D28.34C2^3 - GroupNames

G = D₂₈.₃₄C₂³ order 448 = 2⁶·7

15^th non-split extension by D₂₈ of C₂³ acting via C₂³/C₂²=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂₈.₃₉C₂⁴, D₂₈.₃₄C₂³, 2_-¹⁺⁴⋊₂D₇, Dic₁₄.₃₄C₂³, C₄○D₄⋊₆D₁₄, (C₂×Q₈)⋊₁₃D₁₄, (C₇×D₄).₃₈D₄, C₇⋊₇(D₄○SD₁₆), C₇⋊C₈.₁₈C₂³, (C₇×Q₈).₃₈D₄, D₄⋊D₇⋊₂₂C₂², C₂₈.₂₇₁(C₂×D₄), Q₈⋊D₇⋊₂₀C₂², D₄⋊₈D₁₄⋊₁₀C₂, D₄⋊D₁₄⋊₁₂C₂, C₄.₃₉(C₂³×D₇), D₄.₂₀(C₇⋊D₄), Q₈.Dic₇⋊₁₂C₂, D₄.D₇⋊₂₂C₂², Q₈.₂₀(C₇⋊D₄), (Q₈×C₁₄)⋊₂₃C₂², C₇⋊Q₁₆⋊₁₉C₂², D₄.₂₇(C₂²×D₇), (C₇×D₄).₂₇C₂³, D₄.₈D₁₄⋊₁₀C₂, Q₈.₂₇(C₂²×D₇), (C₇×Q₈).₂₇C₂³, C₂₈.C₂³⋊₁₁C₂, (C₂×C₂₈).₁₂₀C₂³, C₄○D₂₈.₃₃C₂², C₁₄.₁₇₃(C₂²×D₄), C₄.Dic₇⋊₁₈C₂², (C₇×2_-¹⁺⁴)⋊₂C₂, (C₂×D₂₈).₁₈₆C₂², (C₂×C₇⋊C₈)⋊₂₆C₂², (C₂×Q₈⋊D₇)⋊₃₂C₂, C₄.₇₇(C₂×C₇⋊D₄), (C₂×C₁₄).₈₇(C₂×D₄), (C₇×C₄○D₄)⋊₉C₂², C₂².₈(C₂×C₇⋊D₄), C₂.₄₆(C₂²×C₇⋊D₄), (C₂×C₄).₁₀₄(C₂²×D₇), SmallGroup(448,1290)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₈ — D₂₈.₃₄C₂³

Chief series

C₁ — C₇ — C₁₄ — C₂₈ — D₂₈ — C₂×D₂₈ — D₄⋊₈D₁₄ — D₂₈.₃₄C₂³

Lower central

C₇ — C₁₄ — C₂₈ — D₂₈.₃₄C₂³

Upper central

C₁ — C₂ — C₄○D₄ — 2_-¹⁺⁴

Generators and relations for D₂₈.₃₄C₂³
G = < a,b,c,d,e | a²⁸=b²=c²=d²=1, e²=a¹⁴, bab=dad=a^-1, ac=ca, eae^-1=a¹⁵, cbc=a¹⁴b, dbd=a²⁶b, ebe^-1=a⁷b, cd=dc, ce=ec, ede^-1=a²¹d >

Subgroups: 1172 in 258 conjugacy classes, 107 normal (20 characteristic)
C₁, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₇, C₈, C₂×C₄, C₂×C₄, D₄, D₄, D₄, Q₈, Q₈, Q₈, C₂³, D₇, C₁₄, C₁₄, C₂×C₈, M₄(2), D₈, SD₁₆, Q₁₆, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, C₄○D₄, C₄○D₄, Dic₇, C₂₈, C₂₈, C₂₈, D₁₄, C₂×C₁₄, C₂×C₁₄, C₈○D₄, C₂×SD₁₆, C₄○D₈, C₈⋊C₂², C₈.C₂², 2₊¹⁺⁴, 2_-¹⁺⁴, C₇⋊C₈, C₇⋊C₈, Dic₁₄, C₄×D₇, D₂₈, D₂₈, C₇⋊D₄, C₂×C₂₈, C₂×C₂₈, C₇×D₄, C₇×D₄, C₇×D₄, C₇×Q₈, C₇×Q₈, C₇×Q₈, C₂²×D₇, D₄○SD₁₆, C₂×C₇⋊C₈, C₄.Dic₇, D₄⋊D₇, D₄.D₇, Q₈⋊D₇, C₇⋊Q₁₆, C₂×D₂₈, C₄○D₂₈, D₄×D₇, Q₈⋊₂D₇, Q₈×C₁₄, Q₈×C₁₄, C₇×C₄○D₄, C₇×C₄○D₄, C₇×C₄○D₄, C₂×Q₈⋊D₇, C₂₈.C₂³, Q₈.Dic₇, D₄⋊D₁₄, D₄.₈D₁₄, D₄⋊₈D₁₄, C₇×2_-¹⁺⁴, D₂₈.₃₄C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, C₂×D₄, C₂⁴, D₁₄, C₂²×D₄, C₇⋊D₄, C₂²×D₇, D₄○SD₁₆, C₂×C₇⋊D₄, C₂³×D₇, C₂²×C₇⋊D₄, D₂₈.₃₄C₂³

