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## G = C2×C42⋊C2order 64 = 26

### Direct product of C2 and C42⋊C2

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C2×C42⋊C2
G = < a,b,c,d | a2=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, cd=dc >

Subgroups: 201 in 165 conjugacy classes, 129 normal (9 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C23×C4, C2×C42⋊C2
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, C24, C42⋊C2, C23×C4, C2×C4○D4, C2×C42⋊C2

Smallest permutation representation of C2×C42⋊C2
On 32 points
Generators in S32
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 15)(10 16)(11 13)(12 14)(17 21)(18 22)(19 23)(20 24)
(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 31 11 19)(2 32 12 20)(3 29 9 17)(4 30 10 18)(5 15 21 27)(6 16 22 28)(7 13 23 25)(8 14 24 26)
(2 12)(4 10)(6 22)(8 24)(14 26)(16 28)(18 30)(20 32)

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

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

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

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

Matrix representation of C2×C42⋊C2 in GL4(𝔽5) generated by

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

C2×C42⋊C2 in GAP, Magma, Sage, TeX

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

G:=Group("C2xC4^2:C2");
// GroupNames label

G:=SmallGroup(64,195);
// by ID

G=gap.SmallGroup(64,195);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,192,217,86]);
// Polycyclic

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

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