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G = C72:2C8order 392 = 23·72

The semidirect product of C72 and C8 acting via C8/C2=C4

metabelian, soluble, monomial, A-group

Aliases: C72:2C8, (C7xC14).C4, C2.(C72:C4), C7:Dic7.1C2, SmallGroup(392,17)

Series: Derived Chief Lower central Upper central

C1C72 — C72:2C8
C1C72C7xC14C7:Dic7 — C72:2C8
C72 — C72:2C8
C1C2

Generators and relations for C72:2C8
 G = < a,b,c | a7=b7=c8=1, ab=ba, cac-1=a3b-1, cbc-1=a3b4 >

Subgroups: 176 in 20 conjugacy classes, 6 normal (all characteristic)
Quotients: C1, C2, C4, C8, C72:C4, C72:2C8
2C7
2C7
2C7
2C7
49C4
2C14
2C14
2C14
2C14
49C8
14Dic7
14Dic7
14Dic7
14Dic7

Smallest permutation representation of C72:2C8
On 56 points
Generators in S56
(1 48 31 17 13 49 37)(2 50 18 41 38 14 32)(3 39 51 15 19 25 42)(4 26 16 40 43 20 52)(5 44 27 21 9 53 33)(6 54 22 45 34 10 28)(7 35 55 11 23 29 46)(8 30 12 36 47 24 56)
(1 49 17 48 37 13 31)(2 32 14 38 41 18 50)(3 25 15 39 42 19 51)(4 52 20 43 40 16 26)(5 53 21 44 33 9 27)(6 28 10 34 45 22 54)(7 29 11 35 46 23 55)(8 56 24 47 36 12 30)
(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)

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

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

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

32 conjugacy classes

class 1  2 4A4B7A···7L8A8B8C8D14A···14L
order12447···7888814···14
size1149494···4494949494···4

32 irreducible representations

dim111144
type+++-
imageC1C2C4C8C72:C4C72:2C8
kernelC72:2C8C7:Dic7C7xC14C72C2C1
# reps11241212

Matrix representation of C72:2C8 in GL4(F113) generated by

342500
888800
0079112
0010
,
0100
1127900
0001
0011279
,
0010
0001
985500
01500
G:=sub<GL(4,GF(113))| [34,88,0,0,25,88,0,0,0,0,79,1,0,0,112,0],[0,112,0,0,1,79,0,0,0,0,0,112,0,0,1,79],[0,0,98,0,0,0,55,15,1,0,0,0,0,1,0,0] >;

C72:2C8 in GAP, Magma, Sage, TeX

C_7^2\rtimes_2C_8
% in TeX

G:=Group("C7^2:2C8");
// GroupNames label

G:=SmallGroup(392,17);
// by ID

G=gap.SmallGroup(392,17);
# by ID

G:=PCGroup([5,-2,-2,-2,-7,7,10,26,1763,488,5004,4209]);
// Polycyclic

G:=Group<a,b,c|a^7=b^7=c^8=1,a*b=b*a,c*a*c^-1=a^3*b^-1,c*b*c^-1=a^3*b^4>;
// generators/relations

Export

Subgroup lattice of C72:2C8 in TeX

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