D7xC2^2xC4 - GroupNames

G = D₇xC₂²xC₄ order 224 = 2⁵·7

Direct product of C₂²xC₄ and D₇

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

Aliases: D₇xC₂²xC₄, C₂₈:₃C₂³, C₁₄.₂C₂⁴, Dic₇:₃C₂³, D₁₄.₉C₂³, C₂³.₃₄D₁₄, C₇:₁(C₂³xC₄), C₁₄:₁(C₂²xC₄), C₂.₁(C₂³xD₇), (C₂xC₂₈):₁₄C₂², (C₂²xC₂₈):₁₀C₂, (C₂³xD₇).₃C₂, (C₂xC₁₄).₆₃C₂³, (C₂²xDic₇):₁₀C₂, (C₂xDic₇):₁₂C₂², C₂².₂₉(C₂²xD₇), (C₂²xC₁₄).₄₄C₂², (C₂²xD₇).₃₅C₂², (C₂xC₁₄):₆(C₂xC₄), SmallGroup(224,175)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₇ — D₇xC₂²xC₄

Chief series

C₁ — C₇ — C₁₄ — D₁₄ — C₂²xD₇ — C₂³xD₇ — D₇xC₂²xC₄

Lower central

C₇ — D₇xC₂²xC₄

Upper central

C₁ — C₂²xC₄

Generators and relations for D₇xC₂²xC₄
G = < a,b,c,d,e | a²=b²=c⁴=d⁷=e²=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d^-1 >

Subgroups: 782 in 236 conjugacy classes, 145 normal (11 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₂xC₄, C₂xC₄, C₂³, C₂³, D₇, C₁₄, C₁₄, C₂²xC₄, C₂²xC₄, C₂⁴, Dic₇, C₂₈, D₁₄, C₂xC₁₄, C₂³xC₄, C₄xD₇, C₂xDic₇, C₂xC₂₈, C₂²xD₇, C₂²xC₁₄, C₂xC₄xD₇, C₂²xDic₇, C₂²xC₂₈, C₂³xD₇, D₇xC₂²xC₄
Quotients: C₁, C₂, C₄, C₂², C₂xC₄, C₂³, D₇, C₂²xC₄, C₂⁴, D₁₄, C₂³xC₄, C₄xD₇, C₂²xD₇, C₂xC₄xD₇, C₂³xD₇, D₇xC₂²xC₄

Smallest permutation representation of D₇xC₂²xC₄
►On 112 points

Generators in S₁₁₂

(1 69)(2 70)(3 64)(4 65)(5 66)(6 67)(7 68)(8 57)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 78)(16 79)(17 80)(18 81)(19 82)(20 83)(21 84)(22 71)(23 72)(24 73)(25 74)(26 75)(27 76)(28 77)(29 92)(30 93)(31 94)(32 95)(33 96)(34 97)(35 98)(36 85)(37 86)(38 87)(39 88)(40 89)(41 90)(42 91)(43 106)(44 107)(45 108)(46 109)(47 110)(48 111)(49 112)(50 99)(51 100)(52 101)(53 102)(54 103)(55 104)(56 105)

(1 41)(2 42)(3 36)(4 37)(5 38)(6 39)(7 40)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(57 92)(58 93)(59 94)(60 95)(61 96)(62 97)(63 98)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 91)(71 106)(72 107)(73 108)(74 109)(75 110)(76 111)(77 112)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)

(1 20 13 27)(2 21 14 28)(3 15 8 22)(4 16 9 23)(5 17 10 24)(6 18 11 25)(7 19 12 26)(29 43 36 50)(30 44 37 51)(31 45 38 52)(32 46 39 53)(33 47 40 54)(34 48 41 55)(35 49 42 56)(57 71 64 78)(58 72 65 79)(59 73 66 80)(60 74 67 81)(61 75 68 82)(62 76 69 83)(63 77 70 84)(85 99 92 106)(86 100 93 107)(87 101 94 108)(88 102 95 109)(89 103 96 110)(90 104 97 111)(91 105 98 112)

(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)(85 86 87 88 89 90 91)(92 93 94 95 96 97 98)(99 100 101 102 103 104 105)(106 107 108 109 110 111 112)

(1 12)(2 11)(3 10)(4 9)(5 8)(6 14)(7 13)(15 24)(16 23)(17 22)(18 28)(19 27)(20 26)(21 25)(29 38)(30 37)(31 36)(32 42)(33 41)(34 40)(35 39)(43 52)(44 51)(45 50)(46 56)(47 55)(48 54)(49 53)(57 66)(58 65)(59 64)(60 70)(61 69)(62 68)(63 67)(71 80)(72 79)(73 78)(74 84)(75 83)(76 82)(77 81)(85 94)(86 93)(87 92)(88 98)(89 97)(90 96)(91 95)(99 108)(100 107)(101 106)(102 112)(103 111)(104 110)(105 109)

