He3:4M4(2) - GroupNames

G = He₃⋊₄M₄(2) order 432 = 2⁴·3³

The semidirect product of He₃ and M₄(2) acting via M₄(2)/C₂²=C₄

Aliases: He₃⋊₄M₄(2), He₃⋊₂C₈⋊₄C₂, C₂².(He₃⋊C₄), He₃⋊₃C₄.₆C₄, C₃.(C₆².C₄), (C₂²×He₃).₂C₄, He₃⋊₃C₄.₁₀C₂², C₂.₆(C₂×He₃⋊C₄), C₆.₂₈(C₂×C₃²⋊C₄), (C₂×C₆).₁(C₃²⋊C₄), (C₂×He₃).₆(C₂×C₄), (C₂×He₃⋊₃C₄).₈C₂, SmallGroup(432,278)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃ — C₂×He₃ — He₃⋊₄M₄(2)

Chief series

C₁ — C₃ — He₃ — C₂×He₃ — He₃⋊₃C₄ — He₃⋊₂C₈ — He₃⋊₄M₄(2)

Lower central

He₃ — C₂×He₃ — He₃⋊₄M₄(2)

Upper central

C₁ — C₆ — C₂×C₆

Generators and relations for He₃⋊₄M₄(2)
G = < a,b,c,d,e | a³=b³=c³=d⁸=e²=1, ab=ba, cac^-1=ab^-1, dad^-1=abc, ae=ea, bc=cb, bd=db, be=eb, dcd^-1=ac^-1, ce=ec, ede=d⁵ >

Subgroups: 281 in 62 conjugacy classes, 15 normal (13 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₆, C₆, C₈, C₂×C₄, C₃², Dic₃, C₁₂, C₂×C₆, C₂×C₆, M₄(2), C₃×C₆, C₂₄, C₂×Dic₃, C₂×C₁₂, He₃, C₃×Dic₃, C₆², C₃×M₄(2), C₂×He₃, C₂×He₃, C₆×Dic₃, He₃⋊₃C₄, C₂²×He₃, He₃⋊₂C₈, C₂×He₃⋊₃C₄, He₃⋊₄M₄(2)
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, M₄(2), C₃²⋊C₄, C₂×C₃²⋊C₄, He₃⋊C₄, C₆².C₄, C₂×He₃⋊C₄, He₃⋊₄M₄(2)

Smallest permutation representation of He₃⋊₄M₄(2)
►On 72 points

Generators in S₇₂

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

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

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

(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 6)(4 8)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)(34 38)(36 40)(42 46)(44 48)(49 53)(51 55)(58 62)(60 64)(66 70)(68 72)

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

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

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

38 conjugacy classes

class	1	2A	2B	3A	3B	3C	3D	4A	4B	4C	6A	6B	6C	6D	6E	···	6J	8A	8B	8C	8D	12A	12B	12C	12D	12E	12F	24A	···	24H
order	1	2	2	3	3	3	3	4	4	4	6	6	6	6	6	···	6	8	8	8	8	12	12	12	12	12	12	24	···	24
size	1	1	2	1	1	12	12	9	9	18	1	1	2	2	12	···	12	18	18	18	18	9	9	9	9	18	18	18	···	18

38 irreducible representations

dim	1	1	1	1	1	2	3	3	4	4	4	6
type	+	+	+						+	+	-
image	C₁	C₂	C₂	C₄	C₄	M₄(2)	He₃⋊C₄	C₂×He₃⋊C₄	C₃²⋊C₄	C₂×C₃²⋊C₄	C₆².C₄	He₃⋊₄M₄(2)
kernel	He₃⋊₄M₄(2)	He₃⋊₂C₈	C₂×He₃⋊₃C₄	He₃⋊₃C₄	C₂²×He₃	He₃	C₂²	C₂	C₂×C₆	C₆	C₃	C₁
# reps	1	2	1	2	2	2	8	8	2	2	4	4

Matrix representation of He₃⋊₄M₄(2) ►in GL₅(𝔽₇₃)

1	0	0	0	0
0	1	0	0	0
0	0	0	1	0
0	0	0	0	1
0	0	1	0	0

1	0	0	0	0
0	1	0	0	0
0	0	64	0	0
0	0	0	64	0
0	0	0	0	64

1	0	0	0	0
0	1	0	0	0
0	0	0	8	0
0	0	0	0	64
0	0	1	0	0

0	42	0	0	0
15	0	0	0	0
0	0	7	56	7
0	0	10	10	7
0	0	7	10	10

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

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],[1,0,0,0,0,0,1,0,0,0,0,0,64,0,0,0,0,0,64,0,0,0,0,0,64],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,8,0,0,0,0,0,64,0],[0,15,0,0,0,42,0,0,0,0,0,0,7,10,7,0,0,56,10,10,0,0,7,7,10],[1,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;

He₃⋊₄M₄(2) in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_4M_4(2)

% in TeX

G:=Group("He3:4M4(2)");

// GroupNames label

G:=SmallGroup(432,278);

// by ID

G=gap.SmallGroup(432,278);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,28,141,58,3924,298,5381,2539,537]);

// 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=a*b*c,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=a*c^-1,c*e=e*c,e*d*e=d^5>;

// generators/relations

G = He3⋊4M4(2) order 432 = 24·33

The semidirect product of He3 and M4(2) acting via M4(2)/C22=C4

G = He₃⋊₄M₄(2) order 432 = 2⁴·3³

The semidirect product of He₃ and M₄(2) acting via M₄(2)/C₂²=C₄