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G = D7×C4○D4  order 224 = 25·7

Direct product of D7 and C4○D4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D7×C4○D4, D4⋊7D14, Q8⋊6D14, D28⋊10C22, C28.25C23, C14.11C24, D14.6C23, Dic14⋊10C22, Dic7.10C23, (D4×D7)⋊6C2, (C2×C4)⋊7D14, (Q8×D7)⋊6C2, C4○D28⋊7C2, D4⋊2D7⋊6C2, (C2×C28)⋊4C22, Q8⋊2D7⋊6C2, (C7×D4)⋊8C22, (C4×D7)⋊7C22, C7⋊D4⋊4C22, (C7×Q8)⋊7C22, (C2×C14).3C23, C4.25(C22×D7), C2.12(C23×D7), C22.2(C22×D7), (C2×Dic7)⋊10C22, (C22×D7).31C22, (C2×C4×D7)⋊6C2, C7⋊4(C2×C4○D4), (C7×C4○D4)⋊3C2, SmallGroup(224,184)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C14 — D7×C4○D4
Chief series
C1 — C7 — C14 — D14 — C22×D7 — C2×C4×D7 — D7×C4○D4
Lower central
C7 — C14 — D7×C4○D4
Upper central
C1 — C4 — C4○D4

Generators and relations for D7×C4○D4
 G = < a,b,c,d,e | a7=b2=c4=e2=1, d2=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d >

Subgroups: 614 in 164 conjugacy classes, 87 normal (16 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D7, D7, C14, C14, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic7, Dic7, C28, C28, D14, D14, D14, C2×C14, C2×C4○D4, Dic14, C4×D7, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C7×Q8, C22×D7, C2×C4×D7, C4○D28, D4×D7, D4⋊2D7, Q8×D7, Q8⋊2D7, C7×C4○D4, D7×C4○D4
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, C22×D7, C23×D7, D7×C4○D4

Smallest permutation representation of D7×C4○D4
►On 56 points
Generators in S56
(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 12)(2 11)(3 10)(4 9)(5 8)(6 14)(7 13)(15 24)(16 23)(17 22)(18 28)(19 27)(20 26)(21 25)(29 38)(30 37)(31 36)(32 42)(33 41)(34 40)(35 39)(43 52)(44 51)(45 50)(46 56)(47 55)(48 54)(49 53)

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

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

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

D7×C4○D4 is a maximal subgroup of
 C42⋊D14  C56.49C23  D8⋊10D14  SD16⋊D14  D56⋊C22  C14.C25  D14.C24  D28.39C23
D7×C4○D4 is a maximal quotient of
 C42.88D14  C42.188D14  C42⋊8D14  C42⋊10D14  C42.93D14  C42.94D14  C42.95D14  C42.96D14  C42.97D14  C42.98D14  C4×D4⋊2D7  C42.102D14  D4⋊5Dic14  C42.104D14  C4×D4×D7  C42⋊12D14  C42.228D14  D4⋊5D28  C42⋊16D14  C42.229D14  C42.113D14  C42.114D14  C42.122D14  Q8⋊5Dic14  C4×Q8×D7  C4×Q8⋊2D7  Q8⋊5D28  C42.232D14  C42.131D14  C42.132D14  C28⋊(C4○D4)  Dic14⋊20D4  C4⋊C4.178D14  C14.342+ 1+4  C4⋊C4⋊21D14  C14.402+ 1+4  D28⋊20D4  C14.422+ 1+4  C14.432+ 1+4  C14.442+ 1+4  C14.452+ 1+4  (Q8×Dic7)⋊C2  C22⋊Q8⋊25D7  C4⋊C4⋊26D14  D28⋊22D4  Dic14⋊22D4  C14.522+ 1+4  C14.532+ 1+4  C14.202- 1+4  C14.212- 1+4  C14.222- 1+4  C14.232- 1+4  C4⋊C4.197D14  C14.802- 1+4  C14.1212+ 1+4  C14.822- 1+4  C4⋊C4⋊28D14  C14.612+ 1+4  C14.1222+ 1+4  C14.622+ 1+4  C14.832- 1+4  C14.642+ 1+4  C14.842- 1+4  C14.662+ 1+4  C14.672+ 1+4  C42.233D14  C42.137D14  C42.138D14  C42.139D14  D28⋊10D4  Dic14⋊10D4  C42⋊20D14  C42⋊21D14  C42.234D14  C42.143D14  Dic14⋊7Q8  C42.236D14  D28⋊7Q8  C42.237D14  C42.150D14  C42.151D14  C42.152D14  C42.153D14  C42.154D14  C42.159D14  C42.160D14  C42⋊23D14  C42⋊24D14  C42.189D14  C42.161D14  C42.162D14  C42.163D14  C42.164D14  C14.1042- 1+4  (C2×C28)⋊15D4  C14.1452+ 1+4  C14.1072- 1+4  (C2×C28)⋊17D4  C14.1482+ 1+4

50 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J7A7B7C14A14B14C14D···14L28A···28F28G···28O
order1222222222444444444477714141414···1428···2828···28
size112227714141411222771414142222224···42···24···4

50 irreducible representations

dim11111111222224
type++++++++++++
imageC1C2C2C2C2C2C2C2D7C4○D4D14D14D14D7×C4○D4
kernelD7×C4○D4C2×C4×D7C4○D28D4×D7D4⋊2D7Q8×D7Q8⋊2D7C7×C4○D4C4○D4D7C2×C4D4Q8C1
# reps13333111349936

Matrix representation of D7×C4○D4 ►in GL4(𝔽29) generated by

22100
272500
0010
0001
,
252800
15400
00280
00028
,
1000
0100
00170
00017
,
28000
02800
002822
00211
,
28000
02800
002822
0001
G:=sub<GL(4,GF(29))| [22,27,0,0,1,25,0,0,0,0,1,0,0,0,0,1],[25,15,0,0,28,4,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,0,1,0,0,0,0,17,0,0,0,0,17],[28,0,0,0,0,28,0,0,0,0,28,21,0,0,22,1],[28,0,0,0,0,28,0,0,0,0,28,0,0,0,22,1] >;
 

D7×C4○D4 in GAP, Magma, Sage, TeX 

D_7\times C_4\circ D_4
 
% in TeX
 
G:=Group("D7xC4oD4");
 
// GroupNames label
 
G:=SmallGroup(224,184);
 
// by ID
 
G=gap.SmallGroup(224,184);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,86,297,6917]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^7=b^2=c^4=e^2=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=c^2*d>;
 
// generators/relations
 

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