C3xD4xD7 - GroupNames

G = C₃xD₄xD₇ order 336 = 2⁴·3·7

Direct product of C₃, D₄ and D₇

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₃xD₄xD₇, D₂₈:₉C₆, C₁₂:₆D₁₄, C₈₄:₆C₂², C₄₂.₄₂C₂³, C₇:₆(C₆xD₄), C₄:₁(C₆xD₇), C₂₈:₅(C₂xC₆), (C₄xD₇):₆C₆, (C₇xD₄):₈C₆, (C₂xC₆):₄D₁₄, C₂₁:₁₄(C₂xD₄), C₇:D₄:₅C₆, (C₁₂xD₇):₆C₂, D₁₄:₆(C₂xC₆), (C₃xD₂₈):₉C₂, (D₄xC₂₁):₅C₂, C₂²:₂(C₆xD₇), (C₂xC₄₂):₆C₂², Dic₇:₅(C₂xC₆), (C₂²xD₇):₈C₆, (C₆xD₇):₁₀C₂², C₆.₄₂(C₂²xD₇), C₁₄.₁₉(C₂²xC₆), (C₃xDic₇):₈C₂², (C₂xC₆xD₇):₆C₂, C₂.₆(C₂xC₆xD₇), (C₂xC₁₄):₇(C₂xC₆), (C₃xC₇:D₄):₅C₂, SmallGroup(336,178)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₄ — C₃xD₄xD₇

Chief series

C₁ — C₇ — C₁₄ — C₄₂ — C₆xD₇ — C₂xC₆xD₇ — C₃xD₄xD₇

Lower central

C₇ — C₁₄ — C₃xD₄xD₇

Upper central

C₁ — C₆ — C₃xD₄

Generators and relations for C₃xD₄xD₇
G = < a,b,c,d,e | a³=b⁴=c²=d⁷=e²=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ce=ec, ede=d^-1 >

Subgroups: 440 in 108 conjugacy classes, 50 normal (26 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₆, C₆, C₇, C₂xC₄, D₄, D₄, C₂³, C₁₂, C₁₂, C₂xC₆, C₂xC₆, D₇, D₇, C₁₄, C₁₄, C₂xD₄, C₂₁, C₂xC₁₂, C₃xD₄, C₃xD₄, C₂²xC₆, Dic₇, C₂₈, D₁₄, D₁₄, D₁₄, C₂xC₁₄, C₃xD₇, C₃xD₇, C₄₂, C₄₂, C₆xD₄, C₄xD₇, D₂₈, C₇:D₄, C₇xD₄, C₂²xD₇, C₃xDic₇, C₈₄, C₆xD₇, C₆xD₇, C₆xD₇, C₂xC₄₂, D₄xD₇, C₁₂xD₇, C₃xD₂₈, C₃xC₇:D₄, D₄xC₂₁, C₂xC₆xD₇, C₃xD₄xD₇
Quotients: C₁, C₂, C₃, C₂², C₆, D₄, C₂³, C₂xC₆, D₇, C₂xD₄, C₃xD₄, C₂²xC₆, D₁₄, C₃xD₇, C₆xD₄, C₂²xD₇, C₆xD₇, D₄xD₇, C₂xC₆xD₇, C₃xD₄xD₇

Smallest permutation representation of C₃xD₄xD₇
►On 84 points

Generators in S₈₄

(1 34 20)(2 35 21)(3 29 15)(4 30 16)(5 31 17)(6 32 18)(7 33 19)(8 36 22)(9 37 23)(10 38 24)(11 39 25)(12 40 26)(13 41 27)(14 42 28)(43 71 57)(44 72 58)(45 73 59)(46 74 60)(47 75 61)(48 76 62)(49 77 63)(50 78 64)(51 79 65)(52 80 66)(53 81 67)(54 82 68)(55 83 69)(56 84 70)

(1 48 13 55)(2 49 14 56)(3 43 8 50)(4 44 9 51)(5 45 10 52)(6 46 11 53)(7 47 12 54)(15 57 22 64)(16 58 23 65)(17 59 24 66)(18 60 25 67)(19 61 26 68)(20 62 27 69)(21 63 28 70)(29 71 36 78)(30 72 37 79)(31 73 38 80)(32 74 39 81)(33 75 40 82)(34 76 41 83)(35 77 42 84)

