He3:3M4(2) - GroupNames

G = He₃⋊₃M₄(2) order 432 = 2⁴·3³

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

Aliases: He₃⋊₃M₄(2), C₁₂.₉₀S₃², He₃⋊₃C₈⋊₈C₂, C₃²⋊₄C₈⋊₇S₃, (C₃×C₁₂).₃₈D₆, He₃⋊₃C₄.₃C₄, C₃²⋊₂(C₈⋊S₃), C₄.₁₆(C₃²⋊D₆), C₆.₁₃(C₆.D₆), (C₄×He₃).₃₀C₂², C₃.₂(C₁₂.₃₁D₆), (C₃×C₆).₄(C₄×S₃), C₂.₃(He₃⋊(C₂×C₄)), (C₂×He₃).₁₁(C₂×C₄), (C₂×He₃⋊C₂).₃C₄, (C₄×He₃⋊C₂).₁C₂, SmallGroup(432,82)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃ — C₂×He₃ — He₃⋊₃M₄(2)

Chief series

C₁ — C₃ — C₃² — He₃ — C₂×He₃ — C₄×He₃ — He₃⋊₃C₈ — He₃⋊₃M₄(2)

Lower central

He₃ — C₂×He₃ — He₃⋊₃M₄(2)

Upper central

C₁ — C₄

Generators and relations for He₃⋊₃M₄(2)
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, dbd^-1=b^-1, be=eb, cd=dc, ece=c^-1, ede=d⁵ >

Subgroups: 403 in 86 conjugacy classes, 23 normal (13 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², S₃, C₆, C₆, C₈, C₂×C₄, C₃², C₃², Dic₃, C₁₂, C₁₂, D₆, C₂×C₆, M₄(2), C₃×S₃, C₃×C₆, C₃×C₆, C₃⋊C₈, C₂₄, C₄×S₃, C₂×C₁₂, He₃, C₃×Dic₃, C₃×C₁₂, C₃×C₁₂, S₃×C₆, C₈⋊S₃, C₄.Dic₃, He₃⋊C₂, C₂×He₃, C₃×C₃⋊C₈, C₃²⋊₄C₈, S₃×C₁₂, He₃⋊₃C₄, C₄×He₃, C₂×He₃⋊C₂, D₆.Dic₃, He₃⋊₃C₈, C₄×He₃⋊C₂, He₃⋊₃M₄(2)
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, D₆, M₄(2), C₄×S₃, S₃², C₈⋊S₃, C₆.D₆, C₃²⋊D₆, C₁₂.₃₁D₆, He₃⋊(C₂×C₄), He₃⋊₃M₄(2)

Smallest permutation representation of He₃⋊₃M₄(2)
►On 72 points

Generators in S₇₂

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

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

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

(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 60)(18 57)(19 62)(20 59)(21 64)(22 61)(23 58)(24 63)(25 35)(26 40)(27 37)(28 34)(29 39)(30 36)(31 33)(32 38)(42 46)(44 48)(49 71)(50 68)(51 65)(52 70)(53 67)(54 72)(55 69)(56 66)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim	1	1	1	1	1	2	2	2	2	2	4	4	4	6	6	6
type	+	+	+			+	+				+	+		+	-
image	C₁	C₂	C₂	C₄	C₄	S₃	D₆	M₄(2)	C₄×S₃	C₈⋊S₃	S₃²	C₆.D₆	C₁₂.₃₁D₆	C₃²⋊D₆	He₃⋊(C₂×C₄)	He₃⋊₃M₄(2)
kernel	He₃⋊₃M₄(2)	He₃⋊₃C₈	C₄×He₃⋊C₂	He₃⋊₃C₄	C₂×He₃⋊C₂	C₃²⋊₄C₈	C₃×C₁₂	He₃	C₃×C₆	C₃²	C₁₂	C₆	C₃	C₄	C₂	C₁
# reps	1	2	1	2	2	2	2	2	4	8	1	1	2	2	2	4

Matrix representation of He₃⋊₃M₄(2) ►in GL₆(𝔽₇₃)

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

8	0	0	0	0	0
0	8	0	0	0	0
0	0	8	0	0	0
0	0	0	64	0	0
0	0	0	0	64	0
0	0	0	0	0	64

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

0	0	0	1	0	0
0	0	0	0	0	1
0	0	0	0	1	0
46	0	0	0	0	0
0	0	46	0	0	0
0	46	0	0	0	0

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

G:=sub<GL(6,GF(73))| [0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,64],[0,8,0,0,0,0,0,0,1,0,0,0,64,0,0,0,0,0,0,0,0,0,0,8,0,0,0,64,0,0,0,0,0,0,1,0],[0,0,0,46,0,0,0,0,0,0,0,46,0,0,0,0,46,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,72,0] >;

He₃⋊₃M₄(2) in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_3M_4(2)

% in TeX

G:=Group("He3:3M4(2)");

// GroupNames label

G:=SmallGroup(432,82);

// by ID

G=gap.SmallGroup(432,82);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,141,36,58,571,4037,537,14118,7069]);

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

// generators/relations

G = He3⋊3M4(2) order 432 = 24·33

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

G = He₃⋊₃M₄(2) order 432 = 2⁴·3³

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