G:=sub<Sym(112)| (1,69)(2,70)(3,64)(4,65)(5,66)(6,67)(7,68)(8,57)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,92)(30,93)(31,94)(32,95)(33,96)(34,97)(35,98)(36,85)(37,86)(38,87)(39,88)(40,89)(41,90)(42,91)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,112)(50,99)(51,100)(52,101)(53,102)(54,103)(55,104)(56,105), (1,41)(2,42)(3,36)(4,37)(5,38)(6,39)(7,40)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(57,92)(58,93)(59,94)(60,95)(61,96)(62,97)(63,98)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,106)(72,107)(73,108)(74,109)(75,110)(76,111)(77,112)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105), (1,20,13,27)(2,21,14,28)(3,15,8,22)(4,16,9,23)(5,17,10,24)(6,18,11,25)(7,19,12,26)(29,43,36,50)(30,44,37,51)(31,45,38,52)(32,46,39,53)(33,47,40,54)(34,48,41,55)(35,49,42,56)(57,71,64,78)(58,72,65,79)(59,73,66,80)(60,74,67,81)(61,75,68,82)(62,76,69,83)(63,77,70,84)(85,99,92,106)(86,100,93,107)(87,101,94,108)(88,102,95,109)(89,103,96,110)(90,104,97,111)(91,105,98,112), (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)(85,86,87,88,89,90,91)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112), (1,12)(2,11)(3,10)(4,9)(5,8)(6,14)(7,13)(15,24)(16,23)(17,22)(18,28)(19,27)(20,26)(21,25)(29,38)(30,37)(31,36)(32,42)(33,41)(34,40)(35,39)(43,52)(44,51)(45,50)(46,56)(47,55)(48,54)(49,53)(57,66)(58,65)(59,64)(60,70)(61,69)(62,68)(63,67)(71,80)(72,79)(73,78)(74,84)(75,83)(76,82)(77,81)(85,94)(86,93)(87,92)(88,98)(89,97)(90,96)(91,95)(99,108)(100,107)(101,106)(102,112)(103,111)(104,110)(105,109)>;

G:=Group( (1,69)(2,70)(3,64)(4,65)(5,66)(6,67)(7,68)(8,57)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,92)(30,93)(31,94)(32,95)(33,96)(34,97)(35,98)(36,85)(37,86)(38,87)(39,88)(40,89)(41,90)(42,91)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,112)(50,99)(51,100)(52,101)(53,102)(54,103)(55,104)(56,105), (1,41)(2,42)(3,36)(4,37)(5,38)(6,39)(7,40)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(57,92)(58,93)(59,94)(60,95)(61,96)(62,97)(63,98)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,106)(72,107)(73,108)(74,109)(75,110)(76,111)(77,112)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105), (1,20,13,27)(2,21,14,28)(3,15,8,22)(4,16,9,23)(5,17,10,24)(6,18,11,25)(7,19,12,26)(29,43,36,50)(30,44,37,51)(31,45,38,52)(32,46,39,53)(33,47,40,54)(34,48,41,55)(35,49,42,56)(57,71,64,78)(58,72,65,79)(59,73,66,80)(60,74,67,81)(61,75,68,82)(62,76,69,83)(63,77,70,84)(85,99,92,106)(86,100,93,107)(87,101,94,108)(88,102,95,109)(89,103,96,110)(90,104,97,111)(91,105,98,112), (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)(85,86,87,88,89,90,91)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112), (1,12)(2,11)(3,10)(4,9)(5,8)(6,14)(7,13)(15,24)(16,23)(17,22)(18,28)(19,27)(20,26)(21,25)(29,38)(30,37)(31,36)(32,42)(33,41)(34,40)(35,39)(43,52)(44,51)(45,50)(46,56)(47,55)(48,54)(49,53)(57,66)(58,65)(59,64)(60,70)(61,69)(62,68)(63,67)(71,80)(72,79)(73,78)(74,84)(75,83)(76,82)(77,81)(85,94)(86,93)(87,92)(88,98)(89,97)(90,96)(91,95)(99,108)(100,107)(101,106)(102,112)(103,111)(104,110)(105,109) );

