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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

Aliases: C2×C42⋊C2, C4213C22, C22.7C24, C23.28C23, C24.28C22, (C2×C42)⋊3C2, C4⋊C418C22, (C22×C4)⋊10C4, C2.3(C23×C4), (C23×C4).9C2, C4(C42⋊C2), C4.30(C22×C4), (C2×C4).49C23, C23.34(C2×C4), C22.26(C4○D4), C22⋊C4.26C22, (C22×C4).98C22, C22.11(C22×C4), C4(C2×C4⋊C4), C4(C2×C22⋊C4), (C2×C4⋊C4)⋊24C2, (C2×C4)2(C4⋊C4), (C2×C4)⋊11(C2×C4), C2.1(C2×C4○D4), (C2×C4)2(C22⋊C4), (C2×C4)(C42⋊C2), (C2×C22⋊C4).15C2, (C2×C4)(C2×C4⋊C4), SmallGroup(64,195)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C2×C42⋊C2
C1C2C22C23C22×C4C23×C4 — C2×C42⋊C2
C1C2 — C2×C42⋊C2
C1C22×C4 — C2×C42⋊C2
C1C22 — 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 [×6], C2 [×4], C4 [×8], C4 [×8], C22, C22 [×10], C22 [×12], C2×C4 [×36], C2×C4 [×8], C23, C23 [×6], C23 [×4], C42 [×8], C22⋊C4 [×8], C4⋊C4 [×8], C22×C4 [×2], C22×C4 [×16], C24, C2×C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×8], C23×C4, C2×C42⋊C2
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×4], C24, C42⋊C2 [×4], C23×C4, C2×C4○D4 [×2], 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)])

C2×C42⋊C2 is a maximal subgroup of
C42.371D4  C42.42D4  C23.29C42  C24.63D4  C24.132D4  C24.152D4  C24.162C23  C23.15C42  C42.379D4  C42.95D4  C24.53(C2×C4)  C24.169C23  (C22×C4).276D4  C24.69D4  C24.70D4  C24.71D4  C24.73D4  C24.74D4  C427D4  C24.174C23  C24.524C23  C25.85C22  C23.167C24  C4242D4  C439C2  C23.178C24  C23.179C24  C432C2  C23.191C24  C23.192C24  C24.542C23  C24.192C23  C24.545C23  C23.199C24  C42.159D4  C4213D4  C23.224C24  C23.225C24  C23.226C24  C24.208C23  C23.229C24  C23.234C24  C23.236C24  C23.241C24  C23.244C24  C24.217C23  C4215D4  C23.295C24  C42.162D4  C23.311C24  C23.313C24  C24.249C23  C23.315C24  C24.567C23  C24.267C23  C24.268C23  C24.289C23  C24.290C23  C23.374C24  C23.375C24  C24.293C23  C23.377C24  C24.295C23  C23.379C24  C23.382C24  C24.576C23  C23.385C24  C23.398C24  C24.308C23  C23.400C24  C4222D4  C42.183D4  C4223D4  C4225D4  C4226D4  C42.185D4  C4227D4  C4228D4  C42.186D4  C23.524C24  C23.525C24  C42.187D4  C42.188D4  M4(2)○2M4(2)  C24.98D4  C42.257C23  C24.100D4  C42.259C23  C42.262C23  C24.115D4  C24.116D4  C24.117D4  C24.118D4  C2×C4×C4○D4  C22.14C25  C22.38C25  C22.44C25  C22.47C25  C22.64C25  C22.80C25  C22.82C25  C22.83C25  C22.84C25
C2×C42⋊C2 is a maximal quotient of
C2×C4×C22⋊C4  C2×C4×C4⋊C4  C25.85C22  C23.165C24  C23.167C24  C4242D4  C439C2  C4214Q8  C432C2  C23.194C24  C23.195C24  C24.192C23  C24.547C23  C23.201C24  C23.202C24  C23.224C24  C23.225C24  C23.226C24  C23.227C24  C24.208C23  C23.229C24  C23.234C24  C23.235C24  C23.236C24  C23.237C24  C23.238C24  C24.212C23  C42.677C23  C42.259C23  C42.260C23  C42.261C23  C42.262C23  C42.678C23

40 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4AB
order12···222224···44···4
size11···122221···12···2

40 irreducible representations

dim11111112
type++++++
imageC1C2C2C2C2C2C4C4○D4
kernelC2×C42⋊C2C2×C42C2×C22⋊C4C2×C4⋊C4C42⋊C2C23×C4C22×C4C22
# reps122281168

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

4000
0100
0040
0004
,
4000
0300
0004
0010
,
1000
0400
0030
0003
,
4000
0400
0010
0004
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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