He3:3C8 - GroupNames

G = He₃:₃C₈ order 216 = 2³·3³

1^st semidirect product of He₃ and C₈ acting via C₈/C₄=C₂

Aliases: He₃:₃C₈, C₃²:₂C₂₄, (C₃xC₆).C₁₂, C₃²:₄C₈:C₃, C₁₂.₉(C₃xS₃), C₃²:₂(C₃:C₈), (C₃xC₁₂).₂C₆, (C₃xC₁₂).₅S₃, C₂.(C₃²:C₁₂), (C₂xHe₃).₂C₄, (C₄xHe₃).₃C₂, (C₃xC₆).₁Dic₃, C₆.₂(C₃xDic₃), C₄.₂(C₃²:C₆), C₃.₂(C₃xC₃:C₈), SmallGroup(216,14)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — He₃:₃C₈

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — C₃xC₁₂ — C₄xHe₃ — He₃:₃C₈

Lower central

C₃² — He₃:₃C₈

Upper central

C₁ — C₄

Generators and relations for He₃:₃C₈
G = < a,b,c,d | a³=b³=c³=d⁸=1, ab=ba, cac^-1=ab^-1, dad^-1=a^-1, bc=cb, dbd^-1=b^-1, cd=dc >

Subgroups: 92 in 34 conjugacy classes, 17 normal (all characteristic)
Quotients: C₁, C₂, C₃, C₄, S₃, C₆, C₈, Dic₃, C₁₂, C₃xS₃, C₃:C₈, C₂₄, C₃xDic₃, C₃²:C₆, C₃xC₃:C₈, C₃²:C₁₂, He₃:₃C₈

C₁

C₂

C₃

₃C₃

₆C₃

C₄

C₆

₃C₆

₆C₆

C₃²

₂C₃²

₉C₈

C₁₂

₃C₁₂

₆C₁₂

C₃xC₆

₂C₃xC₆

He₃

₃C₃:C₈

₉C₂₄

₉C₃:C₈

C₃xC₁₂

₂C₃xC₁₂

C₂xHe₃

C₃²:₄C₈

₃C₃xC₃:C₈

C₄xHe₃

He₃:₃C₈

Smallest permutation representation of He₃:₃C₈
►On 72 points

Generators in S₇₂

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

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

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

(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)

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

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

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

He₃:₃C₈ is a maximal subgroup of
C₃²:C₆:C₈ He₃:M₄(2) C₁₂.₈₉S₃² He₃:₃M₄(2) He₃:₃D₈ He₃:₄SD₁₆ He₃:₅SD₁₆ He₃:₃Q₁₆ C₈xC₃²:C₆ He₃:₅M₄(2) He₃:₇M₄(2) He₃:₈SD₁₆ He₃:₆D₈ He₃:₆Q₁₆ He₃:₁₀SD₁₆
He₃:₃C₈ is a maximal quotient of
He₃:₃C₁₆

40 conjugacy classes

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

40 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	6	6	6
type	+	+							+	-					+	-
image	C₁	C₂	C₃	C₄	C₆	C₈	C₁₂	C₂₄	S₃	Dic₃	C₃xS₃	C₃:C₈	C₃xDic₃	C₃xC₃:C₈	C₃²:C₆	C₃²:C₁₂	He₃:₃C₈
kernel	He₃:₃C₈	C₄xHe₃	C₃²:₄C₈	C₂xHe₃	C₃xC₁₂	He₃	C₃xC₆	C₃²	C₃xC₁₂	C₃xC₆	C₁₂	C₃²	C₆	C₃	C₄	C₂	C₁
# reps	1	1	2	2	2	4	4	8	1	1	2	2	2	4	1	1	2

Matrix representation of He₃:₃C₈ ►in GL₆(F₇₃)

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

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

9	20	11	64	9	20
11	64	53	62	11	64
11	64	9	20	9	20
53	62	11	64	11	64
9	20	9	20	11	64
11	64	11	64	53	62

G:=sub<GL(6,GF(73))| [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],[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],[0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,72,0,0,0,0,1,72,0,0,0,0],[9,11,11,53,9,11,20,64,64,62,20,64,11,53,9,11,9,11,64,62,20,64,20,64,9,11,9,11,11,53,20,64,20,64,64,62] >;

He₃:₃C₈ in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_3C_8

% in TeX

G:=Group("He3:3C8");

// GroupNames label

G:=SmallGroup(216,14);

// by ID

G=gap.SmallGroup(216,14);

# by ID

G:=PCGroup([6,-2,-3,-2,-2,-3,-3,36,50,1444,1450,5189]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of He₃:₃C₈ in TeX

9	20	11	64	9	20
11	64	53	62	11	64
11	64	9	20	9	20
53	62	11	64	11	64
9	20	9	20	11	64
11	64	11	64	53	62

9	20	11	64	9	20
11	64	53	62	11	64
11	64	9	20	9	20
53	62	11	64	11	64
9	20	9	20	11	64
11	64	11	64	53	62

G = He3:3C8 order 216 = 23·33

1st semidirect product of He3 and C8 acting via C8/C4=C2

G = He₃:₃C₈ order 216 = 2³·3³

1^st semidirect product of He₃ and C₈ acting via C₈/C₄=C₂

9	20	11	64	9	20
11	64	53	62	11	64
11	64	9	20	9	20
53	62	11	64	11	64
9	20	9	20	11	64
11	64	11	64	53	62