	4
69	44	69	44	69	44
40	4	4	29	29	40
69	44	44	33	33	69
40	4	29	40	4	29
69	44	33	69	44	33

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

G:=sub<GL(6,GF(73))| [72,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[40,69,40,69,40,69,4,44,4,44,4,44,40,69,4,44,29,33,4,44,29,33,40,69,40,69,29,33,4,44,4,44,40,69,29,33],[1,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,1,0,0,0,0,0,72,72,0,0] >;

He₃⋊₂SD₁₆ in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_2{\rm SD}_{16}

% in TeX

G:=Group("He3:2SD16");

// GroupNames label

G:=SmallGroup(432,234);

// by ID

G=gap.SmallGroup(432,234);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,85,64,254,135,58,1124,851,298,348,1027,537,14118,7069]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=e^2=1,e*a*e=a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=c*b=b*c,b*d=d*b,e*b*e=b^-1,d*c*d^-1=a^-1*b,e*c*e=c^-1,e*d*e=d^3>;

// generators/relations

Export

Character table of He₃⋊₂SD₁₆ in TeX

40	4	40	4	40	4
69	44	69	44	69	44
40	4	4	29	29	40
69	44	44	33	33	69
40	4	29	40	4	29
69	44	33	69	44	33

40	4	40	4	40	4
69	44	69	44	69	44
40	4	4	29	29	40
69	44	44	33	33	69
40	4	29	40	4	29
69	44	33	69	44	33

G = He3⋊2SD16 order 432 = 24·33

The semidirect product of He3 and SD16 acting via SD16/C2=D4

G = He₃⋊₂SD₁₆ order 432 = 2⁴·3³

The semidirect product of He₃ and SD₁₆ acting via SD₁₆/C₂=D₄

40	4	40	4	40	4
69	44	69	44	69	44
40	4	4	29	29	40
69	44	44	33	33	69
40	4	29	40	4	29
69	44	33	69	44	33