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G = C2×C42order 32 = 25

Abelian group of type [2,4,4]

Aliases: C2×C42, SmallGroup(32,21)

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C2×C42
G = < a,b,c | a2=b4=c4=1, ab=ba, ac=ca, bc=cb >

Subgroups: 54, all normal (4 characteristic)
C1, C2, C4, C22, C22, C2×C4, C23, C42, C22×C4, C2×C42
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, C2×C42

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

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

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

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

C2×C42 is a maximal subgroup of
C22.7C42  C426C4  C424C4  C428C4  C425C4  C429C4  C23.63C23  C24.C22  C23.65C23  C24.3C22  C23.67C23  C4⋊M4(2)  C42.12C4  C42.6C4  C23.36C23  C22.26C24  C23.37C23
C2×C42 is a maximal quotient of
C424C4  C82M4(2)

32 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4X order 1 2 ··· 2 4 ··· 4 size 1 1 ··· 1 1 ··· 1

32 irreducible representations

 dim 1 1 1 1 type + + + image C1 C2 C2 C4 kernel C2×C42 C42 C22×C4 C2×C4 # reps 1 4 3 24

Matrix representation of C2×C42 in GL3(𝔽5) generated by

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

C2×C42 in GAP, Magma, Sage, TeX

C_2\times C_4^2
% in TeX

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

G:=SmallGroup(32,21);
// by ID

G=gap.SmallGroup(32,21);
# by ID

G:=PCGroup([5,-2,2,2,-2,2,40,86]);
// Polycyclic

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

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