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G = C2xD4xD7  order 224 = 25·7

Direct product of C2, D4 and D7

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

Aliases: C2xD4xD7, C28:C23, C23:3D14, D28:7C22, D14:2C23, C14.5C24, Dic7:1C23, (C2xC4):6D14, C14:2(C2xD4), (C2xC14):C23, C7:2(C22xD4), (D4xC14):5C2, C4:1(C22xD7), (C2xD28):11C2, (C2xC28):2C22, (C23xD7):4C2, (C4xD7):3C22, (C7xD4):5C22, C7:D4:1C22, C2.6(C23xD7), C22:1(C22xD7), (C22xC14):4C22, (C2xDic7):8C22, (C22xD7):6C22, (C2xC4xD7):3C2, (C2xC7:D4):9C2, SmallGroup(224,178)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C14 — C2xD4xD7
Chief series
C1 — C7 — C14 — D14 — C22xD7 — C23xD7 — C2xD4xD7
Lower central
C7 — C14 — C2xD4xD7
Upper central
C1 — C22 — C2xD4

Generators and relations for C2xD4xD7
 G = < a,b,c,d,e | a2=b4=c2=d7=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1054 in 236 conjugacy classes, 97 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2xC4, C2xC4, D4, D4, C23, C23, D7, D7, C14, C14, C14, C22xC4, C2xD4, C2xD4, C24, Dic7, C28, D14, D14, C2xC14, C2xC14, C2xC14, C22xD4, C4xD7, D28, C2xDic7, C7:D4, C2xC28, C7xD4, C22xD7, C22xD7, C22xD7, C22xC14, C2xC4xD7, C2xD28, D4xD7, C2xC7:D4, D4xC14, C23xD7, C2xD4xD7
Quotients: C1, C2, C22, D4, C23, D7, C2xD4, C24, D14, C22xD4, C22xD7, D4xD7, C23xD7, C2xD4xD7

Smallest permutation representation of C2xD4xD7
►On 56 points
Generators in S56
(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)

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

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

C2xD4xD7 is a maximal subgroup of
 (D4xD7):C4  D4:D28  D4.6D28  D28:D4  D14:6SD16  C42:11D14  D4:5D28  C24:2D14  C24:3D14  C14.372+ 1+4  C14.382+ 1+4  D28:19D4  C14.402+ 1+4  D28:20D4  C14.1202+ 1+4  C14.1212+ 1+4  C42:18D14  D28:10D4  C42:26D14  D28:11D4  C14.1452+ 1+4
C2xD4xD7 is a maximal quotient of
 C24.27D14  C14.2- 1+4  C42:12D14  C42.228D14  D28:23D4  D28:24D4  Dic14:23D4  Dic14:24D4  C24.56D14  C24:2D14  C24:3D14  C24.33D14  C24.34D14  C28:(C4oD4)  C14.682- 1+4  Dic14:19D4  Dic14:20D4  C14.372+ 1+4  C4:C4:21D14  C14.382+ 1+4  C14.722- 1+4  D28:19D4  C14.402+ 1+4  C14.732- 1+4  D28:20D4  C4:C4:26D14  C14.162- 1+4  C14.172- 1+4  D28:21D4  D28:22D4  Dic14:21D4  Dic14:22D4  C14.792- 1+4  C14.1202+ 1+4  C14.1212+ 1+4  C14.822- 1+4  C4:C4:28D14  C42.233D14  C42:18D14  C42.141D14  D28:10D4  Dic14:10D4  C42:26D14  C42.238D14  D28:11D4  Dic14:11D4  C42.171D14  C42.240D14  D28:12D4  D28:8Q8  D8:13D14  D28.29D4  D28.30D4  D8:10D14  D8:15D14  D8:11D14  D8.10D14  SD16:D14  D8:5D14  D8:6D14  D56:C22  C56.C23  D28.44D4

50 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L2M2N2O4A4B4C4D7A7B7C14A···14I14J···14U28A···28F
order1222222222222222444477714···1414···1428···28
size111122227777141414142214142222···24···44···4

50 irreducible representations

dim1111111222224
type+++++++++++++
imageC1C2C2C2C2C2C2D4D7D14D14D14D4xD7
kernelC2xD4xD7C2xC4xD7C2xD28D4xD7C2xC7:D4D4xC14C23xD7D14C2xD4C2xC4D4C23C2
# reps11182124331266

Matrix representation of C2xD4xD7 ►in GL4(F29) generated by

28000
02800
00280
00028
,
28000
02800
00113
001128
,
1000
0100
0010
001128
,
19100
92800
0010
0001
,
22400
17700
0010
0001
G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[28,0,0,0,0,28,0,0,0,0,1,11,0,0,13,28],[1,0,0,0,0,1,0,0,0,0,1,11,0,0,0,28],[19,9,0,0,1,28,0,0,0,0,1,0,0,0,0,1],[22,17,0,0,4,7,0,0,0,0,1,0,0,0,0,1] >;
 

C2xD4xD7 in GAP, Magma, Sage, TeX 

C_2\times D_4\times D_7
 
% in TeX
 
G:=Group("C2xD4xD7");
 
// GroupNames label
 
G:=SmallGroup(224,178);
 
// by ID
 
G=gap.SmallGroup(224,178);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,159,6917]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^7=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

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