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(D₄×D₇), C₂₈⋊₃(C₂×D₄), D₁₄⋊₈(C₂×D₄), C₄⋊₁D₄⋊₆D₇, (C₄×D₂₈)⋊₄₉C₂, (C₂×D₄)⋊₂₆D₁₄, C₂₈⋊₂D₄⋊₃₆C₂, (C₄×C₂₈)⋊₂₇C₂², C₂³⋊D₁₄⋊₂₇C₂, D₁₄⋊C₄⋊₇₀C₂², (D₄×C₁₄)⋊₃₃C₂², C₄⋊Dic₇⋊₇₄C₂², C₁₄.₉₅(C₂²×D₄), (C₂×C₂₈).₅₀₉C₂³, (C₂×C₁₄).₂₆₁C₂⁴, (C₂³×D₇)⋊₁₃C₂², C₂.₈₁(D₄⋊₆D₁₄), C₂³.D₇⋊₃₇C₂², C₂³.₆₇(C₂²×D₇), (C₂×D₂₈).₂₆₉C₂², (C₂²×C₁₄).₇₅C₂³, C₂².₂₈₂(C₂³×D₇), (C₂×Dic₇).₁₃₆C₂³, (C₂²×D₇).₂₂₉C₂³, (C₂×D₄×D₇)⋊₂₀C₂, C₂.₆₈(C₂×D₄×D₇), (C₇×C₄⋊₁D₄)⋊₈C₂, (C₂×C₄×D₇)⋊₂₉C₂², (C₂×C₇⋊D₄)⋊₂₇C₂², (C₂×C₄).₂₁₄(C₂²×D₇), SmallGroup(448,1170)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₄ — D₂₈⋊₁₁D₄

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂²×D₇ — C₂³×D₇ — C₂×D₄×D₇ — D₂₈⋊₁₁D₄

Lower central

C₇ — C₂×C₁₄ — 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, C₂×C₄, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, D₇, C₁₄, C₁₄, C₁₄, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, Dic₇, C₂₈, C₂₈, D₁₄, D₁₄, C₂×C₁₄, C₂×C₁₄, C₄×D₄, C₂²≀C₂, C₄⋊D₄, C₄⋊₁D₄, C₂²×D₄, C₄×D₇, D₂₈, C₂×Dic₇, C₇⋊D₄, C₂×C₂₈, C₂×C₂₈, C₇×D₄, C₂²×D₇, C₂²×D₇, C₂²×C₁₄, D₄², C₄⋊Dic₇, D₁₄⋊C₄, C₂³.D₇, C₄×C₂₈, C₂×C₄×D₇, C₂×D₂₈, D₄×D₇, C₂×C₇⋊D₄, D₄×C₁₄, C₂³×D₇, C₄×D₂₈, C₂³⋊D₁₄, C₂₈⋊₂D₄, C₇×C₄⋊₁D₄, C₂×D₄×D₇, D₂₈⋊₁₁D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, C₂×D₄, C₂⁴, D₁₄, C₂²×D₄, 2₊¹⁺⁴, C₂²×D₇, D₄², D₄×D₇, C₂³×D₇, C₂×D₄×D₇, 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

D₄×D₇

D₄⋊₆D₁₄

kernel

D₂₈⋊₁₁D₄

C₄×D₂₈

C₂³⋊D₁₄

C₂₈⋊₂D₄

C₇×C₄⋊₁D₄

C₂×D₄×D₇

D₂₈

C₄⋊₁D₄

C₄²

C₂×D₄

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