D56:11C4 - GroupNames

G = D₅₆:₁₁C₄ order 448 = 2⁶·7

The semidirect product of D₅₆ and C₄ acting through Inn(D₅₆)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₅₆:₁₁C₄, C₈.₁₄D₂₈, C₅₆.₆₇D₄, Dic₂₈:₁₁C₄, C₄².₂₆₇D₁₄, (C₄xC₈):₁₀D₇, C₇:₁(C₈oD₈), (C₄xC₅₆):₁₅C₂, C₅₆:C₂:₇C₄, C₈.₂₂(C₄xD₇), C₁₄.₉(C₄xD₄), C₅₆.₅₂(C₂xC₄), C₄.₇₆(C₂xD₂₈), C₂.₁₂(C₄xD₂₈), D₂₈.₁₃(C₂xC₄), (C₂xC₈).₃₂₄D₁₄, C₂₈.₂₉₆(C₂xD₄), C₅₆.C₄:₁₇C₂, Dic₁₄:C₄:₁₅C₂, D₅₆:₇C₂.₁₀C₂, D₂₈.₂C₄:₁₁C₂, (C₄xC₂₈).₃₂₈C₂², C₂₈.₁₀₄(C₂²xC₄), (C₂xC₂₈).₇₉₁C₂³, (C₂xC₅₆).₄₀₇C₂², Dic₁₄.₁₃(C₂xC₄), C₄oD₂₈.₃₅C₂², C₂².₂₀(C₄oD₂₈), C₄.Dic₇.₃₃C₂², C₄.₆₂(C₂xC₄xD₇), (C₂xC₁₄).₆₂(C₄oD₄), (C₂xC₄).₆₇₂(C₂²xD₇), SmallGroup(448,234)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₈ — D₅₆:₁₁C₄

Chief series

C₁ — C₇ — C₁₄ — C₂₈ — C₂xC₂₈ — C₄oD₂₈ — D₅₆:₇C₂ — D₅₆:₁₁C₄

Lower central

C₇ — C₁₄ — C₂₈ — D₅₆:₁₁C₄

Upper central

C₁ — C₈ — C₂xC₈ — C₄xC₈

Generators and relations for D₅₆:₁₁C₄
G = < a,b,c | a⁵⁶=b²=c⁴=1, bab=a^-1, ac=ca, cbc^-1=a⁴²b >

Subgroups: 484 in 106 conjugacy classes, 47 normal (33 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₈, C₈, C₂xC₄, C₂xC₄, D₄, Q₈, D₇, C₁₄, C₁₄, C₄², C₂xC₈, C₂xC₈, M₄(2), D₈, SD₁₆, Q₁₆, C₄oD₄, Dic₇, C₂₈, C₂₈, D₁₄, C₂xC₁₄, C₄xC₈, C₄wrC₂, C₈.C₄, C₈oD₄, C₄oD₈, C₇:C₈, C₅₆, Dic₁₄, C₄xD₇, D₂₈, C₇:D₄, C₂xC₂₈, C₂xC₂₈, C₈oD₈, C₈xD₇, C₈:D₇, C₅₆:C₂, D₅₆, Dic₂₈, C₄.Dic₇, C₄xC₂₈, C₂xC₅₆, C₄oD₂₈, Dic₁₄:C₄, C₅₆.C₄, C₄xC₅₆, D₂₈.₂C₄, D₅₆:₇C₂, D₅₆:₁₁C₄
Quotients: C₁, C₂, C₄, C₂², C₂xC₄, D₄, C₂³, D₇, C₂²xC₄, C₂xD₄, C₄oD₄, D₁₄, C₄xD₄, C₄xD₇, D₂₈, C₂²xD₇, C₈oD₈, C₂xC₄xD₇, C₂xD₂₈, C₄oD₂₈, C₄xD₂₈, D₅₆:₁₁C₄

Smallest permutation representation of D₅₆:₁₁C₄
►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)

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

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

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

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

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

124 conjugacy classes

class	1	2A	2B	2C	2D	4A	4B	4C	···	4G	4H	4I	7A	7B	7C	8A	8B	8C	8D	8E	···	8J	8K	8L	8M	8N	14A	···	14I	28A	···	28AJ	56A	···	56AV
order	1	2	2	2	2	4	4	4	···	4	4	4	7	7	7	8	8	8	8	8	···	8	8	8	8	8	14	···	14	28	···	28	56	···	56
size	1	1	2	28	28	1	1	2	···	2	28	28	2	2	2	1	1	1	1	2	···	2	28	28	28	28	2	···	2	2	···	2	2	···	2

124 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2
type	+	+	+	+	+	+				+	+		+	+		+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	D₄	D₇	C₄oD₄	D₁₄	D₁₄	C₄xD₇	D₂₈	C₈oD₈	C₄oD₂₈	D₅₆:₁₁C₄
kernel	D₅₆:₁₁C₄	Dic₁₄:C₄	C₅₆.C₄	C₄xC₅₆	D₂₈.₂C₄	D₅₆:₇C₂	C₅₆:C₂	D₅₆	Dic₂₈	C₅₆	C₄xC₈	C₂xC₁₄	C₄²	C₂xC₈	C₈	C₈	C₇	C₂²	C₁
# reps	1	2	1	1	2	1	4	2	2	2	3	2	3	6	12	12	8	12	48

Matrix representation of D₅₆:₁₁C₄ ►in GL₂(F₁₁₃) generated by

104	0
0	25

0	25
104	0

112	0
0	15

G:=sub<GL(2,GF(113))| [104,0,0,25],[0,104,25,0],[112,0,0,15] >;

D₅₆:₁₁C₄ in GAP, Magma, Sage, TeX

D_{56}\rtimes_{11}C_4

% in TeX

G:=Group("D56:11C4");

// GroupNames label

G:=SmallGroup(448,234);

// by ID

G=gap.SmallGroup(448,234);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,120,58,136,1684,102,18822]);

// Polycyclic

G:=Group<a,b,c|a^56=b^2=c^4=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^42*b>;

// generators/relations

G = D56:11C4 order 448 = 26·7

The semidirect product of D56 and C4 acting through Inn(D56)

G = D₅₆:₁₁C₄ order 448 = 2⁶·7

The semidirect product of D₅₆ and C₄ acting through Inn(D₅₆)