C56.9Q8 - GroupNames

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

(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 60 71 74 85 88 99 102)(58 101 72 59 86 73 100 87)(61 112 75 70 89 84 103 98)(62 97 76 111 90 69 104 83)(63 82 77 96 91 110 105 68)(64 67 78 81 92 95 106 109)(65 108 79 66 93 80 107 94)

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

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

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

88 conjugacy classes

class	1	2A	2B	4A	4B	4C	4D	4E	4F	4G	7A	7B	7C	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J	14A	14B	14C	14D	14E	14F	16A	16B	16C	16D	16E	16F	16G	16H	28A	···	28F	28G	28H	28I	56A	···	56L	56M	···	56R	112A	···	112X
order	1	2	2	4	4	4	4	4	4	4	7	7	7	8	8	8	8	8	8	8	8	8	8	14	14	14	14	14	14	16	16	16	16	16	16	16	16	28	···	28	28	28	28	56	···	56	56	···	56	112	···	112
size	1	1	2	1	1	2	14	14	14	14	2	2	2	1	1	1	1	2	2	14	14	14	14	2	2	2	4	4	4	4	4	4	4	28	28	28	28	2	···	2	4	4	4	2	···	2	4	···	4	4	···	4

88 irreducible representations

dim	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	C₄	C₄	C₈	D₄	Q₈	D₇	M₄(2)	D₁₄	Dic₁₄	C₇⋊D₄	C₄×D₇	C₈.C₈	C₈×D₇	C₈⋊D₇	C₅₆.₉Q₈
kernel	C₅₆.₉Q₈	C₂₈.C₈	C₈×Dic₇	C₇×M₅(2)	C₂×C₇⋊C₈	C₄×Dic₇	C₇⋊C₈	C₅₆	C₅₆	M₅(2)	C₂×C₁₄	C₂×C₈	C₈	C₈	C₂×C₄	C₇	C₄	C₂²	C₁
# reps	1	1	1	1	2	2	8	1	1	3	2	3	6	6	6	8	12	12	12

Matrix representation of C₅₆.₉Q₈ ►in GL₄(𝔽₁₁₃) generated by

24	1	0	0
112	0	0	0
0	0	95	0
0	0	10	18

112	0	0	0
0	112	0	0
0	0	10	36
0	0	109	103

103	89	0	0
103	10	0	0
0	0	95	0
0	0	43	69

G:=sub<GL(4,GF(113))| [24,112,0,0,1,0,0,0,0,0,95,10,0,0,0,18],[112,0,0,0,0,112,0,0,0,0,10,109,0,0,36,103],[103,103,0,0,89,10,0,0,0,0,95,43,0,0,0,69] >;

C₅₆.₉Q₈ in GAP, Magma, Sage, TeX

C_{56}._9Q_8

% in TeX

G:=Group("C56.9Q8");

// GroupNames label

G:=SmallGroup(448,68);

// by ID

G=gap.SmallGroup(448,68);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,141,36,100,570,136,102,18822]);

// Polycyclic

G:=Group<a,b,c|a^56=1,b^4=a^14,c^2=a^35*b^2,b*a*b^-1=a^29,c*a*c^-1=a^41,c*b*c^-1=a^35*b^3>;

// generators/relations

Export

Subgroup lattice of C₅₆.₉Q₈ in TeX

G = C56.9Q8 order 448 = 26·7

9th non-split extension by C56 of Q8 acting via Q8/C2=C22

G = C₅₆.₉Q₈ order 448 = 2⁶·7

9^th non-split extension by C₅₆ of Q₈ acting via Q₈/C₂=C₂²