C56.93D4 - GroupNames

G = C₅₆.₉₃D₄ order 448 = 2⁶·7

16^th non-split extension by C₅₆ of D₄ acting via D₄/C₂²=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₅₆.₉₃D₄, M₄(2).₃₆D₁₄, D₄⋊D₇⋊₇C₄, C₈○D₄⋊₆D₇, C₇⋊₄(C₈○D₈), Q₈⋊D₇⋊₇C₄, D₄.D₇⋊₇C₄, C₇⋊Q₁₆⋊₇C₄, D₄.₇(C₄×D₇), Q₈.₇(C₄×D₇), C₁₄.₈₁(C₄×D₄), (C₈×Dic₇)⋊₃₂C₂, C₄○D₄.₃₄D₁₄, D₂₈.₁₈(C₂×C₄), (C₂×C₈).₂₇₈D₁₄, C₂₈.₄₄₇(C₂×D₄), C₈.₂₂(C₇⋊D₄), D₂₈⋊₄C₄⋊₁₅C₂, C₂₈.₂₉(C₂²×C₄), D₄⋊₂Dic₇⋊₁₅C₂, D₂₈.₂C₄⋊₁₃C₂, C₂₈.₅₃D₄⋊₁₅C₂, (C₂×C₅₆).₂₃₆C₂², (C₂×C₂₈).₄₂₄C₂³, Dic₁₄.₁₈(C₂×C₄), D₄.₈D₁₄.₄C₂, C₄○D₂₈.₄₄C₂², C₂².₃(C₄○D₂₈), C₄.Dic₇.₄₄C₂², (C₄×Dic₇).₂₃₆C₂², (C₇×M₄(2)).₃₉C₂², C₄.₂₉(C₂×C₄×D₇), (C₇×C₈○D₄)⋊₆C₂, C₇⋊C₈.₁₀(C₂×C₄), C₂.₂₆(C₄×C₇⋊D₄), (C₇×D₄).₁₄(C₂×C₄), C₄.₁₃₈(C₂×C₇⋊D₄), (C₇×Q₈).₁₄(C₂×C₄), (C₂×C₁₄).₉(C₄○D₄), (C₂×C₇⋊C₈).₂₆₇C₂², (C₇×C₄○D₄).₃₉C₂², (C₂×C₄).₅₁₄(C₂²×D₇), SmallGroup(448,678)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₈ — C₅₆.₉₃D₄

Chief series

C₁ — C₇ — C₁₄ — C₂₈ — C₂×C₂₈ — C₄○D₂₈ — D₄.₈D₁₄ — C₅₆.₉₃D₄

Lower central

C₇ — C₁₄ — C₂₈ — C₅₆.₉₃D₄

Upper central

C₁ — C₈ — C₂×C₈ — C₈○D₄

Generators and relations for C₅₆.₉₃D₄
G = < a,b,c | a⁵⁶=c²=1, b⁴=a²⁸, bab^-1=cac=a⁴¹, cbc=a²⁸b³ >

Subgroups: 412 in 106 conjugacy classes, 47 normal (all characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₈, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, D₇, C₁₄, C₁₄, C₄², C₂×C₈, C₂×C₈, M₄(2), M₄(2), D₈, SD₁₆, Q₁₆, C₄○D₄, C₄○D₄, Dic₇, C₂₈, C₂₈, D₁₄, C₂×C₁₄, C₂×C₁₄, C₄×C₈, C₄≀C₂, C₈.C₄, C₈○D₄, C₈○D₄, C₄○D₈, C₇⋊C₈, C₇⋊C₈, C₅₆, C₅₆, Dic₁₄, C₄×D₇, D₂₈, C₂×Dic₇, C₇⋊D₄, C₂×C₂₈, C₂×C₂₈, C₇×D₄, C₇×D₄, C₇×Q₈, C₈○D₈, C₈×D₇, C₈⋊D₇, C₂×C₇⋊C₈, C₄.Dic₇, C₄×Dic₇, D₄⋊D₇, D₄.D₇, Q₈⋊D₇, C₇⋊Q₁₆, C₂×C₅₆, C₂×C₅₆, C₇×M₄(2), C₇×M₄(2), C₄○D₂₈, C₇×C₄○D₄, C₈×Dic₇, C₂₈.₅₃D₄, D₂₈⋊₄C₄, D₄⋊₂Dic₇, D₂₈.₂C₄, D₄.₈D₁₄, C₇×C₈○D₄, C₅₆.₉₃D₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, D₇, C₂²×C₄, C₂×D₄, C₄○D₄, D₁₄, C₄×D₄, C₄×D₇, C₇⋊D₄, C₂²×D₇, C₈○D₈, C₂×C₄×D₇, C₄○D₂₈, C₂×C₇⋊D₄, C₄×C₇⋊D₄, C₅₆.₉₃D₄

