D28:11D4 - GroupNames

G = D₂₈:₁₁D₄ order 448 = 2⁶·7

4^th semidirect product of D₂₈ and D₄ acting via D₄/C₄=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₂₈:₁₁D₄, C₄²:₂₇D₁₄, C₁₄.₇₇2₊¹⁺⁴, C₇:₄D₄², C₄:₂(D₄xD₇), C₂₈:₃(C₂xD₄), D₁₄:₈(C₂xD₄), C₄:₁D₄:₆D₇, (C₄xD₂₈):₄₉C₂, (C₂xD₄):₂₆D₁₄, C₂₈:₂D₄:₃₆C₂, (C₄xC₂₈):₂₇C₂², C₂³:D₁₄:₂₇C₂, D₁₄:C₄:₇₀C₂², (D₄xC₁₄):₃₃C₂², C₄:Dic₇:₇₄C₂², C₁₄.₉₅(C₂²xD₄), (C₂xC₂₈).₅₀₉C₂³, (C₂xC₁₄).₂₆₁C₂⁴, (C₂³xD₇):₁₃C₂², C₂.₈₁(D₄:₆D₁₄), C₂³.D₇:₃₇C₂², C₂³.₆₇(C₂²xD₇), (C₂xD₂₈).₂₆₉C₂², (C₂²xC₁₄).₇₅C₂³, C₂².₂₈₂(C₂³xD₇), (C₂xDic₇).₁₃₆C₂³, (C₂²xD₇).₂₂₉C₂³, (C₂xD₄xD₇):₂₀C₂, C₂.₆₈(C₂xD₄xD₇), (C₇xC₄:₁D₄):₈C₂, (C₂xC₄xD₇):₂₉C₂², (C₂xC₇:D₄):₂₇C₂², (C₂xC₄).₂₁₄(C₂²xD₇), SmallGroup(448,1170)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₁₄ — D₂₈:₁₁D₄

Chief series

C₁ — C₇ — C₁₄ — C₂xC₁₄ — C₂²xD₇ — C₂³xD₇ — C₂xD₄xD₇ — D₂₈:₁₁D₄

Lower central

C₇ — C₂xC₁₄ — D₂₈:₁₁D₄

Upper central

C₁ — C₂² — C₄:₁D₄

Generators and relations for D₂₈:₁₁D₄
G = < a,b,c,d | a²⁸=b²=c⁴=d²=1, bab=a^-1, ac=ca, dad=a¹⁵, cbc^-1=a¹⁴b, bd=db, dcd=c^-1 >

Subgroups: 2572 in 428 conjugacy classes, 115 normal (13 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₂xC₄, C₂xC₄, C₂xC₄, D₄, C₂³, C₂³, D₇, C₁₄, C₁₄, C₁₄, C₄², C₂²:C₄, C₄:C₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂⁴, Dic₇, C₂₈, C₂₈, D₁₄, D₁₄, C₂xC₁₄, C₂xC₁₄, C₄xD₄, C₂²wrC₂, C₄:D₄, C₄:₁D₄, C₂²xD₄, C₄xD₇, D₂₈, C₂xDic₇, C₇:D₄, C₂xC₂₈, C₂xC₂₈, C₇xD₄, C₂²xD₇, C₂²xD₇, C₂²xC₁₄, D₄², C₄:Dic₇, D₁₄:C₄, C₂³.D₇, C₄xC₂₈, C₂xC₄xD₇, C₂xD₂₈, D₄xD₇, C₂xC₇:D₄, D₄xC₁₄, C₂³xD₇, C₄xD₂₈, C₂³:D₁₄, C₂₈:₂D₄, C₇xC₄:₁D₄, C₂xD₄xD₇, D₂₈:₁₁D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, C₂xD₄, C₂⁴, D₁₄, C₂²xD₄, 2₊¹⁺⁴, C₂²xD₇, D₄², D₄xD₇, C₂³xD₇, C₂xD₄xD₇, D₄:₆D₁₄, D₂₈:₁₁D₄

Smallest permutation representation of D₂₈:₁₁D₄
►On 112 points

Generators in S₁₁₂

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

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

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

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

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

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

67 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	···	2O	4A	4B	4C	4D	4E	4F	4G	4H	4I	7A	7B	7C	14A	···	14I	14J	···	14U	28A	···	28R
order	1	2	2	2	2	2	2	2	2	···	2	4	4	4	4	4	4	4	4	4	7	7	7	14	···	14	14	···	14	28	···	28
size	1	1	1	1	4	4	4	4	14	···	14	2	2	2	2	4	28	28	28	28	2	2	2	2	···	2	8	···	8	4	···	4

67 irreducible representations

dim

type

image

C₁

C₂

D₄

D₇

D₁₄

2₊¹⁺⁴

D₄xD₇

D₄:₆D₁₄

kernel

D₂₈:₁₁D₄

C₄xD₂₈

C₂³:D₁₄

C₂₈:₂D₄

C₇xC₄:₁D₄

C₂xD₄xD₇

D₂₈

C₄:₁D₄

C₄²

C₂xD₄

C₁₄

C₄

C₂

# reps

Matrix representation of D₂₈:₁₁D₄ ►in GL₆(F₂₉)

8	26	0	0	0	0
3	28	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	0	28
0	0	0	0	1	0

8	26	0	0	0	0
21	21	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	0	1
0	0	0	0	1	0

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

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

G:=sub<GL(6,GF(29))| [8,3,0,0,0,0,26,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,28,0],[8,21,0,0,0,0,26,21,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,28,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

D₂₈:₁₁D₄ in GAP, Magma, Sage, TeX

D_{28}\rtimes_{11}D_4

% in TeX

G:=Group("D28:11D4");

// GroupNames label

G:=SmallGroup(448,1170);

// by ID

G=gap.SmallGroup(448,1170);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,1571,570,297,136,18822]);

// Polycyclic

G:=Group<a,b,c,d|a^28=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^15,c*b*c^-1=a^14*b,b*d=d*b,d*c*d=c^-1>;

// generators/relations

G = D28:11D4 order 448 = 26·7

4th semidirect product of D28 and D4 acting via D4/C4=C2

G = D₂₈:₁₁D₄ order 448 = 2⁶·7

4^th semidirect product of D₂₈ and D₄ acting via D₄/C₄=C₂