D28.32C2^3 - GroupNames

G = D₂₈.₃₂C₂³ order 448 = 2⁶·7

13^rd non-split extension by D₂₈ of C₂³ acting via C₂³/C₂²=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂₈.₃₇C₂⁴, D₂₈.₃₂C₂³, 2₊¹⁺⁴⋊₃D₇, Dic₁₄.₃₂C₂³, C₇⋊₅(D₄○D₈), C₄○D₄⋊₅D₁₄, (C₂×D₄)⋊₁₆D₁₄, (C₇×D₄).₃₆D₄, C₇⋊C₈.₁₆C₂³, (C₇×Q₈).₃₆D₄, D₄⋊₈D₁₄⋊₉C₂, D₄⋊D₇⋊₂₀C₂², C₂₈.₂₆₉(C₂×D₄), Q₈⋊D₇⋊₁₉C₂², D₄⋊D₁₄⋊₁₁C₂, C₄.₃₇(C₂³×D₇), D₄.₈D₁₄⋊₈C₂, C₄○D₂₈⋊₁₀C₂², D₄.₁₈(C₇⋊D₄), (C₂×D₂₈)⋊₃₉C₂², (D₄×C₁₄)⋊₂₄C₂², Q₈.Dic₇⋊₁₀C₂, D₄.D₇⋊₁₉C₂², Q₈.₁₈(C₇⋊D₄), D₄.₂₅(C₂²×D₇), (C₇×D₄).₂₅C₂³, C₇⋊Q₁₆⋊₂₁C₂², D₄.D₁₄⋊₁₁C₂, (C₇×Q₈).₂₅C₂³, Q₈.₂₅(C₂²×D₇), (C₂×C₂₈).₁₁₈C₂³, C₁₄.₁₇₁(C₂²×D₄), C₄.Dic₇⋊₁₆C₂², (C₇×2₊¹⁺⁴)⋊₂C₂, (C₂×D₄⋊D₇)⋊₃₂C₂, (C₂×C₇⋊C₈)⋊₂₄C₂², C₄.₇₅(C₂×C₇⋊D₄), (C₂×C₁₄).₈₅(C₂×D₄), (C₇×C₄○D₄)⋊₈C₂², C₂².₆(C₂×C₇⋊D₄), C₂.₄₄(C₂²×C₇⋊D₄), (C₂×C₄).₁₀₂(C₂²×D₇), SmallGroup(448,1288)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₈ — D₂₈.₃₂C₂³

Chief series

C₁ — C₇ — C₁₄ — C₂₈ — D₂₈ — C₂×D₂₈ — D₄⋊₈D₁₄ — D₂₈.₃₂C₂³

Lower central

C₇ — C₁₄ — C₂₈ — D₂₈.₃₂C₂³

Upper central

C₁ — C₂ — C₄○D₄ — 2₊¹⁺⁴

Generators and relations for D₂₈.₃₂C₂³
G = < a,b,c,d,e | a²⁸=b²=c²=d²=e²=1, bab=dad=a^-1, ac=ca, eae=a¹⁵, cbc=a¹⁴b, dbd=a²⁶b, ebe=a⁷b, cd=dc, ce=ec, ede=a²¹d >

Subgroups: 1236 in 268 conjugacy classes, 107 normal (20 characteristic)
C₁, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₇, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, D₇, C₁₄, C₁₄, C₂×C₈, M₄(2), D₈, SD₁₆, Q₁₆, C₂×D₄, C₂×D₄, C₄○D₄, C₄○D₄, C₄○D₄, Dic₇, C₂₈, C₂₈, C₂₈, D₁₄, C₂×C₁₄, C₂×C₁₄, C₈○D₄, C₂×D₈, C₄○D₈, 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×Q₈, C₂²×D₇, C₂²×C₁₄, D₄○D₈, C₂×C₇⋊C₈, C₄.Dic₇, D₄⋊D₇, D₄.D₇, Q₈⋊D₇, C₇⋊Q₁₆, C₂×D₂₈, C₄○D₂₈, D₄×D₇, Q₈⋊₂D₇, D₄×C₁₄, D₄×C₁₄, C₇×C₄○D₄, C₇×C₄○D₄, C₇×C₄○D₄, C₂×D₄⋊D₇, D₄.D₁₄, Q₈.Dic₇, D₄⋊D₁₄, D₄.₈D₁₄, D₄⋊₈D₁₄, C₇×2₊¹⁺⁴, D₂₈.₃₂C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, C₂×D₄, C₂⁴, D₁₄, C₂²×D₄, C₇⋊D₄, C₂²×D₇, D₄○D₈, C₂×C₇⋊D₄, C₂³×D₇, C₂²×C₇⋊D₄, D₂₈.₃₂C₂³