Smallest permutation representation of C₅₆.₉₃D₄
►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 50 43 36 29 22 15 8)(2 35 44 21 30 7 16 49)(3 20 45 6 31 48 17 34)(4 5 46 47 32 33 18 19)(9 42 51 28 37 14 23 56)(10 27 52 13 38 55 24 41)(11 12 53 54 39 40 25 26)(57 104 71 62 85 76 99 90)(58 89 72 103 86 61 100 75)(59 74 73 88 87 102 101 60)(63 70 77 84 91 98 105 112)(64 111 78 69 92 83 106 97)(65 96 79 110 93 68 107 82)(66 81 80 95 94 109 108 67)

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

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

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

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

88 conjugacy classes

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

88 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	4
type	+	+	+	+	+	+	+	+					+	+		+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₄	D₄	D₇	C₄○D₄	D₁₄	D₁₄	D₁₄	C₇⋊D₄	C₄×D₇	C₄×D₇	C₈○D₈	C₄○D₂₈	C₅₆.₉₃D₄
kernel	C₅₆.₉₃D₄	C₈×Dic₇	C₂₈.₅₃D₄	D₂₈⋊₄C₄	D₄⋊₂Dic₇	D₂₈.₂C₄	D₄.₈D₁₄	C₇×C₈○D₄	D₄⋊D₇	D₄.D₇	Q₈⋊D₇	C₇⋊Q₁₆	C₅₆	C₈○D₄	C₂×C₁₄	C₂×C₈	M₄(2)	C₄○D₄	C₈	D₄	Q₈	C₇	C₂²	C₁
# reps	1	1	1	1	1	1	1	1	2	2	2	2	2	3	2	3	3	3	12	6	6	8	12	12

Matrix representation of C₅₆.₉₃D₄ ►in GL₄(𝔽₁₁₃) generated by

80	112	0	0
72	9	0	0
0	0	95	0
0	0	0	95

112	103	0	0
0	1	0	0
0	0	95	0
0	0	0	69

1	10	0	0
0	112	0	0
0	0	0	69
0	0	95	0

G:=sub<GL(4,GF(113))| [80,72,0,0,112,9,0,0,0,0,95,0,0,0,0,95],[112,0,0,0,103,1,0,0,0,0,95,0,0,0,0,69],[1,0,0,0,10,112,0,0,0,0,0,95,0,0,69,0] >;

C₅₆.₉₃D₄ in GAP, Magma, Sage, TeX

C_{56}._{93}D_4

% in TeX

G:=Group("C56.93D4");

// GroupNames label

G:=SmallGroup(448,678);

// by ID

G=gap.SmallGroup(448,678);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,58,136,1684,851,102,18822]);

// Polycyclic

G:=Group<a,b,c|a^56=c^2=1,b^4=a^28,b*a*b^-1=c*a*c=a^41,c*b*c=a^28*b^3>;

// generators/relations

G = C56.93D4 order 448 = 26·7

16th non-split extension by C56 of D4 acting via D4/C22=C2

G = C₅₆.₉₃D₄ order 448 = 2⁶·7

16^th non-split extension by C₅₆ of D₄ acting via D₄/C₂²=C₂