Smallest permutation representation of D₂₈.₃₄C₂³
►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 35)(30 34)(31 33)(36 56)(37 55)(38 54)(39 53)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(57 62)(58 61)(59 60)(63 84)(64 83)(65 82)(66 81)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)(73 74)(85 95)(86 94)(87 93)(88 92)(89 91)(96 112)(97 111)(98 110)(99 109)(100 108)(101 107)(102 106)(103 105)

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

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

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

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

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

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

73 conjugacy classes

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

73 irreducible representations

dim

type

image

C₁

C₂

D₄

D₇

D₁₄

C₇⋊D₄

D₄○SD₁₆

D₂₈.₃₄C₂³

kernel

D₂₈.₃₄C₂³

C₂×Q₈⋊D₇

C₂₈.C₂³

Q₈.Dic₇

D₄⋊D₁₄

D₄.₈D₁₄

D₄⋊₈D₁₄

C₇×2_-¹⁺⁴

C₇×D₄

C₇×Q₈

2_-¹⁺⁴

C₂×Q₈

C₄○D₄

D₄

Q₈

C₇

C₁

# reps

Matrix representation of D₂₈.₃₄C₂³ ►in GL₆(𝔽₁₁₃)

25	34	0	0	0	0
79	112	0	0	0	0
0	0	0	1	0	0
0	0	112	0	0	0
0	0	112	112	1	2
0	0	0	1	112	112

25	34	0	0	0	0
88	88	0	0	0	0
0	0	0	112	0	0
0	0	112	0	0	0
0	0	0	0	112	0
0	0	112	112	1	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	100	100	26	26
0	0	100	100	0	26
0	0	100	13	0	0
0	0	100	87	13	26

112	0	0	0	0	0
34	1	0	0	0	0
0	0	100	100	0	26
0	0	100	100	26	26
0	0	13	100	0	0
0	0	87	100	13	26

112	0	0	0	0	0
0	112	0	0	0	0
0	0	0	0	1	0
0	0	1	1	112	111
0	0	112	0	0	0
0	0	1	1	0	112

G:=sub<GL(6,GF(113))| [25,79,0,0,0,0,34,112,0,0,0,0,0,0,0,112,112,0,0,0,1,0,112,1,0,0,0,0,1,112,0,0,0,0,2,112],[25,88,0,0,0,0,34,88,0,0,0,0,0,0,0,112,0,112,0,0,112,0,0,112,0,0,0,0,112,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,100,100,100,100,0,0,100,100,13,87,0,0,26,0,0,13,0,0,26,26,0,26],[112,34,0,0,0,0,0,1,0,0,0,0,0,0,100,100,13,87,0,0,100,100,100,100,0,0,0,26,0,13,0,0,26,26,0,26],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,1,112,1,0,0,0,1,0,1,0,0,1,112,0,0,0,0,0,111,0,112] >;

D₂₈.₃₄C₂³ in GAP, Magma, Sage, TeX

D_{28}._{34}C_2^3

% in TeX

G:=Group("D28.34C2^3");

// GroupNames label

G:=SmallGroup(448,1290);

// by ID

G=gap.SmallGroup(448,1290);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,387,675,136,1684,235,102,18822]);

// Polycyclic

G:=Group<a,b,c,d,e|a^28=b^2=c^2=d^2=1,e^2=a^14,b*a*b=d*a*d=a^-1,a*c=c*a,e*a*e^-1=a^15,c*b*c=a^14*b,d*b*d=a^26*b,e*b*e^-1=a^7*b,c*d=d*c,c*e=e*c,e*d*e^-1=a^21*d>;

// generators/relations

G = D28.34C23 order 448 = 26·7

15th non-split extension by D28 of C23 acting via C23/C22=C2

G = D₂₈.₃₄C₂³ order 448 = 2⁶·7

15^th non-split extension by D₂₈ of C₂³ acting via C₂³/C₂²=C₂