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G = C23.1D14order 224 = 25·7

1st non-split extension by C23 of D14 acting via D14/C7=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.1D14, C22.2D28, (C2xDic7):C4, (C22xD7):C4, C7:1(C23:C4), C22:C4:1D7, (C2xC14).27D4, C23.D7:1C2, C22.3(C4xD7), C2.4(D14:C4), C14.2(C22:C4), C22.8(C7:D4), (C22xC14).5C22, (C7xC22:C4):1C2, (C2xC14).1(C2xC4), (C2xC7:D4).1C2, SmallGroup(224,12)

Series: Derived Chief Lower central Upper central

C1C2xC14 — C23.1D14
C1C7C14C2xC14C22xC14C2xC7:D4 — C23.1D14
C7C14C2xC14 — C23.1D14
C1C2C23C22:C4

Generators and relations for C23.1D14
 G = < a,b,c,d | a2=b2=c28=1, d2=a, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=ac-1 >

Subgroups: 262 in 52 conjugacy classes, 19 normal (all characteristic)
Quotients: C1, C2, C4, C22, C2xC4, D4, D7, C22:C4, D14, C23:C4, C4xD7, D28, C7:D4, D14:C4, C23.1D14
2C2
2C2
2C2
28C2
4C4
4C22
14C22
14C4
28C4
28C22
2C14
2C14
2C14
4D7
2C2xC4
7C23
7C2xC4
14C2xC4
14D4
14D4
2Dic7
2D14
4Dic7
4D14
4C2xC14
4C28
7C22:C4
7C2xD4
2C2xC28
2C7:D4
2C7:D4
2C2xDic7
7C23:C4

Smallest permutation representation of C23.1D14
On 56 points
Generators in S56
(1 29)(3 31)(5 33)(7 35)(9 37)(11 39)(13 41)(15 43)(17 45)(19 47)(21 49)(23 51)(25 53)(27 55)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 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 35 29 7)(2 34)(3 5 31 33)(6 30)(8 28)(9 55 37 27)(10 54)(11 25 39 53)(12 24)(13 51 41 23)(14 50)(15 21 43 49)(16 20)(17 47 45 19)(18 46)(22 42)(26 38)(36 56)(40 52)(44 48)

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

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

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

C23.1D14 is a maximal subgroup of
C23:C4:5D7  C23:D28  C23.5D28  D7xC23:C4  (C2xD28):13C4  C24:D14  C22:C4:D14
C23.1D14 is a maximal quotient of
C14.C4wrC2  C4:Dic7:C4  C23.30D28  (C22xD7):C8  (C2xDic7):C8  C22.2D56  C7:C2wrC4  (C2xC28).D4  C23.D28  C23.2D28  C23.3D28  C23.4D28  (C2xC4).D28  (C2xQ8).D14  C24.2D14

41 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E7A7B7C14A···14I14J···14O28A···28L
order1222224444477714···1414···1428···28
size1122228442828282222···24···44···4

41 irreducible representations

dim11111122222244
type+++++++++
imageC1C2C2C2C4C4D4D7D14C4xD7D28C7:D4C23:C4C23.1D14
kernelC23.1D14C23.D7C7xC22:C4C2xC7:D4C2xDic7C22xD7C2xC14C22:C4C23C22C22C22C7C1
# reps11112223366616

Matrix representation of C23.1D14 in GL4(F29) generated by

28033
0281212
0010
0001
,
28000
02800
00280
00028
,
1214122
1927512
19199
18181010
,
13151022
8162212
001910
002210
G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,3,12,1,0,3,12,0,1],[28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[12,19,19,18,14,27,1,18,1,5,9,10,22,12,9,10],[13,8,0,0,15,16,0,0,10,22,19,22,22,12,10,10] >;

C23.1D14 in GAP, Magma, Sage, TeX

C_2^3._1D_{14}
% in TeX

G:=Group("C2^3.1D14");
// GroupNames label

G:=SmallGroup(224,12);
// by ID

G=gap.SmallGroup(224,12);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,121,31,362,297,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^28=1,d^2=a,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a*c^-1>;
// generators/relations

Export

Subgroup lattice of C23.1D14 in TeX

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