D56:10C4 - GroupNames

G = D₅₆:₁₀C₄ order 448 = 2⁶·7

10^th semidirect product of D₅₆ and C₄ acting via C₄/C₂=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₅₆:₁₀C₄, Dic₂₈:₁₀C₄, M₄(2).₂₆D₁₄, C₇:C₈.₃₅D₄, C₅₆:C₂:₄C₄, C₅₆:C₄:₁C₂, C₈.₁₁(C₄xD₇), C₇:₃(C₈.₂₆D₄), C₅₆.₂₉(C₂xC₄), C₄.₂₁₂(D₄xD₇), C₈.C₄:₄D₇, C₁₄.₅₅(C₄xD₄), (C₂xC₈).₆₉D₁₄, D₂₈:₄C₄:₇C₂, D₂₈.C₄:₁₀C₂, D₂₈.₁₀(C₂xC₄), C₂₈.₃₇₁(C₂xD₄), D₅₆:₇C₂.₄C₂, C₂₈.₅₄(C₂²xC₄), (C₂xC₅₆).₄₁C₂², (C₂xC₂₈).₃₁₀C₂³, Dic₁₄.₁₀(C₂xC₄), C₄oD₂₈.₁₇C₂², C₂.₁₅(D₂₈:C₄), C₂².₁(Q₈:₂D₇), (C₄xDic₇).₄₀C₂², (C₇xM₄(2)).₂₀C₂², C₄.₄₆(C₂xC₄xD₇), (C₇xC₈.C₄):₄C₂, (C₂xC₇:C₈).₇₈C₂², (C₂xC₁₄).₁(C₄oD₄), (C₂xC₄).₄₁₃(C₂²xD₇), SmallGroup(448,428)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₈ — D₅₆:₁₀C₄

Chief series

C₁ — C₇ — C₁₄ — C₂₈ — C₂xC₂₈ — C₄oD₂₈ — D₅₆:₇C₂ — D₅₆:₁₀C₄

Lower central

C₇ — C₁₄ — C₂₈ — D₅₆:₁₀C₄

Upper central

C₁ — C₄ — C₂xC₄ — C₈.C₄

Generators and relations for D₅₆:₁₀C₄
G = < a,b,c | a⁵⁶=b²=c⁴=1, bab=a^-1, cac^-1=a¹³, cbc^-1=a²⁶b >

Subgroups: 508 in 104 conjugacy classes, 45 normal (27 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₈, C₈, C₂xC₄, C₂xC₄, D₄, Q₈, D₇, C₁₄, C₁₄, C₄², C₂xC₈, C₂xC₈, M₄(2), M₄(2), D₈, SD₁₆, Q₁₆, C₄oD₄, Dic₇, C₂₈, D₁₄, C₂xC₁₄, C₈:C₄, C₄wrC₂, C₈.C₄, C₈oD₄, C₄oD₈, C₇:C₈, C₅₆, C₅₆, Dic₁₄, C₄xD₇, D₂₈, C₂xDic₇, C₇:D₄, C₂xC₂₈, C₈.₂₆D₄, C₈xD₇, C₈:D₇, C₅₆:C₂, D₅₆, Dic₂₈, C₂xC₇:C₈, C₄xDic₇, C₂xC₅₆, C₇xM₄(2), C₄oD₂₈, C₅₆:C₄, D₂₈:₄C₄, C₇xC₈.C₄, D₅₆:₇C₂, D₂₈.C₄, D₅₆:₁₀C₄
Quotients: C₁, C₂, C₄, C₂², C₂xC₄, D₄, C₂³, D₇, C₂²xC₄, C₂xD₄, C₄oD₄, D₁₄, C₄xD₄, C₄xD₇, C₂²xD₇, C₈.₂₆D₄, C₂xC₄xD₇, D₄xD₇, Q₈:₂D₇, D₂₈:C₄, D₅₆:₁₀C₄

