C2xD4xHe3 - GroupNames

G = C₂xD₄xHe₃ order 432 = 2⁴·3³

Direct product of C₂, D₄ and He₃

direct product, metabelian, nilpotent (class 2), monomial

Aliases: C₂xD₄xHe₃, C₁₂.₂₇C₆², (C₆xC₁₂):₆C₆, C₄:(C₂²xHe₃), C₆²:₃(C₂xC₆), (C₂xC₆²):₁C₆, (D₄xC₃²):₅C₆, C₃²:₁₁(C₆xD₄), (C₂xC₆).₈C₆², C₂³:₂(C₂xHe₃), C₆.₂₁(C₂xC₆²), (C₂³xHe₃):₁C₂, (C₆xD₄).₂C₃², C₆.₂₂(D₄xC₃²), C₂.₂(C₂³xHe₃), (C₄xHe₃):₁₀C₂², C₂²:₂(C₂²xHe₃), (C₂xHe₃).₄₀C₂³, (C₂²xHe₃):₆C₂², (D₄xC₃xC₆):C₃, C₃.₂(D₄xC₃xC₆), (C₃xC₆):₆(C₃xD₄), (C₃xC₁₂):₄(C₂xC₆), (C₂xC₄xHe₃):₁₀C₂, (C₂xC₄):₂(C₂xHe₃), (C₂xC₁₂).₁₇(C₃xC₆), (C₃xD₄).₁₀(C₃xC₆), (C₃xC₆).₃₁(C₂²xC₆), (C₂²xC₆).₁₇(C₃xC₆), SmallGroup(432,404)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆ — C₂xD₄xHe₃

Chief series

C₁ — C₃ — C₆ — C₃xC₆ — C₂xHe₃ — C₂²xHe₃ — D₄xHe₃ — C₂xD₄xHe₃

Lower central

C₁ — C₆ — C₂xD₄xHe₃

Upper central

C₁ — C₂xC₆ — C₂xD₄xHe₃

Generators and relations for C₂xD₄xHe₃
G = < a,b,c,d,e,f | a²=b⁴=c²=d³=e³=f³=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b^-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf^-1=de^-1, ef=fe >

Subgroups: 665 in 297 conjugacy classes, 133 normal (15 characteristic)
C₁, C₂, C₂, C₂, C₃, C₃, C₄, C₂², C₂², C₂², C₆, C₆, C₆, C₂xC₄, D₄, C₂³, C₃², C₁₂, C₁₂, C₂xC₆, C₂xC₆, C₂xC₆, C₂xD₄, C₃xC₆, C₃xC₆, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₃xD₄, C₂²xC₆, C₂²xC₆, He₃, C₃xC₁₂, C₆², C₆², C₆xD₄, C₆xD₄, C₂xHe₃, C₂xHe₃, C₂xHe₃, C₆xC₁₂, D₄xC₃², C₂xC₆², C₄xHe₃, C₂²xHe₃, C₂²xHe₃, C₂²xHe₃, D₄xC₃xC₆, C₂xC₄xHe₃, D₄xHe₃, C₂³xHe₃, C₂xD₄xHe₃
Quotients: C₁, C₂, C₃, C₂², C₆, D₄, C₂³, C₃², C₂xC₆, C₂xD₄, C₃xC₆, C₃xD₄, C₂²xC₆, He₃, C₆², C₆xD₄, C₂xHe₃, D₄xC₃², C₂xC₆², C₂²xHe₃, D₄xC₃xC₆, D₄xHe₃, C₂³xHe₃, C₂xD₄xHe₃

Smallest permutation representation of C₂xD₄xHe₃
►On 72 points

Generators in S₇₂

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

(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)

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

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

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

(17 35 21)(18 36 22)(19 33 23)(20 34 24)(25 43 59)(26 44 60)(27 41 57)(28 42 58)(37 71 55)(38 72 56)(39 69 53)(40 70 54)(45 49 63)(46 50 64)(47 51 61)(48 52 62)

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

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

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

110 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	···	3J	4A	4B	6A	···	6F	6G	···	6N	6O	···	6AL	6AM	···	6BR	12A	12B	12C	12D	12E	···	12T
order	1	2	2	2	2	2	2	2	3	3	3	···	3	4	4	6	···	6	6	···	6	6	···	6	6	···	6	12	12	12	12	12	···	12
size	1	1	1	1	2	2	2	2	1	1	3	···	3	2	2	1	···	1	2	···	2	3	···	3	6	···	6	2	2	2	2	6	···	6

110 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	3	3	3	3	6
type	+	+	+	+					+
image	C₁	C₂	C₂	C₂	C₃	C₆	C₆	C₆	D₄	C₃xD₄	He₃	C₂xHe₃	C₂xHe₃	C₂xHe₃	D₄xHe₃
kernel	C₂xD₄xHe₃	C₂xC₄xHe₃	D₄xHe₃	C₂³xHe₃	D₄xC₃xC₆	C₆xC₁₂	D₄xC₃²	C₂xC₆²	C₂xHe₃	C₃xC₆	C₂xD₄	C₂xC₄	D₄	C₂³	C₂
# reps	1	1	4	2	8	8	32	16	2	16	2	2	8	4	4

Matrix representation of C₂xD₄xHe₃ ►in GL₅(F₁₃)

12	0	0	0	0
0	12	0	0	0
0	0	12	0	0
0	0	0	12	0
0	0	0	0	12

12	8	0	0	0
3	1	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

12	8	0	0	0
0	1	0	0	0
0	0	12	0	0
0	0	0	12	0
0	0	0	0	12

3	0	0	0	0
0	3	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	3	0	0
0	0	0	3	0
0	0	0	0	3

9	0	0	0	0
0	9	0	0	0
0	0	1	0	0
0	0	0	9	0
0	0	0	0	3

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[12,3,0,0,0,8,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,8,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[3,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3],[9,0,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,3] >;

C₂xD₄xHe₃ in GAP, Magma, Sage, TeX

C_2\times D_4\times {\rm He}_3

% in TeX

G:=Group("C2xD4xHe3");

// GroupNames label

G:=SmallGroup(432,404);

// by ID

G=gap.SmallGroup(432,404);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-3,1037,760]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^4=c^2=d^3=e^3=f^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f^-1=d*e^-1,e*f=f*e>;

// generators/relations

G = C2xD4xHe3 order 432 = 24·33

Direct product of C2, D4 and He3

G = C₂xD₄xHe₃ order 432 = 2⁴·3³

Direct product of C₂, D₄ and He₃