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G = D4:8D14order 224 = 25·7

4th semidirect product of D4 and D14 acting through Inn(D4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4:8D14, Q8:7D14, C7:22+ 1+4, D28:11C22, C28.26C23, C14.12C24, D14.7C23, Dic7.7C23, Dic14:12C22, C4oD4:3D7, (D4xD7):5C2, (C2xC4):4D14, C4oD28:8C2, (C2xD28):13C2, (C2xC28):5C22, Q8:2D7:5C2, (C7xD4):9C22, (C4xD7):2C22, C7:D4:5C22, (C7xQ8):8C22, (C2xC14).4C23, C2.13(C23xD7), C4.33(C22xD7), (C22xD7):4C22, C22.3(C22xD7), (C7xC4oD4):4C2, SmallGroup(224,185)

Series: Derived Chief Lower central Upper central

C1C14 — D4:8D14
C1C7C14D14C22xD7D4xD7 — D4:8D14
C7C14 — D4:8D14
C1C2C4oD4

Generators and relations for D4:8D14
 G = < a,b,c,d | a4=b2=c14=d2=1, bab=dad=a-1, ac=ca, cbc-1=a2b, bd=db, dcd=c-1 >

Subgroups: 742 in 166 conjugacy classes, 85 normal (12 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, D7, C14, C14, C2xD4, C4oD4, C4oD4, Dic7, C28, C28, D14, D14, C2xC14, 2+ 1+4, Dic14, C4xD7, D28, C7:D4, C2xC28, C7xD4, C7xQ8, C22xD7, C2xD28, C4oD28, D4xD7, Q8:2D7, C7xC4oD4, D4:8D14
Quotients: C1, C2, C22, C23, D7, C24, D14, 2+ 1+4, C22xD7, C23xD7, D4:8D14

Smallest permutation representation of D4:8D14
On 56 points
Generators in S56
(1 48 41 28)(2 49 42 15)(3 50 29 16)(4 51 30 17)(5 52 31 18)(6 53 32 19)(7 54 33 20)(8 55 34 21)(9 56 35 22)(10 43 36 23)(11 44 37 24)(12 45 38 25)(13 46 39 26)(14 47 40 27)
(1 28)(2 49)(3 16)(4 51)(5 18)(6 53)(7 20)(8 55)(9 22)(10 43)(11 24)(12 45)(13 26)(14 47)(15 42)(17 30)(19 32)(21 34)(23 36)(25 38)(27 40)(29 50)(31 52)(33 54)(35 56)(37 44)(39 46)(41 48)
(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)
(1 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 28)(8 27)(9 26)(10 25)(11 24)(12 23)(13 22)(14 21)(29 52)(30 51)(31 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 56)(40 55)(41 54)(42 53)

G:=sub<Sym(56)| (1,48,41,28)(2,49,42,15)(3,50,29,16)(4,51,30,17)(5,52,31,18)(6,53,32,19)(7,54,33,20)(8,55,34,21)(9,56,35,22)(10,43,36,23)(11,44,37,24)(12,45,38,25)(13,46,39,26)(14,47,40,27), (1,28)(2,49)(3,16)(4,51)(5,18)(6,53)(7,20)(8,55)(9,22)(10,43)(11,24)(12,45)(13,26)(14,47)(15,42)(17,30)(19,32)(21,34)(23,36)(25,38)(27,40)(29,50)(31,52)(33,54)(35,56)(37,44)(39,46)(41,48), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(29,52)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,56)(40,55)(41,54)(42,53)>;

G:=Group( (1,48,41,28)(2,49,42,15)(3,50,29,16)(4,51,30,17)(5,52,31,18)(6,53,32,19)(7,54,33,20)(8,55,34,21)(9,56,35,22)(10,43,36,23)(11,44,37,24)(12,45,38,25)(13,46,39,26)(14,47,40,27), (1,28)(2,49)(3,16)(4,51)(5,18)(6,53)(7,20)(8,55)(9,22)(10,43)(11,24)(12,45)(13,26)(14,47)(15,42)(17,30)(19,32)(21,34)(23,36)(25,38)(27,40)(29,50)(31,52)(33,54)(35,56)(37,44)(39,46)(41,48), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(29,52)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,56)(40,55)(41,54)(42,53) );