Smallest permutation representation of D₅₆:₁₀C₄
►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 91)(2 90)(3 89)(4 88)(5 87)(6 86)(7 85)(8 84)(9 83)(10 82)(11 81)(12 80)(13 79)(14 78)(15 77)(16 76)(17 75)(18 74)(19 73)(20 72)(21 71)(22 70)(23 69)(24 68)(25 67)(26 66)(27 65)(28 64)(29 63)(30 62)(31 61)(32 60)(33 59)(34 58)(35 57)(36 112)(37 111)(38 110)(39 109)(40 108)(41 107)(42 106)(43 105)(44 104)(45 103)(46 102)(47 101)(48 100)(49 99)(50 98)(51 97)(52 96)(53 95)(54 94)(55 93)(56 92)

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

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

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

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

64 conjugacy classes

class	1	2A	2B	2C	2D	4A	4B	4C	4D	4E	4F	4G	7A	7B	7C	8A	···	8F	8G	8H	8I	8J	14A	14B	14C	14D	14E	14F	28A	···	28F	28G	28H	28I	56A	···	56L	56M	···	56X
order	1	2	2	2	2	4	4	4	4	4	4	4	7	7	7	8	···	8	8	8	8	8	14	14	14	14	14	14	28	···	28	28	28	28	56	···	56	56	···	56
size	1	1	2	28	28	1	1	2	28	28	28	28	2	2	2	4	···	4	14	14	14	14	2	2	2	4	4	4	2	···	2	4	4	4	4	···	4	8	···	8

64 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	4	4	4	4
type	+	+	+	+	+	+				+	+		+	+			+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	D₄	D₇	C₄oD₄	D₁₄	D₁₄	C₄xD₇	C₈.₂₆D₄	D₄xD₇	Q₈:₂D₇	D₅₆:₁₀C₄
kernel	D₅₆:₁₀C₄	C₅₆:C₄	D₂₈:₄C₄	C₇xC₈.C₄	D₅₆:₇C₂	D₂₈.C₄	C₅₆:C₂	D₅₆	Dic₂₈	C₇:C₈	C₈.C₄	C₂xC₁₄	C₂xC₈	M₄(2)	C₈	C₇	C₄	C₂²	C₁
# reps	1	1	2	1	1	2	4	2	2	2	3	2	3	6	12	2	3	3	12

Matrix representation of D₅₆:₁₀C₄ ►in GL₆(F₁₁₃)

24	112	0	0	0	0
1	0	0	0	0	0
0	0	0	0	98	0
0	0	0	0	0	98
0	0	1	0	0	0
0	0	0	112	0	0

24	112	0	0	0	0
10	89	0	0	0	0
0	0	0	112	0	0
0	0	112	0	0	0
0	0	0	0	0	15
0	0	0	0	98	0

98	0	0	0	0	0
92	15	0	0	0	0
0	0	98	0	0	0
0	0	0	1	0	0
0	0	0	0	15	0
0	0	0	0	0	112

G:=sub<GL(6,GF(113))| [24,1,0,0,0,0,112,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,98,0,0,0,0,0,0,98,0,0],[24,10,0,0,0,0,112,89,0,0,0,0,0,0,0,112,0,0,0,0,112,0,0,0,0,0,0,0,0,98,0,0,0,0,15,0],[98,92,0,0,0,0,0,15,0,0,0,0,0,0,98,0,0,0,0,0,0,1,0,0,0,0,0,0,15,0,0,0,0,0,0,112] >;

D₅₆:₁₀C₄ in GAP, Magma, Sage, TeX

D_{56}\rtimes_{10}C_4

% in TeX

G:=Group("D56:10C4");

// GroupNames label

G:=SmallGroup(448,428);

// by ID

G=gap.SmallGroup(448,428);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,120,219,58,136,1684,438,102,18822]);

// Polycyclic

G:=Group<a,b,c|a^56=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^26*b>;

// generators/relations

G = D56:10C4 order 448 = 26·7

10th semidirect product of D56 and C4 acting via C4/C2=C2

G = D₅₆:₁₀C₄ order 448 = 2⁶·7

10^th semidirect product of D₅₆ and C₄ acting via C₄/C₂=C₂