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G = C56.16Q8order 448 = 26·7

6th non-split extension by C56 of Q8 acting via Q8/C4=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C56.1C8, C8.29D28, C56.65D4, C56.16Q8, C8.14Dic14, C42.9Dic7, C8.1(C7⋊C8), (C4×C8).11D7, C14.4(C4⋊C8), C71(C8.C8), C28.37(C2×C8), (C4×C28).14C4, (C2×C56).18C4, (C4×C56).13C2, C28.39(C4⋊C4), (C2×C8).9Dic7, (C2×C8).321D14, C2.5(C28⋊C8), C28.C8.5C2, C4.19(C4⋊Dic7), (C2×C56).397C22, (C2×C14).19M4(2), C22.2(C4.Dic7), C4.8(C2×C7⋊C8), (C2×C28).301(C2×C4), (C2×C4).69(C2×Dic7), SmallGroup(448,20)

Series: Derived Chief Lower central Upper central

C1C28 — C56.16Q8
C1C7C14C28C56C2×C56C28.C8 — C56.16Q8
C7C14C28 — C56.16Q8
C1C8C2×C8C4×C8

Generators and relations for C56.16Q8
 G = < a,b,c | a56=1, b4=a28, c2=a21b2, ab=ba, cac-1=a13, cbc-1=b3 >

2C2
2C4
2C4
2C14
2C2×C4
2C28
2C28
14C16
14C16
2C2×C28
7M5(2)
7M5(2)
2C7⋊C16
2C7⋊C16
7C8.C8

Smallest permutation representation of C56.16Q8
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 51 44 37 30 23 16 9)(3 52 45 38 31 24 17 10)(4 53 46 39 32 25 18 11)(5 54 47 40 33 26 19 12)(6 55 48 41 34 27 20 13)(7 56 49 42 35 28 21 14)(57 64 71 78 85 92 99 106)(58 65 72 79 86 93 100 107)(59 66 73 80 87 94 101 108)(60 67 74 81 88 95 102 109)(61 68 75 82 89 96 103 110)(62 69 76 83 90 97 104 111)(63 70 77 84 91 98 105 112)
(1 106 8 85 15 64 22 99 29 78 36 57 43 92 50 71)(2 63 9 98 16 77 23 112 30 91 37 70 44 105 51 84)(3 76 10 111 17 90 24 69 31 104 38 83 45 62 52 97)(4 89 11 68 18 103 25 82 32 61 39 96 46 75 53 110)(5 102 12 81 19 60 26 95 33 74 40 109 47 88 54 67)(6 59 13 94 20 73 27 108 34 87 41 66 48 101 55 80)(7 72 14 107 21 86 28 65 35 100 42 79 49 58 56 93)

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

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

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

124 conjugacy classes

class 1 2A2B4A4B4C···4G7A7B7C8A8B8C8D8E···8J14A···14I16A···16H28A···28AJ56A···56AV
order122444···477788888···814···1416···1628···2856···56
size112112···222211112···22···228···282···22···2

124 irreducible representations

dim1111112222222222222
type++++-+--+-+
imageC1C2C2C4C4C8D4Q8D7M4(2)Dic7Dic7D14C7⋊C8Dic14D28C8.C8C4.Dic7C56.16Q8
kernelC56.16Q8C28.C8C4×C56C4×C28C2×C56C56C56C56C4×C8C2×C14C42C2×C8C2×C8C8C8C8C7C22C1
# reps1212281132333126681248

Matrix representation of C56.16Q8 in GL2(𝔽113) generated by

510
0100
,
950
044
,
01
690
G:=sub<GL(2,GF(113))| [51,0,0,100],[95,0,0,44],[0,69,1,0] >;

C56.16Q8 in GAP, Magma, Sage, TeX

C_{56}._{16}Q_8
% in TeX

G:=Group("C56.16Q8");
// GroupNames label

G:=SmallGroup(448,20);
// by ID

G=gap.SmallGroup(448,20);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,64,100,1123,136,102,18822]);
// Polycyclic

G:=Group<a,b,c|a^56=1,b^4=a^28,c^2=a^21*b^2,a*b=b*a,c*a*c^-1=a^13,c*b*c^-1=b^3>;
// generators/relations

Export

Subgroup lattice of C56.16Q8 in TeX

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