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G = Q8.10D14order 224 = 25·7

1st non-split extension by Q8 of D14 acting through Inn(Q8)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q8.10D14, C712- 1+4, C28.24C23, C14.10C24, D14.5C23, D28.13C22, Dic7.6C23, Dic14.13C22, (C2×Q8)⋊5D7, (Q8×D7)⋊4C2, C4○D286C2, (Q8×C14)⋊7C2, Q82D74C2, (C2×C4).24D14, (C4×D7).5C22, C4.24(C22×D7), C2.11(C23×D7), C7⋊D4.2C22, (C2×C14).68C23, (C2×C28).48C22, (C7×Q8).10C22, C22.7(C22×D7), SmallGroup(224,183)

Series: Derived Chief Lower central Upper central

C1C14 — Q8.10D14
C1C7C14D14C4×D7Q8×D7 — Q8.10D14
C7C14 — Q8.10D14
C1C2C2×Q8

Generators and relations for Q8.10D14
 G = < a,b,c,d | a4=1, b2=c14=d2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=dbd-1=a2b, dcd-1=c13 >

Subgroups: 510 in 146 conjugacy classes, 85 normal (9 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, D4, Q8, Q8, D7, C14, C14, C2×Q8, C2×Q8, C4○D4, Dic7, C28, D14, C2×C14, 2- 1+4, Dic14, C4×D7, D28, C7⋊D4, C2×C28, C7×Q8, C4○D28, Q8×D7, Q82D7, Q8×C14, Q8.10D14
Quotients: C1, C2, C22, C23, D7, C24, D14, 2- 1+4, C22×D7, C23×D7, Q8.10D14

Smallest permutation representation of Q8.10D14
On 112 points
Generators in S112
(1 54 15 40)(2 55 16 41)(3 56 17 42)(4 29 18 43)(5 30 19 44)(6 31 20 45)(7 32 21 46)(8 33 22 47)(9 34 23 48)(10 35 24 49)(11 36 25 50)(12 37 26 51)(13 38 27 52)(14 39 28 53)(57 91 71 105)(58 92 72 106)(59 93 73 107)(60 94 74 108)(61 95 75 109)(62 96 76 110)(63 97 77 111)(64 98 78 112)(65 99 79 85)(66 100 80 86)(67 101 81 87)(68 102 82 88)(69 103 83 89)(70 104 84 90)
(1 84 15 70)(2 71 16 57)(3 58 17 72)(4 73 18 59)(5 60 19 74)(6 75 20 61)(7 62 21 76)(8 77 22 63)(9 64 23 78)(10 79 24 65)(11 66 25 80)(12 81 26 67)(13 68 27 82)(14 83 28 69)(29 93 43 107)(30 108 44 94)(31 95 45 109)(32 110 46 96)(33 97 47 111)(34 112 48 98)(35 99 49 85)(36 86 50 100)(37 101 51 87)(38 88 52 102)(39 103 53 89)(40 90 54 104)(41 105 55 91)(42 92 56 106)
(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 89 15 103)(2 102 16 88)(3 87 17 101)(4 100 18 86)(5 85 19 99)(6 98 20 112)(7 111 21 97)(8 96 22 110)(9 109 23 95)(10 94 24 108)(11 107 25 93)(12 92 26 106)(13 105 27 91)(14 90 28 104)(29 66 43 80)(30 79 44 65)(31 64 45 78)(32 77 46 63)(33 62 47 76)(34 75 48 61)(35 60 49 74)(36 73 50 59)(37 58 51 72)(38 71 52 57)(39 84 53 70)(40 69 54 83)(41 82 55 68)(42 67 56 81)

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

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

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

Q8.10D14 is a maximal subgroup of
D28.4D4  D28.5D4  D28.14D4  D28.15D4  D28.29D4  D28.30D4  C56.C23  D28.44D4  C14.C25  D7×2- 1+4  D28.39C23
Q8.10D14 is a maximal quotient of
C14.72+ 1+4  C14.82+ 1+4  C14.2- 1+4  C14.2+ 1+4  C14.52- 1+4  C14.62- 1+4  C42.122D14  Q85Dic14  C42.125D14  C42.126D14  Q85D28  C42.132D14  C42.133D14  C42.134D14  C14.152- 1+4  C14.162- 1+4  C14.172- 1+4  D2822D4  Dic1422D4  C14.202- 1+4  C14.212- 1+4  C14.222- 1+4  C14.232- 1+4  C14.242- 1+4  C14.582+ 1+4  C14.262- 1+4  C42.147D14  C42.150D14  C42.151D14  C42.154D14  C42.157D14  C42.158D14  Dic148Q8  C42.171D14  D2812D4  D288Q8  C42.174D14  C42.176D14  C42.177D14  C42.178D14  C42.180D14  C14.422- 1+4  Q8×C7⋊D4  C14.442- 1+4  C14.452- 1+4

47 conjugacy classes

class 1 2A2B2C2D2E2F4A···4F4G4H4I4J7A7B7C14A···14I28A···28R
order12222224···4444477714···1428···28
size112141414142···2141414142222···24···4

47 irreducible representations

dim1111122244
type++++++++-
imageC1C2C2C2C2D7D14D142- 1+4Q8.10D14
kernelQ8.10D14C4○D28Q8×D7Q82D7Q8×C14C2×Q8C2×C4Q8C7C1
# reps16441391216

Matrix representation of Q8.10D14 in GL4(𝔽29) generated by

00415
002825
251400
1400
,
22280
142708
80277
08152
,
00910
00923
201900
20600
,
1818237
911236
6221818
623911
G:=sub<GL(4,GF(29))| [0,0,25,1,0,0,14,4,4,28,0,0,15,25,0,0],[2,14,8,0,22,27,0,8,8,0,27,15,0,8,7,2],[0,0,20,20,0,0,19,6,9,9,0,0,10,23,0,0],[18,9,6,6,18,11,22,23,23,23,18,9,7,6,18,11] >;

Q8.10D14 in GAP, Magma, Sage, TeX

Q_8._{10}D_{14}
% in TeX

G:=Group("Q8.10D14");
// GroupNames label

G:=SmallGroup(224,183);
// by ID

G=gap.SmallGroup(224,183);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,188,86,579,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^4=1,b^2=c^14=d^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=c^13>;
// generators/relations

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