D28.39C2^3 - GroupNames

G = D₂₈.₃₉C₂³ order 448 = 2⁶·7

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂₈.₅₃C₂⁴, C₁₄.₁₈C₂⁵, D₁₄.₁₀C₂⁴, D₂₈.₃₉C₂³, 2_-¹⁺⁴⋊₄D₇, Dic₇.₁₃C₂⁴, Dic₁₄.₄₀C₂³, C₄○D₄⋊₁₃D₁₄, (C₂×Q₈)⋊₂₅D₁₄, (D₄×D₇)⋊₁₅C₂², (C₂×C₁₄).₉C₂⁴, D₄⋊₈D₁₄⋊₁₂C₂, (Q₈×D₇)⋊₁₈C₂², C₄.₅₀(C₂³×D₇), C₂.₁₉(D₇×C₂⁴), C₇⋊D₄.₅C₂³, C₄○D₂₈⋊₁₅C₂², (C₂×D₂₈)⋊₄₁C₂², C₇⋊₃(C₂.C₂⁵), (Q₈×C₁₄)⋊₂₅C₂², (C₇×D₄).₃₃C₂³, D₄.₃₃(C₂²×D₇), (C₄×D₇).₂₂C₂³, Q₈.₃₄(C₂²×D₇), (C₇×Q₈).₃₄C₂³, D₄⋊₂D₇⋊₁₉C₂², C₂².₆(C₂³×D₇), (C₂×C₂₈).₁₂₄C₂³, Q₈.₁₀D₁₄⋊₈C₂, Q₈⋊₂D₇⋊₁₇C₂², (C₇×2_-¹⁺⁴)⋊₅C₂, (C₂×Dic₇).₃₀₀C₂³, (C₂²×D₇).₁₄₄C₂³, (D₇×C₄○D₄)⋊₁₀C₂, (C₂×C₄×D₇)⋊₃₈C₂², (C₂×Q₈⋊₂D₇)⋊₂₂C₂, (C₇×C₄○D₄)⋊₁₃C₂², (C₂×C₄).₁₀₈(C₂²×D₇), SmallGroup(448,1382)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₄ — D₂₈.₃₉C₂³

Chief series

C₁ — C₇ — C₁₄ — D₁₄ — C₂²×D₇ — C₂×C₄×D₇ — D₇×C₄○D₄ — D₂₈.₃₉C₂³

Lower central

C₇ — C₁₄ — D₂₈.₃₉C₂³

Upper central

C₁ — C₂ — 2_-¹⁺⁴

Generators and relations for D₂₈.₃₉C₂³
G = < a,b,c,d,e | a²⁸=b²=c²=d²=e²=1, bab=a^-1, ac=ca, ad=da, eae=a¹³, cbc=a¹⁴b, bd=db, ebe=a²⁶b, dcd=ece=a¹⁴c, de=ed >

Subgroups: 3252 in 810 conjugacy classes, 443 normal (8 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, D₇, C₁₄, C₁₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, C₄○D₄, Dic₇, Dic₇, C₂₈, D₁₄, D₁₄, C₂×C₁₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, 2_-¹⁺⁴, Dic₁₄, C₄×D₇, D₂₈, C₂×Dic₇, C₇⋊D₄, C₂×C₂₈, C₇×D₄, C₇×Q₈, C₂²×D₇, C₂.C₂⁵, C₂×C₄×D₇, C₂×D₂₈, C₄○D₂₈, D₄×D₇, D₄⋊₂D₇, Q₈×D₇, Q₈⋊₂D₇, Q₈×C₁₄, C₇×C₄○D₄, C₂×Q₈⋊₂D₇, Q₈.₁₀D₁₄, D₇×C₄○D₄, D₄⋊₈D₁₄, C₇×2_-¹⁺⁴, D₂₈.₃₉C₂³
Quotients: C₁, C₂, C₂², C₂³, D₇, C₂⁴, D₁₄, C₂⁵, C₂²×D₇, C₂.C₂⁵, C₂³×D₇, D₇×C₂⁴, D₂₈.₃₉C₂³