G=PermutationGroup([[(1,69),(2,70),(3,64),(4,65),(5,66),(6,67),(7,68),(8,57),(9,58),(10,59),(11,60),(12,61),(13,62),(14,63),(15,78),(16,79),(17,80),(18,81),(19,82),(20,83),(21,84),(22,71),(23,72),(24,73),(25,74),(26,75),(27,76),(28,77),(29,92),(30,93),(31,94),(32,95),(33,96),(34,97),(35,98),(36,85),(37,86),(38,87),(39,88),(40,89),(41,90),(42,91),(43,106),(44,107),(45,108),(46,109),(47,110),(48,111),(49,112),(50,99),(51,100),(52,101),(53,102),(54,103),(55,104),(56,105)], [(1,41),(2,42),(3,36),(4,37),(5,38),(6,39),(7,40),(8,29),(9,30),(10,31),(11,32),(12,33),(13,34),(14,35),(15,50),(16,51),(17,52),(18,53),(19,54),(20,55),(21,56),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49),(57,92),(58,93),(59,94),(60,95),(61,96),(62,97),(63,98),(64,85),(65,86),(66,87),(67,88),(68,89),(69,90),(70,91),(71,106),(72,107),(73,108),(74,109),(75,110),(76,111),(77,112),(78,99),(79,100),(80,101),(81,102),(82,103),(83,104),(84,105)], [(1,20,13,27),(2,21,14,28),(3,15,8,22),(4,16,9,23),(5,17,10,24),(6,18,11,25),(7,19,12,26),(29,43,36,50),(30,44,37,51),(31,45,38,52),(32,46,39,53),(33,47,40,54),(34,48,41,55),(35,49,42,56),(57,71,64,78),(58,72,65,79),(59,73,66,80),(60,74,67,81),(61,75,68,82),(62,76,69,83),(63,77,70,84),(85,99,92,106),(86,100,93,107),(87,101,94,108),(88,102,95,109),(89,103,96,110),(90,104,97,111),(91,105,98,112)], [(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),(85,86,87,88,89,90,91),(92,93,94,95,96,97,98),(99,100,101,102,103,104,105),(106,107,108,109,110,111,112)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,14),(7,13),(15,24),(16,23),(17,22),(18,28),(19,27),(20,26),(21,25),(29,38),(30,37),(31,36),(32,42),(33,41),(34,40),(35,39),(43,52),(44,51),(45,50),(46,56),(47,55),(48,54),(49,53),(57,66),(58,65),(59,64),(60,70),(61,69),(62,68),(63,67),(71,80),(72,79),(73,78),(74,84),(75,83),(76,82),(77,81),(85,94),(86,93),(87,92),(88,98),(89,97),(90,96),(91,95),(99,108),(100,107),(101,106),(102,112),(103,111),(104,110),(105,109)]])

D₇xC₂²xC₄ is a maximal subgroup of
C₂².₅₈(D₄xD₇) (C₂xC₄):₉D₂₈ D₁₄:C₄² D₁₄:(C₄:C₄) D₁₄:C₄:C₄ D₁₄:M₄(2) C₂⁴.₁₂D₁₄ C₄:(D₁₄:C₄) D₁₄:C₄:₆C₄ D₁₄:₆M₄(2) C₄²:₈D₁₄ C₄²:₁₂D₁₄ C₄:C₄:₂₁D₁₄ C₄:C₄:₂₆D₁₄ C₄:C₄:₂₈D₁₄ (C₂xC₂₈):₁₅D₄
D₇xC₂²xC₄ is a maximal quotient of
C₂⁴.₂₄D₁₄ C₁₄.₈2₊¹⁺⁴ C₄².₈₇D₁₄ C₄²:₇D₁₄ C₄².₁₈₈D₁₄ C₄².₉₁D₁₄ C₄²:₁₁D₁₄ C₄².₁₀₈D₁₄ C₄².₁₂₅D₁₄ C₄².₁₂₆D₁₄ C₂₈.₇₀C₂⁴ C₅₆.₄₉C₂³

80 conjugacy classes

class	1	2A	···	2G	2H	···	2O	4A	···	4H	4I	···	4P	7A	7B	7C	14A	···	14U	28A	···	28X
order	1	2	···	2	2	···	2	4	···	4	4	···	4	7	7	7	14	···	14	28	···	28
size	1	1	···	1	7	···	7	1	···	1	7	···	7	2	2	2	2	···	2	2	···	2

80 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2
type	+	+	+	+	+		+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₄	D₇	D₁₄	D₁₄	C₄xD₇
kernel	D₇xC₂²xC₄	C₂xC₄xD₇	C₂²xDic₇	C₂²xC₂₈	C₂³xD₇	C₂²xD₇	C₂²xC₄	C₂xC₄	C₂³	C₂²
# reps	1	12	1	1	1	16	3	18	3	24

Matrix representation of D₇xC₂²xC₄ ►in GL₄(F₂₉) generated by

1	0	0	0
0	28	0	0
0	0	1	0
0	0	0	1

28	0	0	0
0	1	0	0
0	0	1	0
0	0	0	1

1	0	0	0
0	17	0	0
0	0	17	0
0	0	0	17

1	0	0	0
0	1	0	0
0	0	28	1
0	0	20	8

28	0	0	0
0	1	0	0
0	0	1	0
0	0	9	28

G:=sub<GL(4,GF(29))| [1,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[28,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,17,0,0,0,0,17,0,0,0,0,17],[1,0,0,0,0,1,0,0,0,0,28,20,0,0,1,8],[28,0,0,0,0,1,0,0,0,0,1,9,0,0,0,28] >;

D₇xC₂²xC₄ in GAP, Magma, Sage, TeX

D_7\times C_2^2\times C_4

% in TeX

G:=Group("D7xC2^2xC4");

// GroupNames label

G:=SmallGroup(224,175);

// by ID

G=gap.SmallGroup(224,175);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,69,6917]);

// Polycyclic

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

// generators/relations

G = D7xC22xC4 order 224 = 25·7

Direct product of C22xC4 and D7

G = D₇xC₂²xC₄ order 224 = 2⁵·7

Direct product of C₂²xC₄ and D₇