D56:8C4 - GroupNames

G = D₅₆⋊₈C₄ order 448 = 2⁶·7

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₅₆⋊₈C₄, C₈.₂₂D₂₈, C₅₆.₃₈D₄, Dic₂₈⋊₈C₄, C₂₈.₃SD₁₆, C₈.₁(C₄×D₇), C₄.Q₈⋊₁D₇, C₇⋊₁(D₈⋊₂C₄), C₅₆.₂₁(C₂×C₄), (C₂×C₁₄).₃₀D₈, (C₂×C₈).₄₁D₁₄, (C₂×C₂₈).₈₈D₄, C₄.₇(Q₈⋊D₇), C₂₈.C₈⋊₄C₂, C₄.₁(D₁₄⋊C₄), D₅₆⋊₇C₂.₆C₂, C₂₈.₁(C₂²⋊C₄), (C₂×C₅₆).₄₇C₂², C₂².₈(D₄⋊D₇), C₂.₆(C₁₄.D₈), C₁₄.₄(D₄⋊C₄), (C₇×C₄.Q₈)⋊₁C₂, (C₂×C₄).₁₆(C₇⋊D₄), SmallGroup(448,45)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₅₆ — D₅₆⋊₈C₄

Chief series

C₁ — C₇ — C₁₄ — C₂₈ — C₂×C₂₈ — C₂×C₅₆ — D₅₆⋊₇C₂ — D₅₆⋊₈C₄

Lower central

C₇ — C₁₄ — C₂₈ — C₅₆ — D₅₆⋊₈C₄

Upper central

C₁ — C₂ — C₂×C₄ — C₂×C₈ — C₄.Q₈

Generators and relations for D₅₆⋊₈C₄
G = < a,b,c | a⁵⁶=b²=c⁴=1, bab=a^-1, cac^-1=a⁴³, cbc^-1=a⁷b >

Subgroups: 364 in 58 conjugacy classes, 25 normal (all characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₈, C₂×C₄, C₂×C₄, D₄, Q₈, D₇, C₁₄, C₁₄, C₁₆, C₄⋊C₄, C₂×C₈, D₈, SD₁₆, Q₁₆, C₄○D₄, Dic₇, C₂₈, C₂₈, D₁₄, C₂×C₁₄, C₄.Q₈, M₅(2), C₄○D₈, C₅₆, Dic₁₄, C₄×D₇, D₂₈, C₇⋊D₄, C₂×C₂₈, C₂×C₂₈, D₈⋊₂C₄, C₇⋊C₁₆, C₅₆⋊C₂, D₅₆, Dic₂₈, C₇×C₄⋊C₄, C₂×C₅₆, C₄○D₂₈, C₂₈.C₈, C₇×C₄.Q₈, D₅₆⋊₇C₂, D₅₆⋊₈C₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, D₇, C₂²⋊C₄, D₈, SD₁₆, D₁₄, D₄⋊C₄, C₄×D₇, D₂₈, C₇⋊D₄, D₈⋊₂C₄, D₁₄⋊C₄, D₄⋊D₇, Q₈⋊D₇, C₁₄.D₈, D₅₆⋊₈C₄

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

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

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

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

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

58 conjugacy classes

class	1	2A	2B	2C	4A	4B	4C	4D	4E	7A	7B	7C	8A	8B	8C	14A	···	14I	16A	16B	16C	16D	28A	···	28F	28G	···	28R	56A	···	56L
order	1	2	2	2	4	4	4	4	4	7	7	7	8	8	8	14	···	14	16	16	16	16	28	···	28	28	···	28	56	···	56
size	1	1	2	56	2	2	8	8	56	2	2	2	2	2	4	2	···	2	28	28	28	28	4	···	4	8	···	8	4	···	4

58 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	4	4	4	4
type	+	+	+	+			+	+	+		+	+		+			+	+
image	C₁	C₂	C₂	C₂	C₄	C₄	D₄	D₄	D₇	SD₁₆	D₈	D₁₄	C₄×D₇	D₂₈	C₇⋊D₄	D₈⋊₂C₄	Q₈⋊D₇	D₄⋊D₇	D₅₆⋊₈C₄
kernel	D₅₆⋊₈C₄	C₂₈.C₈	C₇×C₄.Q₈	D₅₆⋊₇C₂	D₅₆	Dic₂₈	C₅₆	C₂×C₂₈	C₄.Q₈	C₂₈	C₂×C₁₄	C₂×C₈	C₈	C₈	C₂×C₄	C₇	C₄	C₂²	C₁
# reps	1	1	1	1	2	2	1	1	3	2	2	3	6	6	6	2	3	3	12

Matrix representation of D₅₆⋊₈C₄ ►in GL₄(𝔽₁₁₃) generated by

91	91	0	0
22	91	0	0
0	0	95	18
0	0	95	95

0	0	95	18
0	0	95	95
91	91	0	0
22	91	0	0

1	0	0	0
0	112	0	0
0	0	100	13
0	0	13	13

G:=sub<GL(4,GF(113))| [91,22,0,0,91,91,0,0,0,0,95,95,0,0,18,95],[0,0,91,22,0,0,91,91,95,95,0,0,18,95,0,0],[1,0,0,0,0,112,0,0,0,0,100,13,0,0,13,13] >;

D₅₆⋊₈C₄ in GAP, Magma, Sage, TeX

D_{56}\rtimes_8C_4

% in TeX

G:=Group("D56:8C4");

// GroupNames label

G:=SmallGroup(448,45);

// by ID

G=gap.SmallGroup(448,45);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,141,36,758,675,794,192,1684,851,102,18822]);

// Polycyclic

G:=Group<a,b,c|a^56=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^43,c*b*c^-1=a^7*b>;

// generators/relations

G = D56⋊8C4 order 448 = 26·7

8th semidirect product of D56 and C4 acting via C4/C2=C2

G = D₅₆⋊₈C₄ order 448 = 2⁶·7

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