Smallest permutation representation of D₂₈.₃₉C₂³
►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 46)(30 45)(31 44)(32 43)(33 42)(34 41)(35 40)(36 39)(37 38)(47 56)(48 55)(49 54)(50 53)(51 52)(57 80)(58 79)(59 78)(60 77)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(81 84)(82 83)(85 104)(86 103)(87 102)(88 101)(89 100)(90 99)(91 98)(92 97)(93 96)(94 95)(105 112)(106 111)(107 110)(108 109)

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

(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)

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

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

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

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

85 conjugacy classes

class	1	2A	2B	···	2F	2G	···	2P	4A	···	4J	4K	4L	4M	···	4Q	7A	7B	7C	14A	14B	14C	14D	···	14R	28A	···	28AD
order	1	2	2	···	2	2	···	2	4	···	4	4	4	4	···	4	7	7	7	14	14	14	14	···	14	28	···	28
size	1	1	2	···	2	14	···	14	2	···	2	7	7	14	···	14	2	2	2	2	2	2	4	···	4	4	···	4

85 irreducible representations

dim	1	1	1	1	1	1	2	2	2	4	8
type	+	+	+	+	+	+	+	+	+		+
image	C₁	C₂	C₂	C₂	C₂	C₂	D₇	D₁₄	D₁₄	C₂.C₂⁵	D₂₈.₃₉C₂³
kernel	D₂₈.₃₉C₂³	C₂×Q₈⋊₂D₇	Q₈.₁₀D₁₄	D₇×C₄○D₄	D₄⋊₈D₁₄	C₇×2_-¹⁺⁴	2_-¹⁺⁴	C₂×Q₈	C₄○D₄	C₇	C₁
# reps	1	5	5	10	10	1	3	15	30	2	3

Matrix representation of D₂₈.₃₉C₂³ ►in GL₆(𝔽₂₉)

10	7	0	0	0	0
22	1	0	0	0	0
0	0	12	0	0	0
0	0	5	17	12	12
0	0	0	0	0	12
0	0	0	0	12	0

10	7	0	0	0	0
19	19	0	0	0	0
0	0	17	17	12	0
0	0	24	12	17	17
0	0	0	0	0	17
0	0	0	0	12	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	28	0	0	0
0	0	0	0	0	1
0	0	27	28	1	1
0	0	0	1	0	0

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

28	0	0	0	0	0
22	1	0	0	0	0
0	0	1	0	28	28
0	0	27	28	1	1
0	0	0	0	0	28
0	0	0	0	28	0

G:=sub<GL(6,GF(29))| [10,22,0,0,0,0,7,1,0,0,0,0,0,0,12,5,0,0,0,0,0,17,0,0,0,0,0,12,0,12,0,0,0,12,12,0],[10,19,0,0,0,0,7,19,0,0,0,0,0,0,17,24,0,0,0,0,17,12,0,0,0,0,12,17,0,12,0,0,0,17,17,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,27,0,0,0,0,0,28,1,0,0,0,0,1,0,0,0,0,1,1,0],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,28,0,28,0,0,0,28,0,0,28],[28,22,0,0,0,0,0,1,0,0,0,0,0,0,1,27,0,0,0,0,0,28,0,0,0,0,28,1,0,28,0,0,28,1,28,0] >;

D₂₈.₃₉C₂³ in GAP, Magma, Sage, TeX

D_{28}._{39}C_2^3

% in TeX

G:=Group("D28.39C2^3");

// GroupNames label

G:=SmallGroup(448,1382);

// by ID

G=gap.SmallGroup(448,1382);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,570,1684,438,102,18822]);

// Polycyclic

G:=Group<a,b,c,d,e|a^28=b^2=c^2=d^2=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e=a^13,c*b*c=a^14*b,b*d=d*b,e*b*e=a^26*b,d*c*d=e*c*e=a^14*c,d*e=e*d>;

// generators/relations

G = D28.39C23 order 448 = 26·7

20th non-split extension by D28 of C23 acting via C23/C22=C2

G = D₂₈.₃₉C₂³ order 448 = 2⁶·7

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