Smallest permutation representation of D₂₈.₃₂C₂³
►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 28)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(29 41)(30 40)(31 39)(32 38)(33 37)(34 36)(42 56)(43 55)(44 54)(45 53)(46 52)(47 51)(48 50)(57 78)(58 77)(59 76)(60 75)(61 74)(62 73)(63 72)(64 71)(65 70)(66 69)(67 68)(79 84)(80 83)(81 82)(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 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 81)(22 82)(23 83)(24 84)(25 57)(26 58)(27 59)(28 60)(29 91)(30 92)(31 93)(32 94)(33 95)(34 96)(35 97)(36 98)(37 99)(38 100)(39 101)(40 102)(41 103)(42 104)(43 105)(44 106)(45 107)(46 108)(47 109)(48 110)(49 111)(50 112)(51 85)(52 86)(53 87)(54 88)(55 89)(56 90)

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

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

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

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

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

73 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	2J	4A	4B	4C	4D	4E	4F	7A	7B	7C	8A	8B	8C	8D	8E	14A	14B	14C	14D	···	14AD	28A	···	28R
order	1	2	2	2	2	2	2	2	2	2	2	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	4	4	28	28	28	2	2	2	2	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○D₈

D₂₈.₃₂C₂³

kernel

D₂₈.₃₂C₂³

C₂×D₄⋊D₇

D₄.D₁₄

Q₈.Dic₇

D₄⋊D₁₄

D₄.₈D₁₄

D₄⋊₈D₁₄

C₇×2₊¹⁺⁴

C₇×D₄

C₇×Q₈

2₊¹⁺⁴

C₂×D₄

C₄○D₄

D₄

Q₈

C₇

C₁

# reps

Matrix representation of D₂₈.₃₂C₂³ ►in GL₆(𝔽₁₁₃)

80	104	0	0	0	0
9	112	0	0	0	0
0	0	0	1	0	0
0	0	112	0	0	0
0	0	0	0	0	1
0	0	0	0	112	0

80	104	0	0	0	0
33	33	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	81	0	112
0	0	81	0	112	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	106	0
0	0	0	1	0	106
0	0	0	0	112	0
0	0	0	0	0	112

112	0	0	0	0	0
104	1	0	0	0	0
0	0	1	0	0	0
0	0	0	112	0	0
0	0	0	0	1	0
0	0	0	0	0	112

112	0	0	0	0	0
0	112	0	0	0	0
0	0	82	82	0	0
0	0	82	31	0	0
0	0	0	0	82	82
0	0	0	0	82	31

G:=sub<GL(6,GF(113))| [80,9,0,0,0,0,104,112,0,0,0,0,0,0,0,112,0,0,0,0,1,0,0,0,0,0,0,0,0,112,0,0,0,0,1,0],[80,33,0,0,0,0,104,33,0,0,0,0,0,0,0,1,0,81,0,0,1,0,81,0,0,0,0,0,0,112,0,0,0,0,112,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,106,0,112,0,0,0,0,106,0,112],[112,104,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,112],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,82,82,0,0,0,0,82,31,0,0,0,0,0,0,82,82,0,0,0,0,82,31] >;

D₂₈.₃₂C₂³ in GAP, Magma, Sage, TeX

D_{28}._{32}C_2^3

% in TeX

G:=Group("D28.32C2^3");

// GroupNames label

G:=SmallGroup(448,1288);

// by ID

G=gap.SmallGroup(448,1288);

# by ID

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

// Polycyclic

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

// generators/relations

G = D28.32C23 order 448 = 26·7

13rd non-split extension by D28 of C23 acting via C23/C22=C2

G = D₂₈.₃₂C₂³ order 448 = 2⁶·7

13^rd non-split extension by D₂₈ of C₂³ acting via C₂³/C₂²=C₂