C28:2D4 - GroupNames

G = C₂₈:₂D₄ order 224 = 2⁵·7

2^nd semidirect product of C₂₈ and D₄ acting via D₄/C₂=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂₈:₂D₄, D₁₄:₃D₄, C₂³.₈D₁₄, (C₂xD₄):₄D₇, (D₄xC₁₄):₃C₂, C₇:₄(C₄:D₄), C₄:₂(C₇:D₄), C₂.₂₆(D₄xD₇), C₄:Dic₇:₁₄C₂, (C₂xC₄).₅₁D₁₄, C₁₄.₅₀(C₂xD₄), C₂³.D₇:₁₁C₂, C₁₄.₃₁(C₄oD₄), (C₂xC₁₄).₅₃C₂³, (C₂xC₂₈).₃₄C₂², C₂.₁₇(D₄:₂D₇), C₂².₆₀(C₂²xD₇), (C₂²xC₁₄).₂₀C₂², (C₂xDic₇).₁₉C₂², (C₂²xD₇).₂₆C₂², (C₂xC₄xD₇):₂C₂, (C₂xC₇:D₄):₅C₂, C₂.₁₄(C₂xC₇:D₄), SmallGroup(224,133)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₁₄ — C₂₈:₂D₄

Chief series

C₁ — C₇ — C₁₄ — C₂xC₁₄ — C₂²xD₇ — C₂xC₄xD₇ — C₂₈:₂D₄

Lower central

C₇ — C₂xC₁₄ — C₂₈:₂D₄

Upper central

C₁ — C₂² — C₂xD₄

Generators and relations for C₂₈:₂D₄
G = < a,b,c | a²⁸=b⁴=c²=1, bab^-1=a^-1, cac=a¹³, cbc=b^-1 >

Subgroups: 382 in 94 conjugacy classes, 35 normal (19 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₂xC₄, C₂xC₄, D₄, C₂³, C₂³, D₇, C₁₄, C₁₄, C₂²:C₄, C₄:C₄, C₂²xC₄, C₂xD₄, C₂xD₄, Dic₇, C₂₈, D₁₄, D₁₄, C₂xC₁₄, C₂xC₁₄, C₄:D₄, C₄xD₇, C₂xDic₇, C₂xDic₇, C₇:D₄, C₂xC₂₈, C₇xD₄, C₂²xD₇, C₂²xC₁₄, C₄:Dic₇, C₂³.D₇, C₂xC₄xD₇, C₂xC₇:D₄, D₄xC₁₄, C₂₈:₂D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, C₂xD₄, C₄oD₄, D₁₄, C₄:D₄, C₇:D₄, C₂²xD₇, D₄xD₇, D₄:₂D₇, C₂xC₇:D₄, C₂₈:₂D₄

Smallest permutation representation of C₂₈:₂D₄
►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 53 99 65)(2 52 100 64)(3 51 101 63)(4 50 102 62)(5 49 103 61)(6 48 104 60)(7 47 105 59)(8 46 106 58)(9 45 107 57)(10 44 108 84)(11 43 109 83)(12 42 110 82)(13 41 111 81)(14 40 112 80)(15 39 85 79)(16 38 86 78)(17 37 87 77)(18 36 88 76)(19 35 89 75)(20 34 90 74)(21 33 91 73)(22 32 92 72)(23 31 93 71)(24 30 94 70)(25 29 95 69)(26 56 96 68)(27 55 97 67)(28 54 98 66)

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

dim

type

image

C₁

C₂

D₄

D₇

C₄oD₄

D₁₄

C₇:D₄

D₄xD₇

D₄:₂D₇

kernel

C₂₈:₂D₄

C₄:Dic₇

C₂³.D₇

C₂xC₄xD₇

C₂xC₇:D₄

D₄xC₁₄

C₂₈

D₁₄

C₂xD₄

C₁₄

C₂xC₄

C₂³

C₄

C₂

# reps

Matrix representation of C₂₈:₂D₄ ►in GL₄(F₂₉) generated by

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

20	15	0	0
10	9	0	0
0	0	28	27
0	0	1	1

1	0	0	0
7	28	0	0
0	0	28	0
0	0	1	1

G:=sub<GL(4,GF(29))| [1,7,0,0,22,10,0,0,0,0,12,17,0,0,0,17],[20,10,0,0,15,9,0,0,0,0,28,1,0,0,27,1],[1,7,0,0,0,28,0,0,0,0,28,1,0,0,0,1] >;

C₂₈:₂D₄ in GAP, Magma, Sage, TeX

C_{28}\rtimes_2D_4

% in TeX

G:=Group("C28:2D4");

// GroupNames label

G:=SmallGroup(224,133);

// by ID

G=gap.SmallGroup(224,133);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,218,188,6917]);

// Polycyclic

G:=Group<a,b,c|a^28=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^13,c*b*c=b^-1>;

// generators/relations

G = C28:2D4 order 224 = 25·7

2nd semidirect product of C28 and D4 acting via D4/C2=C22

G = C₂₈:₂D₄ order 224 = 2⁵·7

2^nd semidirect product of C₂₈ and D₄ acting via D₄/C₂=C₂²