D4xC28 - GroupNames

G = D₄xC₂₈ order 224 = 2⁵·7

Direct product of C₂₈ and D₄

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: D₄xC₂₈, C₄²:₄C₁₄, C₄:C₄:₇C₁₄, C₄:₁(C₂xC₂₈), C₂₈:₆(C₂xC₄), (C₄xC₂₈):₁₁C₂, C₂.₃(D₄xC₁₄), C₂²:C₄:₆C₁₄, (C₂²xC₂₈):₄C₂, C₂²:₁(C₂xC₂₈), (C₂²xC₄):₂C₁₄, (C₂xD₄).₇C₁₄, C₁₄.₆₆(C₂xD₄), (D₄xC₁₄).₁₄C₂, C₂.₄(C₂²xC₂₈), C₂³.₇(C₂xC₁₄), C₁₄.₃₉(C₄oD₄), (C₂xC₁₄).₇₃C₂³, C₁₄.₃₂(C₂²xC₄), (C₂xC₂₈).₁₂₁C₂², C₂².₇(C₂²xC₁₄), (C₂²xC₁₄).₂₆C₂², (C₇xC₄:C₄):₁₆C₂, (C₂xC₁₄):₄(C₂xC₄), (C₂xC₂₈)o(D₄xC₁₄), C₂.₂(C₇xC₄oD₄), (C₇xC₂²:C₄):₁₄C₂, (C₂xC₄).₁₅(C₂xC₁₄), (C₂xC₂₈)o(C₇xC₄:C₄), (C₂xC₂₈)o(C₇xC₂²:C₄), SmallGroup(224,153)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂ — D₄xC₂₈

Chief series

C₁ — C₂ — C₂² — C₂xC₁₄ — C₂xC₂₈ — C₇xC₂²:C₄ — D₄xC₂₈

Lower central

C₁ — C₂ — D₄xC₂₈

Upper central

C₁ — C₂xC₂₈ — D₄xC₂₈

Generators and relations for D₄xC₂₈
G = < a,b,c | a²⁸=b⁴=c²=1, ab=ba, ac=ca, cbc=b^-1 >

Subgroups: 124 in 94 conjugacy classes, 64 normal (24 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₇, C₂xC₄, C₂xC₄, C₂xC₄, D₄, C₂³, C₁₄, C₁₄, C₄², C₂²:C₄, C₄:C₄, C₂²xC₄, C₂xD₄, C₂₈, C₂₈, C₂xC₁₄, C₂xC₁₄, C₂xC₁₄, C₄xD₄, C₂xC₂₈, C₂xC₂₈, C₂xC₂₈, C₇xD₄, C₂²xC₁₄, C₄xC₂₈, C₇xC₂²:C₄, C₇xC₄:C₄, C₂²xC₂₈, D₄xC₁₄, D₄xC₂₈
Quotients: C₁, C₂, C₄, C₂², C₇, C₂xC₄, D₄, C₂³, C₁₄, C₂²xC₄, C₂xD₄, C₄oD₄, C₂₈, C₂xC₁₄, C₄xD₄, C₂xC₂₈, C₇xD₄, C₂²xC₁₄, C₂²xC₂₈, D₄xC₁₄, C₇xC₄oD₄, D₄xC₂₈

Smallest permutation representation of D₄xC₂₈
►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)

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

(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(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)

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

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

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

D₄xC₂₈ is a maximal subgroup of
C₂₈.₅₇D₈ C₂₈.₅₀D₈ C₂₈.₃₈SD₁₆ D₄.₃Dic₁₄ C₄².₄₇D₁₄ C₂₈:₃M₄(2) C₄².₄₈D₁₄ C₂₈:₇D₈ D₄.₁D₂₈ C₄².₅₁D₁₄ D₄.₂D₂₈ C₄².₁₀₂D₁₄ D₄:₅Dic₁₄ C₄².₁₀₄D₁₄ C₄².₁₀₅D₁₄ C₄².₁₀₆D₁₄ D₄:₆Dic₁₄ C₄²:₁₁D₁₄ C₄².₁₀₈D₁₄ C₄²:₁₂D₁₄ C₄².₂₂₈D₁₄ D₂₈:₂₃D₄ D₂₈:₂₄D₄ Dic₁₄:₂₃D₄ Dic₁₄:₂₄D₄ D₄:₅D₂₈ D₄:₆D₂₈ C₄²:₁₆D₁₄ C₄².₂₂₉D₁₄ C₄².₁₁₃D₁₄ C₄².₁₁₄D₁₄ C₄²:₁₇D₁₄ C₄².₁₁₅D₁₄ C₄².₁₁₆D₁₄ C₄².₁₁₇D₁₄ C₄².₁₁₈D₁₄ C₄².₁₁₉D₁₄

140 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	···	4L	7A	···	7F	14A	···	14R	14S	···	14AP	28A	···	28X	28Y	···	28BT
order	1	2	2	2	2	2	2	2	4	4	4	4	4	···	4	7	···	7	14	···	14	14	···	14	28	···	28	28	···	28
size	1	1	1	1	2	2	2	2	1	1	1	1	2	···	2	1	···	1	1	···	1	2	···	2	1	···	1	2	···	2

140 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2
type	+	+	+	+	+	+									+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₇	C₁₄	C₁₄	C₁₄	C₁₄	C₁₄	C₂₈	D₄	C₄oD₄	C₇xD₄	C₇xC₄oD₄
kernel	D₄xC₂₈	C₄xC₂₈	C₇xC₂²:C₄	C₇xC₄:C₄	C₂²xC₂₈	D₄xC₁₄	C₇xD₄	C₄xD₄	C₄²	C₂²:C₄	C₄:C₄	C₂²xC₄	C₂xD₄	D₄	C₂₈	C₁₄	C₄	C₂
# reps	1	1	2	1	2	1	8	6	6	12	6	12	6	48	2	2	12	12

Matrix representation of D₄xC₂₈ ►in GL₃(F₂₉) generated by

12	0	0
0	20	0
0	0	20

1	0	0
0	1	2
0	28	28

1	0	0
0	1	0
0	28	28

G:=sub<GL(3,GF(29))| [12,0,0,0,20,0,0,0,20],[1,0,0,0,1,28,0,2,28],[1,0,0,0,1,28,0,0,28] >;

D₄xC₂₈ in GAP, Magma, Sage, TeX

D_4\times C_{28}

% in TeX

G:=Group("D4xC28");

// GroupNames label

G:=SmallGroup(224,153);

// by ID

G=gap.SmallGroup(224,153);

# by ID

G:=PCGroup([6,-2,-2,-2,-7,-2,-2,672,697,518]);

// Polycyclic

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

// generators/relations

G = D4xC28 order 224 = 25·7

Direct product of C28 and D4

G = D₄xC₂₈ order 224 = 2⁵·7

Direct product of C₂₈ and D₄