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## G = C2×D72order 392 = 23·72

### Direct product of C2, D7 and D7

Aliases: C2×D72, C72⋊C23, C141D14, C7⋊D7⋊C22, (C7×C14)⋊C22, (C7×D7)⋊C22, (D7×C14)⋊5C2, C71(C22×D7), (C2×C7⋊D7)⋊4C2, SmallGroup(392,41)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C72 — C2×D72
 Chief series C1 — C7 — C72 — C7×D7 — D72 — C2×D72
 Lower central C72 — C2×D72
 Upper central C1 — C2

Generators and relations for C2×D72
G = < a,b,c,d,e | a2=b7=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: 754 in 79 conjugacy classes, 28 normal (6 characteristic)
C1, C2, C2, C22, C7, C7, C23, D7, D7, C14, C14, D14, D14, C2×C14, C72, C22×D7, C7×D7, C7⋊D7, C7×C14, D72, D7×C14, C2×C7⋊D7, C2×D72
Quotients: C1, C2, C22, C23, D7, D14, C22×D7, D72, C2×D72

Permutation representations of C2×D72
On 28 points - transitive group 28T52
Generators in S28
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)
(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)
(1 26)(2 25)(3 24)(4 23)(5 22)(6 28)(7 27)(8 19)(9 18)(10 17)(11 16)(12 15)(13 21)(14 20)
(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)
(1 19)(2 20)(3 21)(4 15)(5 16)(6 17)(7 18)(8 26)(9 27)(10 28)(11 22)(12 23)(13 24)(14 25)

G:=sub<Sym(28)| (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28), (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), (1,26)(2,25)(3,24)(4,23)(5,22)(6,28)(7,27)(8,19)(9,18)(10,17)(11,16)(12,15)(13,21)(14,20), (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), (1,19)(2,20)(3,21)(4,15)(5,16)(6,17)(7,18)(8,26)(9,27)(10,28)(11,22)(12,23)(13,24)(14,25)>;

G:=Group( (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28), (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), (1,26)(2,25)(3,24)(4,23)(5,22)(6,28)(7,27)(8,19)(9,18)(10,17)(11,16)(12,15)(13,21)(14,20), (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), (1,19)(2,20)(3,21)(4,15)(5,16)(6,17)(7,18)(8,26)(9,27)(10,28)(11,22)(12,23)(13,24)(14,25) );

G=PermutationGroup([[(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28)], [(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)], [(1,26),(2,25),(3,24),(4,23),(5,22),(6,28),(7,27),(8,19),(9,18),(10,17),(11,16),(12,15),(13,21),(14,20)], [(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)], [(1,19),(2,20),(3,21),(4,15),(5,16),(6,17),(7,18),(8,26),(9,27),(10,28),(11,22),(12,23),(13,24),(14,25)]])

G:=TransitiveGroup(28,52);

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 7A ··· 7F 7G ··· 7O 14A ··· 14F 14G ··· 14O 14P ··· 14AA order 1 2 2 2 2 2 2 2 7 ··· 7 7 ··· 7 14 ··· 14 14 ··· 14 14 ··· 14 size 1 1 7 7 7 7 49 49 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 14 ··· 14

50 irreducible representations

 dim 1 1 1 1 2 2 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 D7 D14 D14 D72 C2×D72 kernel C2×D72 D72 D7×C14 C2×C7⋊D7 D14 D7 C14 C2 C1 # reps 1 4 2 1 6 12 6 9 9

Matrix representation of C2×D72 in GL4(𝔽29) generated by

 28 0 0 0 0 28 0 0 0 0 28 0 0 0 0 28
,
 11 26 0 0 15 25 0 0 0 0 1 0 0 0 0 1
,
 11 26 0 0 11 18 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 25 3 0 0 14 11
,
 1 0 0 0 0 1 0 0 0 0 28 0 0 0 24 1
G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[11,15,0,0,26,25,0,0,0,0,1,0,0,0,0,1],[11,11,0,0,26,18,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,25,14,0,0,3,11],[1,0,0,0,0,1,0,0,0,0,28,24,0,0,0,1] >;

C2×D72 in GAP, Magma, Sage, TeX

C_2\times D_7^2
% in TeX

G:=Group("C2xD7^2");
// GroupNames label

G:=SmallGroup(392,41);
// by ID

G=gap.SmallGroup(392,41);
# by ID

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

G:=Group<a,b,c,d,e|a^2=b^7=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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