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## G = C7×C7⋊D4order 392 = 23·72

### Direct product of C7 and C7⋊D4

Aliases: C7×C7⋊D4, C14C2, C726D4, Dic7⋊C14, D142C14, C1422C2, C14.21D14, C72(C7×D4), C22⋊(C7×D7), (C2×C14)⋊1D7, (C2×C14)⋊2C14, (D7×C14)⋊4C2, C2.5(D7×C14), C14.5(C2×C14), (C7×Dic7)⋊4C2, (C7×C14).10C22, SmallGroup(392,27)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C7×C7⋊D4
 Chief series C1 — C7 — C14 — C7×C14 — D7×C14 — C7×C7⋊D4
 Lower central C7 — C14 — C7×C7⋊D4
 Upper central C1 — C14 — C2×C14

Generators and relations for C7×C7⋊D4
G = < a,b,c,d | a7=b7=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Permutation representations of C7×C7⋊D4
On 28 points - transitive group 28T47
Generators in S28
(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 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 21 20 19 18 17 16)(22 28 27 26 25 24 23)
(1 22 8 15)(2 23 9 16)(3 24 10 17)(4 25 11 18)(5 26 12 19)(6 27 13 20)(7 28 14 21)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)

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

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

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

G:=TransitiveGroup(28,47);

119 conjugacy classes

 class 1 2A 2B 2C 4 7A ··· 7F 7G ··· 7AA 14A ··· 14F 14G ··· 14BW 14BX ··· 14CC 28A ··· 28F order 1 2 2 2 4 7 ··· 7 7 ··· 7 14 ··· 14 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 2 14 14 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 14 ··· 14 14 ··· 14

119 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C7 C14 C14 C14 D4 D7 D14 C7⋊D4 C7×D4 C7×D7 D7×C14 C7×C7⋊D4 kernel C7×C7⋊D4 C7×Dic7 D7×C14 C142 C7⋊D4 Dic7 D14 C2×C14 C72 C2×C14 C14 C7 C7 C22 C2 C1 # reps 1 1 1 1 6 6 6 6 1 3 3 6 6 18 18 36

Matrix representation of C7×C7⋊D4 in GL2(𝔽29) generated by

 20 0 0 20
,
 20 0 0 16
,
 0 1 28 0
,
 0 1 1 0
G:=sub<GL(2,GF(29))| [20,0,0,20],[20,0,0,16],[0,28,1,0],[0,1,1,0] >;

C7×C7⋊D4 in GAP, Magma, Sage, TeX

C_7\times C_7\rtimes D_4
% in TeX

G:=Group("C7xC7:D4");
// GroupNames label

G:=SmallGroup(392,27);
// by ID

G=gap.SmallGroup(392,27);
# by ID

G:=PCGroup([5,-2,-2,-7,-2,-7,301,8404]);
// Polycyclic

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

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