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## G = C4×C22⋊C4order 64 = 26

### Direct product of C4 and C22⋊C4

direct product, p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C4×C22⋊C4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C22⋊C4
 Lower central C1 — C2 — C4×C22⋊C4
 Upper central C1 — C22×C4 — C4×C22⋊C4
 Jennings C1 — C23 — C4×C22⋊C4

Generators and relations for C4×C22⋊C4
G = < a,b,c,d | a4=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >

Subgroups: 185 in 129 conjugacy classes, 73 normal (11 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C22×C4, C22×C4, C22×C4, C24, C2.C42, C2×C42, C2×C22⋊C4, C23×C4, C4×C22⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C42, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4, C4×C22⋊C4

Smallest permutation representation of C4×C22⋊C4
On 32 points
Generators in S32
(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)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 24)(10 21)(11 22)(12 23)(25 32)(26 29)(27 30)(28 31)
(1 5)(2 6)(3 7)(4 8)(9 30)(10 31)(11 32)(12 29)(13 17)(14 18)(15 19)(16 20)(21 28)(22 25)(23 26)(24 27)
(1 24 13 30)(2 21 14 31)(3 22 15 32)(4 23 16 29)(5 27 17 9)(6 28 18 10)(7 25 19 11)(8 26 20 12)

G:=sub<Sym(32)| (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), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,24)(10,21)(11,22)(12,23)(25,32)(26,29)(27,30)(28,31), (1,5)(2,6)(3,7)(4,8)(9,30)(10,31)(11,32)(12,29)(13,17)(14,18)(15,19)(16,20)(21,28)(22,25)(23,26)(24,27), (1,24,13,30)(2,21,14,31)(3,22,15,32)(4,23,16,29)(5,27,17,9)(6,28,18,10)(7,25,19,11)(8,26,20,12)>;

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), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,24)(10,21)(11,22)(12,23)(25,32)(26,29)(27,30)(28,31), (1,5)(2,6)(3,7)(4,8)(9,30)(10,31)(11,32)(12,29)(13,17)(14,18)(15,19)(16,20)(21,28)(22,25)(23,26)(24,27), (1,24,13,30)(2,21,14,31)(3,22,15,32)(4,23,16,29)(5,27,17,9)(6,28,18,10)(7,25,19,11)(8,26,20,12) );

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)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,24),(10,21),(11,22),(12,23),(25,32),(26,29),(27,30),(28,31)], [(1,5),(2,6),(3,7),(4,8),(9,30),(10,31),(11,32),(12,29),(13,17),(14,18),(15,19),(16,20),(21,28),(22,25),(23,26),(24,27)], [(1,24,13,30),(2,21,14,31),(3,22,15,32),(4,23,16,29),(5,27,17,9),(6,28,18,10),(7,25,19,11),(8,26,20,12)]])

40 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I ··· 4AB order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2

40 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 type + + + + + + image C1 C2 C2 C2 C2 C4 C4 D4 C4○D4 kernel C4×C22⋊C4 C2.C42 C2×C42 C2×C22⋊C4 C23×C4 C22⋊C4 C22×C4 C2×C4 C22 # reps 1 2 2 2 1 16 8 4 4

Matrix representation of C4×C22⋊C4 in GL4(𝔽5) generated by

 3 0 0 0 0 1 0 0 0 0 4 0 0 0 0 4
,
 4 0 0 0 0 1 0 0 0 0 4 0 0 0 2 1
,
 1 0 0 0 0 1 0 0 0 0 4 0 0 0 0 4
,
 1 0 0 0 0 3 0 0 0 0 3 3 0 0 0 2
G:=sub<GL(4,GF(5))| [3,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,1,0,0,0,0,4,2,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,3,0,0,0,0,3,0,0,0,3,2] >;

C4×C22⋊C4 in GAP, Magma, Sage, TeX

C_4\times C_2^2\rtimes C_4
% in TeX

G:=Group("C4xC2^2:C4");
// GroupNames label

G:=SmallGroup(64,58);
// by ID

G=gap.SmallGroup(64,58);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,2,96,121,199,86]);
// Polycyclic

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

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