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(C₄×D₇), C₇⋊₃(C₈.₂₆D₄), C₅₆.₂₉(C₂×C₄), C₄.₂₁₂(D₄×D₇), C₈.C₄⋊₄D₇, C₁₄.₅₅(C₄×D₄), (C₂×C₈).₆₉D₁₄, D₂₈⋊₄C₄⋊₇C₂, D₂₈.C₄⋊₁₀C₂, D₂₈.₁₀(C₂×C₄), C₂₈.₃₇₁(C₂×D₄), D₅₆⋊₇C₂.₄C₂, C₂₈.₅₄(C₂²×C₄), (C₂×C₅₆).₄₁C₂², (C₂×C₂₈).₃₁₀C₂³, Dic₁₄.₁₀(C₂×C₄), C₄○D₂₈.₁₇C₂², C₂.₁₅(D₂₈⋊C₄), C₂².₁(Q₈⋊₂D₇), (C₄×Dic₇).₄₀C₂², (C₇×M₄(2)).₂₀C₂², C₄.₄₆(C₂×C₄×D₇), (C₇×C₈.C₄)⋊₄C₂, (C₂×C₇⋊C₈).₇₈C₂², (C₂×C₁₄).₁(C₄○D₄), (C₂×C₄).₄₁₃(C₂²×D₇), SmallGroup(448,428)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₈ — D₅₆⋊₁₀C₄

Chief series

C₁ — C₇ — C₁₄ — C₂₈ — C₂×C₂₈ — C₄○D₂₈ — D₅₆⋊₇C₂ — D₅₆⋊₁₀C₄

Lower central

C₇ — C₁₄ — C₂₈ — D₅₆⋊₁₀C₄

Upper central

C₁ — C₄ — C₂×C₄ — 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, C₂×C₄, C₂×C₄, D₄, Q₈, D₇, C₁₄, C₁₄, C₄², C₂×C₈, C₂×C₈, M₄(2), M₄(2), D₈, SD₁₆, Q₁₆, C₄○D₄, Dic₇, C₂₈, D₁₄, C₂×C₁₄, C₈⋊C₄, C₄≀C₂, C₈.C₄, C₈○D₄, C₄○D₈, C₇⋊C₈, C₅₆, C₅₆, Dic₁₄, C₄×D₇, D₂₈, C₂×Dic₇, C₇⋊D₄, C₂×C₂₈, C₈.₂₆D₄, C₈×D₇, C₈⋊D₇, C₅₆⋊C₂, D₅₆, Dic₂₈, C₂×C₇⋊C₈, C₄×Dic₇, C₂×C₅₆, C₇×M₄(2), C₄○D₂₈, C₅₆⋊C₄, D₂₈⋊₄C₄, C₇×C₈.C₄, D₅₆⋊₇C₂, D₂₈.C₄, D₅₆⋊₁₀C₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, D₇, C₂²×C₄, C₂×D₄, C₄○D₄, D₁₄, C₄×D₄, C₄×D₇, C₂²×D₇, C₈.₂₆D₄, C₂×C₄×D₇, D₄×D₇, 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	C₄○D₄	D₁₄	D₁₄	C₄×D₇	C₈.₂₆D₄	D₄×D₇	Q₈⋊₂D₇	D₅₆⋊₁₀C₄
kernel	D₅₆⋊₁₀C₄	C₅₆⋊C₄	D₂₈⋊₄C₄	C₇×C₈.C₄	D₅₆⋊₇C₂	D₂₈.C₄	C₅₆⋊C₂	D₅₆	Dic₂₈	C₇⋊C₈	C₈.C₄	C₂×C₁₄	C₂×C₈	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