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G = C4×C4○D4order 64 = 26

Direct product of C4 and C4○D4

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

Aliases: C4×C4○D4, D4C42, Q8C42, C22.10C24, C23.30C23, C42.87C22, C42(C4×D4), C42(C4×Q8), D46(C2×C4), Q86(C2×C4), C42(C4×D4), C42(C2×D4), (C4×D4)⋊23C2, C42(C4×Q8), C42(C2×Q8), C42(C4⋊C4), (C2×C42)⋊8C2, (C4×Q8)⋊18C2, C2.6(C23×C4), C4⋊C4.79C22, C42(C22⋊C4), C4.18(C22×C4), C42(C42⋊C2), C42⋊C219C2, (C2×C4).159C23, (C2×D4).76C22, C22.1(C22×C4), (C2×Q8).71C22, C42(C42⋊C2), C22⋊C4.27C22, (C22×C4).121C22, (C2×C4)⋊8(C2×C4), C2.4(C2×C4○D4), C42(C2×C4○D4), (C2×C4○D4).13C2, SmallGroup(64,198)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C4×C4○D4
C1C2C22C2×C4C42C2×C42 — C4×C4○D4
C1C2 — C4×C4○D4
C1C42 — C4×C4○D4
C1C22 — C4×C4○D4

Generators and relations for C4×C4○D4
 G = < a,b,c,d | a4=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

Subgroups: 185 in 155 conjugacy classes, 125 normal (8 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×12], C4 [×6], C22, C22 [×6], C22 [×6], C2×C4, C2×C4 [×23], C2×C4 [×12], D4 [×12], Q8 [×4], C23 [×3], C42, C42 [×9], C22⋊C4 [×6], C4⋊C4 [×6], C22×C4 [×9], C2×D4 [×3], C2×Q8, C4○D4 [×8], C2×C42 [×3], C42⋊C2 [×3], C4×D4 [×6], C4×Q8 [×2], C2×C4○D4, C4×C4○D4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×4], C24, C23×C4, C2×C4○D4 [×2], C4×C4○D4

Smallest permutation representation of C4×C4○D4
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 17 15 8)(2 18 16 5)(3 19 13 6)(4 20 14 7)(9 29 22 27)(10 30 23 28)(11 31 24 25)(12 32 21 26)
(1 6 15 19)(2 7 16 20)(3 8 13 17)(4 5 14 18)(9 31 22 25)(10 32 23 26)(11 29 24 27)(12 30 21 28)
(1 24)(2 21)(3 22)(4 23)(5 32)(6 29)(7 30)(8 31)(9 13)(10 14)(11 15)(12 16)(17 25)(18 26)(19 27)(20 28)

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

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

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

C4×C4○D4 is a maximal subgroup of
(C2×C42)⋊C4  D4.5C42  2- 1+45C4  C42.674C23  C42.260C23  C42.261C23  C42.678C23  C42.290C23  D46M4(2)  Q86M4(2)  C42.697C23  C42.698C23  D48M4(2)  Q87M4(2)  C22.14C25  C22.33C25  C22.44C25  C22.49C25  C22.50C25  C22.64C25  C22.69C25  C22.70C25  C22.71C25  C22.72C25  C22.87C25  C22.88C25  C22.89C25  C22.92C25  C22.93C25  C22.97C25  C22.98C25  C22.99C25  C22.100C25  C22.101C25  C22.103C25  C22.104C25  C22.106C25  C22.107C25  C22.110C25  C22.113C25
 C42.D2p: C42.455D4  C42.397D4  C42.374D4  D44M4(2)  D4.C42  C42.102D4  C42.426D4  C42.383D4 ...
C4×C4○D4 is a maximal quotient of
D4×C42  Q8×C42  C4242D4  C439C2  C4214Q8  C23.178C24  C23.179C24  C432C2  C23.214C24  C23.215C24  C24.203C23  C24.204C23  C23.218C24  C24.205C23  C24.549C23  C23.223C24  C23.225C24  C24.208C23  C23.229C24  C23.231C24  C23.233C24  C23.235C24  C23.238C24  C24.212C23  C23.241C24  C24.217C23  C24.218C23  C23.250C24  C23.252C24  C23.253C24  C24.221C23  C23.255C24  C24.223C23  C42.290C23  C42.291C23  C42.292C23  C42.293C23  C42.294C23  D46M4(2)  Q86M4(2)
 C4⋊C4.D2p: C23.244C24  C24.220C23  C42.188D6  C42.188D10  C42.188D14 ...

40 conjugacy classes

class 1 2A2B2C2D···2I4A···4L4M···4AD
order12222···24···44···4
size11112···21···12···2

40 irreducible representations

dim11111112
type++++++
imageC1C2C2C2C2C2C4C4○D4
kernelC4×C4○D4C2×C42C42⋊C2C4×D4C4×Q8C2×C4○D4C4○D4C4
# reps133621168

Matrix representation of C4×C4○D4 in GL3(𝔽5) generated by

300
020
002
,
400
020
002
,
400
020
043
,
100
041
001
G:=sub<GL(3,GF(5))| [3,0,0,0,2,0,0,0,2],[4,0,0,0,2,0,0,0,2],[4,0,0,0,2,4,0,0,3],[1,0,0,0,4,0,0,1,1] >;

C4×C4○D4 in GAP, Magma, Sage, TeX

C_4\times C_4\circ D_4
% in TeX

G:=Group("C4xC4oD4");
// GroupNames label

G:=SmallGroup(64,198);
// by ID

G=gap.SmallGroup(64,198);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,192,217,158,69]);
// Polycyclic

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

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