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G = C7⋊Q16order 112 = 24·7

The semidirect product of C7 and Q16 acting via Q16/Q8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C72Q16, Q8.D7, C4.4D14, C14.10D4, C28.4C22, Dic14.2C2, C7⋊C8.C2, (C7×Q8).1C2, C2.7(C7⋊D4), SmallGroup(112,17)

Series: Derived Chief Lower central Upper central

C1C28 — C7⋊Q16
C1C7C14C28Dic14 — C7⋊Q16
C7C14C28 — C7⋊Q16
C1C2C4Q8

Generators and relations for C7⋊Q16
 G = < a,b,c | a7=b8=1, c2=b4, bab-1=a-1, ac=ca, cbc-1=b-1 >

2C4
14C4
7C8
7Q8
2Dic7
2C28
7Q16

Character table of C7⋊Q16

 class 124A4B4C7A7B7C8A8B14A14B14C28A28B28C28D28E28F28G28H28I
 size 1124282221414222444444444
ρ11111111111111111111111    trivial
ρ2111-11111-1-11111-1-1-1-1-11-11    linear of order 2
ρ31111-1111-1-1111111111111    linear of order 2
ρ4111-1-1111111111-1-1-1-1-11-11    linear of order 2
ρ522-20022200222-200000-20-2    orthogonal lifted from D4
ρ6222-20ζ7473ζ767ζ757200ζ767ζ7572ζ7473ζ75727572747374737572767ζ7473767ζ767    orthogonal lifted from D14
ρ7222-20ζ7572ζ7473ζ76700ζ7473ζ767ζ7572ζ767767757275727677473ζ75727473ζ7473    orthogonal lifted from D14
ρ822220ζ767ζ7572ζ747300ζ7572ζ7473ζ767ζ7473ζ7473ζ767ζ767ζ7473ζ7572ζ767ζ7572ζ7572    orthogonal lifted from D7
ρ922220ζ7572ζ7473ζ76700ζ7473ζ767ζ7572ζ767ζ767ζ7572ζ7572ζ767ζ7473ζ7572ζ7473ζ7473    orthogonal lifted from D7
ρ10222-20ζ767ζ7572ζ747300ζ7572ζ7473ζ767ζ7473747376776774737572ζ7677572ζ7572    orthogonal lifted from D14
ρ1122220ζ7473ζ767ζ757200ζ767ζ7572ζ7473ζ7572ζ7572ζ7473ζ7473ζ7572ζ767ζ7473ζ767ζ767    orthogonal lifted from D7
ρ122-20002222-2-2-2-2000000000    symplectic lifted from Q16, Schur index 2
ρ132-2000222-22-2-2-2000000000    symplectic lifted from Q16, Schur index 2
ρ1422-200ζ7473ζ767ζ757200ζ767ζ7572ζ7473757275727473ζ7473ζ7572ζ7677473767767    complex lifted from C7⋊D4
ρ1522-200ζ767ζ7572ζ747300ζ7572ζ7473ζ7677473ζ7473ζ7677677473ζ757276775727572    complex lifted from C7⋊D4
ρ1622-200ζ7473ζ767ζ757200ζ767ζ7572ζ74737572ζ7572ζ7473747375727677473ζ767767    complex lifted from C7⋊D4
ρ1722-200ζ7572ζ7473ζ76700ζ7473ζ767ζ7572767767ζ75727572ζ76774737572ζ74737473    complex lifted from C7⋊D4
ρ1822-200ζ767ζ7572ζ747300ζ7572ζ7473ζ76774737473767ζ767ζ74737572767ζ75727572    complex lifted from C7⋊D4
ρ1922-200ζ7572ζ7473ζ76700ζ7473ζ767ζ7572767ζ7677572ζ7572767ζ7473757274737473    complex lifted from C7⋊D4
ρ204-400074+2ζ7376+2ζ775+2ζ7200-2ζ76-2ζ7-2ζ75-2ζ72-2ζ74-2ζ73000000000    symplectic faithful, Schur index 2
ρ214-400076+2ζ775+2ζ7274+2ζ7300-2ζ75-2ζ72-2ζ74-2ζ73-2ζ76-2ζ7000000000    symplectic faithful, Schur index 2
ρ224-400075+2ζ7274+2ζ7376+2ζ700-2ζ74-2ζ73-2ζ76-2ζ7-2ζ75-2ζ72000000000    symplectic faithful, Schur index 2

Smallest permutation representation of C7⋊Q16
Regular action on 112 points
Generators in S112
(1 102 91 12 23 54 36)(2 37 55 24 13 92 103)(3 104 93 14 17 56 38)(4 39 49 18 15 94 97)(5 98 95 16 19 50 40)(6 33 51 20 9 96 99)(7 100 89 10 21 52 34)(8 35 53 22 11 90 101)(25 74 43 66 63 105 88)(26 81 106 64 67 44 75)(27 76 45 68 57 107 82)(28 83 108 58 69 46 77)(29 78 47 70 59 109 84)(30 85 110 60 71 48 79)(31 80 41 72 61 111 86)(32 87 112 62 65 42 73)
(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 25 5 29)(2 32 6 28)(3 31 7 27)(4 30 8 26)(9 69 13 65)(10 68 14 72)(11 67 15 71)(12 66 16 70)(17 61 21 57)(18 60 22 64)(19 59 23 63)(20 58 24 62)(33 83 37 87)(34 82 38 86)(35 81 39 85)(36 88 40 84)(41 89 45 93)(42 96 46 92)(43 95 47 91)(44 94 48 90)(49 110 53 106)(50 109 54 105)(51 108 55 112)(52 107 56 111)(73 99 77 103)(74 98 78 102)(75 97 79 101)(76 104 80 100)

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

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

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

C7⋊Q16 is a maximal subgroup of
SD16⋊D7  SD163D7  D7×Q16  Q16⋊D7  C28.C23  D4.8D14  D4.9D14  Q8.2F7  C21⋊Q16  C7⋊Dic12  C217Q16  Q8.D21
C7⋊Q16 is a maximal quotient of
C28.Q8  C14.Q16  Q8⋊Dic7  C21⋊Q16  C7⋊Dic12  C217Q16

Matrix representation of C7⋊Q16 in GL4(𝔽113) generated by

112100
328000
0010
0001
,
713500
1114200
00051
003151
,
3410100
687900
005722
006856
G:=sub<GL(4,GF(113))| [112,32,0,0,1,80,0,0,0,0,1,0,0,0,0,1],[71,111,0,0,35,42,0,0,0,0,0,31,0,0,51,51],[34,68,0,0,101,79,0,0,0,0,57,68,0,0,22,56] >;

C7⋊Q16 in GAP, Magma, Sage, TeX

C_7\rtimes Q_{16}
% in TeX

G:=Group("C7:Q16");
// GroupNames label

G:=SmallGroup(112,17);
// by ID

G=gap.SmallGroup(112,17);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-7,40,61,46,182,97,42,2404]);
// Polycyclic

G:=Group<a,b,c|a^7=b^8=1,c^2=b^4,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C7⋊Q16 in TeX
Character table of C7⋊Q16 in TeX

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