G=PermutationGroup([[(1,48,41,28),(2,49,42,15),(3,50,29,16),(4,51,30,17),(5,52,31,18),(6,53,32,19),(7,54,33,20),(8,55,34,21),(9,56,35,22),(10,43,36,23),(11,44,37,24),(12,45,38,25),(13,46,39,26),(14,47,40,27)], [(1,28),(2,49),(3,16),(4,51),(5,18),(6,53),(7,20),(8,55),(9,22),(10,43),(11,24),(12,45),(13,26),(14,47),(15,42),(17,30),(19,32),(21,34),(23,36),(25,38),(27,40),(29,50),(31,52),(33,54),(35,56),(37,44),(39,46),(41,48)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,28),(8,27),(9,26),(10,25),(11,24),(12,23),(13,22),(14,21),(29,52),(30,51),(31,50),(32,49),(33,48),(34,47),(35,46),(36,45),(37,44),(38,43),(39,56),(40,55),(41,54),(42,53)]])

D4:8D14 is a maximal subgroup of
D4:4D28  M4(2):D14  D28:18D4  D28.39D4  D4.11D28  D4.12D28  D8:15D14  D8:11D14  D8:5D14  C56.C23  D28.32C23  D28.34C23  C14.C25  D7x2+ 1+4  D28.39C23
D4:8D14 is a maximal quotient of
C42.90D14  C42:7D14  C42.91D14  C42:9D14  C42:10D14  C42.95D14  C42.97D14  C42.99D14  C42.100D14  D4:6Dic14  C42:11D14  D4xD28  D28:23D4  Dic14:24D4  C42:17D14  C42.116D14  C42.117D14  C42.119D14  Q8xDic14  C42.126D14  Q8:6D28  D28:10Q8  C42.133D14  C42.136D14  C14.372+ 1+4  C14.382+ 1+4  D28:19D4  C14.462+ 1+4  C14.1152+ 1+4  C14.472+ 1+4  C14.482+ 1+4  C14.172- 1+4  D28:21D4  C14.512+ 1+4  C14.1182+ 1+4  C14.242- 1+4  C14.562+ 1+4  C14.262- 1+4  C14.1202+ 1+4  C14.1212+ 1+4  C14.612+ 1+4  C14.1222+ 1+4  C14.662+ 1+4  C14.852- 1+4  C14.682+ 1+4  C14.862- 1+4  C42:18D14  D28:10D4  C42:20D14  C42.143D14  C42.144D14  C42:22D14  C42.145D14  C42.148D14  D28:7Q8  C42.150D14  C42.153D14  C42.156D14  C42.157D14  C42.158D14  C42:23D14  C42:24D14  C42.161D14  C42.163D14  C42.164D14  C42:25D14  C42.165D14  C14.1062- 1+4  C14.1452+ 1+4  C14.1462+ 1+4  C14.1082- 1+4  C14.1482+ 1+4

47 conjugacy classes

class 1 2A2B2C2D2E···2J4A4B4C4D4E4F7A7B7C14A14B14C14D···14L28A···28F28G···28O
order122222···244444477714141414···1428···2828···28
size1122214···14222214142222224···42···24···4

47 irreducible representations

dim111111222244
type++++++++++++
imageC1C2C2C2C2C2D7D14D14D142+ 1+4D4:8D14
kernelD4:8D14C2xD28C4oD28D4xD7Q8:2D7C7xC4oD4C4oD4C2xC4D4Q8C7C1
# reps133621399316

Matrix representation of D4:8D14 in GL4(F29) generated by

276623
42025
00823
00621
,
276623
42025
1014823
014621
,
2619121
3077
002210
001910
,
223236
1527250
00238
00216
G:=sub<GL(4,GF(29))| [27,4,0,0,6,2,0,0,6,0,8,6,23,25,23,21],[27,4,10,0,6,2,14,14,6,0,8,6,23,25,23,21],[26,3,0,0,19,0,0,0,1,7,22,19,21,7,10,10],[2,15,0,0,23,27,0,0,23,25,23,21,6,0,8,6] >;

D4:8D14 in GAP, Magma, Sage, TeX

D_4\rtimes_8D_{14}
% in TeX

G:=Group("D4:8D14");
// GroupNames label

G:=SmallGroup(224,185);
// by ID

G=gap.SmallGroup(224,185);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,188,579,69,6917]);
// Polycyclic

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

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