(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)

(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 73 74 75 76 77)(78 79 80 81 82 83 84)

(1 7)(2 6)(3 5)(8 10)(11 14)(12 13)(15 17)(18 21)(19 20)(22 24)(25 28)(26 27)(29 31)(32 35)(33 34)(36 38)(39 42)(40 41)(43 45)(46 49)(47 48)(50 52)(53 56)(54 55)(57 59)(60 63)(61 62)(64 66)(67 70)(68 69)(71 73)(74 77)(75 76)(78 80)(81 84)(82 83)

G:=sub<Sym(84)| (1,34,20)(2,35,21)(3,29,15)(4,30,16)(5,31,17)(6,32,18)(7,33,19)(8,36,22)(9,37,23)(10,38,24)(11,39,25)(12,40,26)(13,41,27)(14,42,28)(43,71,57)(44,72,58)(45,73,59)(46,74,60)(47,75,61)(48,76,62)(49,77,63)(50,78,64)(51,79,65)(52,80,66)(53,81,67)(54,82,68)(55,83,69)(56,84,70), (1,48,13,55)(2,49,14,56)(3,43,8,50)(4,44,9,51)(5,45,10,52)(6,46,11,53)(7,47,12,54)(15,57,22,64)(16,58,23,65)(17,59,24,66)(18,60,25,67)(19,61,26,68)(20,62,27,69)(21,63,28,70)(29,71,36,78)(30,72,37,79)(31,73,38,80)(32,74,39,81)(33,75,40,82)(34,76,41,83)(35,77,42,84), (43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84), (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)(71,72,73,74,75,76,77)(78,79,80,81,82,83,84), (1,7)(2,6)(3,5)(8,10)(11,14)(12,13)(15,17)(18,21)(19,20)(22,24)(25,28)(26,27)(29,31)(32,35)(33,34)(36,38)(39,42)(40,41)(43,45)(46,49)(47,48)(50,52)(53,56)(54,55)(57,59)(60,63)(61,62)(64,66)(67,70)(68,69)(71,73)(74,77)(75,76)(78,80)(81,84)(82,83)>;

G:=Group( (1,34,20)(2,35,21)(3,29,15)(4,30,16)(5,31,17)(6,32,18)(7,33,19)(8,36,22)(9,37,23)(10,38,24)(11,39,25)(12,40,26)(13,41,27)(14,42,28)(43,71,57)(44,72,58)(45,73,59)(46,74,60)(47,75,61)(48,76,62)(49,77,63)(50,78,64)(51,79,65)(52,80,66)(53,81,67)(54,82,68)(55,83,69)(56,84,70), (1,48,13,55)(2,49,14,56)(3,43,8,50)(4,44,9,51)(5,45,10,52)(6,46,11,53)(7,47,12,54)(15,57,22,64)(16,58,23,65)(17,59,24,66)(18,60,25,67)(19,61,26,68)(20,62,27,69)(21,63,28,70)(29,71,36,78)(30,72,37,79)(31,73,38,80)(32,74,39,81)(33,75,40,82)(34,76,41,83)(35,77,42,84), (43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84), (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)(71,72,73,74,75,76,77)(78,79,80,81,82,83,84), (1,7)(2,6)(3,5)(8,10)(11,14)(12,13)(15,17)(18,21)(19,20)(22,24)(25,28)(26,27)(29,31)(32,35)(33,34)(36,38)(39,42)(40,41)(43,45)(46,49)(47,48)(50,52)(53,56)(54,55)(57,59)(60,63)(61,62)(64,66)(67,70)(68,69)(71,73)(74,77)(75,76)(78,80)(81,84)(82,83) );

