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G = C72:2Q8order 392 = 23·72

The semidirect product of C72 and Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial

Aliases: C72:2Q8, Dic7.D7, C7:1Dic14, C14.5D14, C2.5D72, C7:Dic7.2C2, (C7xC14).5C22, (C7xDic7).1C2, SmallGroup(392,22)

Series: Derived Chief Lower central Upper central

C1C7xC14 — C72:2Q8
C1C7C72C7xC14C7xDic7 — C72:2Q8
C72C7xC14 — C72:2Q8
C1C2

Generators and relations for C72:2Q8
 G = < a,b,c,d | a7=b7=c4=1, d2=c2, ab=ba, cac-1=a-1, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 222 in 33 conjugacy classes, 14 normal (6 characteristic)
Quotients: C1, C2, C22, Q8, D7, D14, Dic14, D72, C72:2Q8
2C7
2C7
2C7
7C4
7C4
49C4
2C14
2C14
2C14
49Q8
7Dic7
7C28
7Dic7
7C28
14Dic7
14Dic7
14Dic7
7Dic14
7Dic14

Smallest permutation representation of C72:2Q8
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 7 6 5 4 3 2)(8 14 13 12 11 10 9)(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 49 48 47 46 45 44)(50 56 55 54 53 52 51)
(1 27 8 20)(2 26 9 19)(3 25 10 18)(4 24 11 17)(5 23 12 16)(6 22 13 15)(7 28 14 21)(29 48 36 55)(30 47 37 54)(31 46 38 53)(32 45 39 52)(33 44 40 51)(34 43 41 50)(35 49 42 56)
(1 36 8 29)(2 37 9 30)(3 38 10 31)(4 39 11 32)(5 40 12 33)(6 41 13 34)(7 42 14 35)(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)

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

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

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

47 conjugacy classes

class 1  2 4A4B4C7A···7F7G···7O14A···14F14G···14O28A···28L
order124447···77···714···1414···1428···28
size111414982···24···42···24···414···14

47 irreducible representations

dim111222244
type+++-++-+-
imageC1C2C2Q8D7D14Dic14D72C72:2Q8
kernelC72:2Q8C7xDic7C7:Dic7C72Dic7C14C7C2C1
# reps1211661299

Matrix representation of C72:2Q8 in GL6(F29)

100000
010000
001000
000100
0000814
00001828
,
100000
010000
0001000
0026300
000010
000001
,
010000
2800000
0028000
0002800
0000280
0000111
,
13270000
27160000
0031900
00242600
000010
000001

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,18,0,0,0,0,14,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,26,0,0,0,0,10,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,28,0,0,0,0,1,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,11,0,0,0,0,0,1],[13,27,0,0,0,0,27,16,0,0,0,0,0,0,3,24,0,0,0,0,19,26,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C72:2Q8 in GAP, Magma, Sage, TeX

C_7^2\rtimes_2Q_8
% in TeX

G:=Group("C7^2:2Q8");
// GroupNames label

G:=SmallGroup(392,22);
// by ID

G=gap.SmallGroup(392,22);
# by ID

G:=PCGroup([5,-2,-2,-2,-7,-7,20,61,26,488,8404]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^7=c^4=1,d^2=c^2,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

Export

Subgroup lattice of C72:2Q8 in TeX

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