He3:M4(2) - GroupNames

G = He₃⋊M₄(2) order 432 = 2⁴·3³

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

Aliases: He₃⋊₂M₄(2), C₁₂.₈₈S₃², He₃⋊₄C₈⋊₅C₂, He₃⋊₃C₈⋊₆C₂, C₃²⋊₄C₈⋊₅S₃, (C₃×C₁₂).₃₆D₆, C₃²⋊C₁₂.₁C₄, C₆.₁₂(S₃×Dic₃), C₃²⋊(C₄.Dic₃), C₃²⋊₁(C₈⋊S₃), C₃⋊Dic₃.₂Dic₃, C₄.₁₄(C₃²⋊D₆), C₃.₂(D₆.Dic₃), (C₄×He₃).₂₈C₂², (C₄×C₃⋊S₃).₃S₃, (C₃×C₆).₂(C₄×S₃), (C₄×C₃²⋊C₆).₁C₂, (C₂×C₃²⋊C₆).₂C₄, (C₂×He₃).₉(C₂×C₄), (C₂×C₃⋊S₃).₂Dic₃, (C₃×C₆).₂(C₂×Dic₃), C₂.₃(C₆.S₃²), SmallGroup(432,77)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃ — C₂×He₃ — He₃⋊M₄(2)

Chief series

C₁ — C₃ — C₃² — He₃ — C₂×He₃ — C₄×He₃ — C₄×C₃²⋊C₆ — He₃⋊M₄(2)

Lower central

He₃ — C₂×He₃ — He₃⋊M₄(2)

Upper central

C₁ — C₄

Generators and relations for He₃⋊M₄(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, bd=db, ebe=b^-1, dcd^-1=c^-1, ce=ec, ede=d⁵ >

Subgroups: 371 in 78 conjugacy classes, 25 normal (all 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⋊S₃, C₃×C₆, C₃×C₆, C₃⋊C₈, C₂₄, C₄×S₃, C₂×C₁₂, He₃, C₃×Dic₃, C₃⋊Dic₃, C₃×C₁₂, C₃×C₁₂, S₃×C₆, C₂×C₃⋊S₃, C₈⋊S₃, C₄.Dic₃, C₃²⋊C₆, C₂×He₃, C₃×C₃⋊C₈, C₃²⋊₄C₈, S₃×C₁₂, C₄×C₃⋊S₃, C₃²⋊C₁₂, C₄×He₃, C₂×C₃²⋊C₆, D₆.Dic₃, C₁₂.₃₁D₆, He₃⋊₃C₈, He₃⋊₄C₈, C₄×C₃²⋊C₆, He₃⋊M₄(2)
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, Dic₃, D₆, M₄(2), C₄×S₃, C₂×Dic₃, S₃², C₈⋊S₃, C₄.Dic₃, S₃×Dic₃, C₃²⋊D₆, D₆.Dic₃, C₆.S₃², He₃⋊M₄(2)

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

Generators in S₇₂

(9 60 22)(10 23 61)(11 62 24)(12 17 63)(13 64 18)(14 19 57)(15 58 20)(16 21 59)(41 52 69)(42 70 53)(43 54 71)(44 72 55)(45 56 65)(46 66 49)(47 50 67)(48 68 51)

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

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

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

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

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

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

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	1	2	2	2	2	2	2	2	2	2	4	4	4	6	6	6
type	+	+	+	+			+	+	-	+	-					+	-		+
image	C₁	C₂	C₂	C₂	C₄	C₄	S₃	S₃	Dic₃	D₆	Dic₃	M₄(2)	C₄×S₃	C₈⋊S₃	C₄.Dic₃	S₃²	S₃×Dic₃	D₆.Dic₃	C₃²⋊D₆	C₆.S₃²	He₃⋊M₄(2)
kernel	He₃⋊M₄(2)	He₃⋊₃C₈	He₃⋊₄C₈	C₄×C₃²⋊C₆	C₃²⋊C₁₂	C₂×C₃²⋊C₆	C₃²⋊₄C₈	C₄×C₃⋊S₃	C₃⋊Dic₃	C₃×C₁₂	C₂×C₃⋊S₃	He₃	C₃×C₆	C₃²	C₃²	C₁₂	C₆	C₃	C₄	C₂	C₁
# reps	1	1	1	1	2	2	1	1	1	2	1	2	2	4	4	1	1	2	2	2	4

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

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

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

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

3	6	0	0	0	0
67	70	0	0	0	0
0	0	0	0	3	6
0	0	0	0	67	70
0	0	3	6	0	0
0	0	67	70	0	0

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

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[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],[3,67,0,0,0,0,6,70,0,0,0,0,0,0,0,0,3,67,0,0,0,0,6,70,0,0,3,67,0,0,0,0,6,70,0,0],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;

He₃⋊M₄(2) in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes M_4(2)

% in TeX

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

// GroupNames label

G:=SmallGroup(432,77);

// by ID

G=gap.SmallGroup(432,77);

# by ID

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

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

// generators/relations

G = He3⋊M4(2) order 432 = 24·33

1st semidirect product of He3 and M4(2) acting via M4(2)/C4=C22

G = He₃⋊M₄(2) order 432 = 2⁴·3³

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