G=PermutationGroup([[(1,34,20),(2,35,21),(3,29,15),(4,30,16),(5,31,17),(6,32,18),(7,33,19),(8,36,22),(9,37,23),(10,38,24),(11,39,25),(12,40,26),(13,41,27),(14,42,28),(43,71,57),(44,72,58),(45,73,59),(46,74,60),(47,75,61),(48,76,62),(49,77,63),(50,78,64),(51,79,65),(52,80,66),(53,81,67),(54,82,68),(55,83,69),(56,84,70)], [(1,48,13,55),(2,49,14,56),(3,43,8,50),(4,44,9,51),(5,45,10,52),(6,46,11,53),(7,47,12,54),(15,57,22,64),(16,58,23,65),(17,59,24,66),(18,60,25,67),(19,61,26,68),(20,62,27,69),(21,63,28,70),(29,71,36,78),(30,72,37,79),(31,73,38,80),(32,74,39,81),(33,75,40,82),(34,76,41,83),(35,77,42,84)], [(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70),(71,78),(72,79),(73,80),(74,81),(75,82),(76,83),(77,84)], [(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),(71,72,73,74,75,76,77),(78,79,80,81,82,83,84)], [(1,7),(2,6),(3,5),(8,10),(11,14),(12,13),(15,17),(18,21),(19,20),(22,24),(25,28),(26,27),(29,31),(32,35),(33,34),(36,38),(39,42),(40,41),(43,45),(46,49),(47,48),(50,52),(53,56),(54,55),(57,59),(60,63),(61,62),(64,66),(67,70),(68,69),(71,73),(74,77),(75,76),(78,80),(81,84),(82,83)]])

75 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	4A	4B	6A	6B	6C	6D	6E	6F	6G	6H	6I	6J	6K	6L	6M	6N	7A	7B	7C	12A	12B	12C	12D	14A	14B	14C	14D	···	14I	21A	···	21F	28A	28B	28C	42A	···	42F	42G	···	42R	84A	···	84F
order	1	2	2	2	2	2	2	2	3	3	4	4	6	6	6	6	6	6	6	6	6	6	6	6	6	6	7	7	7	12	12	12	12	14	14	14	14	···	14	21	···	21	28	28	28	42	···	42	42	···	42	84	···	84
size	1	1	2	2	7	7	14	14	1	1	2	14	1	1	2	2	2	2	7	7	7	7	14	14	14	14	2	2	2	2	2	14	14	2	2	2	4	···	4	2	···	2	4	4	4	2	···	2	4	···	4	4	···	4

75 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+	+	+							+	+		+	+				+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₆	C₆	D₄	D₇	C₃xD₄	D₁₄	D₁₄	C₃xD₇	C₆xD₇	C₆xD₇	D₄xD₇	C₃xD₄xD₇
kernel	C₃xD₄xD₇	C₁₂xD₇	C₃xD₂₈	C₃xC₇:D₄	D₄xC₂₁	C₂xC₆xD₇	D₄xD₇	C₄xD₇	D₂₈	C₇:D₄	C₇xD₄	C₂²xD₇	C₃xD₇	C₃xD₄	D₇	C₁₂	C₂xC₆	D₄	C₄	C₂²	C₃	C₁
# reps	1	1	1	2	1	2	2	2	2	4	2	4	2	3	4	3	6	6	6	12	3	6

Matrix representation of C₃xD₄xD₇ ►in GL₄(F₃₃₇) generated by

208	0	0	0
0	208	0	0
0	0	128	0
0	0	0	128

336	0	0	0
0	336	0	0
0	0	1	2
0	0	336	336

1	0	0	0
0	1	0	0
0	0	1	0
0	0	336	336

303	1	0	0
336	0	0	0
0	0	1	0
0	0	0	1

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

G:=sub<GL(4,GF(337))| [208,0,0,0,0,208,0,0,0,0,128,0,0,0,0,128],[336,0,0,0,0,336,0,0,0,0,1,336,0,0,2,336],[1,0,0,0,0,1,0,0,0,0,1,336,0,0,0,336],[303,336,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;

C₃xD₄xD₇ in GAP, Magma, Sage, TeX

C_3\times D_4\times D_7

% in TeX

G:=Group("C3xD4xD7");

// GroupNames label

G:=SmallGroup(336,178);

// by ID

G=gap.SmallGroup(336,178);

# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-7,260,10373]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^4=c^2=d^7=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;

// generators/relations

G = C3xD4xD7 order 336 = 24·3·7

Direct product of C3, D4 and D7

G = C₃xD₄xD₇ order 336 = 2⁴·3·7

Direct product of C₃, D₄ and D₇