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G = C22×C4○D4order 64 = 26

Direct product of C22 and C4○D4

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

Aliases: C22×C4○D4, D43C23, C2.3C25, Q83C23, C4.18C24, C22.1C24, C24.33C22, C23.49C23, C4(C22×D4), D4(C22×C4), Q8(C22×C4), C4(C22×Q8), (C23×C4)⋊8C2, (C2×C4)⋊5C23, (C22×D4)⋊13C2, (C2×D4)⋊19C22, (C2×Q8)⋊19C22, (C22×Q8)⋊11C2, (C22×C4)⋊20C22, C4(C2×C4○D4), (C2×C4)2(C2×D4), (C2×C4)2(C2×Q8), (C2×C4)(C4○D4), (C22×C4)(C2×Q8), (C2×C4)(C22×Q8), (C22×C4)(C22×Q8), (C2×C4)(C2×C4○D4), SmallGroup(64,263)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C22×C4○D4
C1C2C22C23C22×C4C23×C4 — C22×C4○D4
C1C2 — C22×C4○D4
C1C22×C4 — C22×C4○D4
C1C2 — C22×C4○D4

Generators and relations for C22×C4○D4
 G = < a,b,c,d,e | a2=b2=c4=e2=1, d2=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d >

Subgroups: 505 in 445 conjugacy classes, 385 normal (6 characteristic)
C1, C2, C2 [×6], C2 [×12], C4 [×16], C22 [×19], C22 [×36], C2×C4 [×72], D4 [×48], Q8 [×16], C23, C23 [×18], C23 [×12], C22×C4, C22×C4 [×39], C2×D4 [×36], C2×Q8 [×12], C4○D4 [×64], C24 [×3], C23×C4 [×3], C22×D4 [×3], C22×Q8, C2×C4○D4 [×24], C22×C4○D4
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C4○D4 [×4], C24 [×31], C2×C4○D4 [×6], C25, C22×C4○D4

Smallest permutation representation of C22×C4○D4
On 32 points
Generators in S32
(1 9)(2 10)(3 11)(4 12)(5 22)(6 23)(7 24)(8 21)(13 18)(14 19)(15 20)(16 17)(25 32)(26 29)(27 30)(28 31)
(1 23)(2 24)(3 21)(4 22)(5 12)(6 9)(7 10)(8 11)(13 26)(14 27)(15 28)(16 25)(17 32)(18 29)(19 30)(20 31)
(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 29 3 31)(2 30 4 32)(5 16 7 14)(6 13 8 15)(9 26 11 28)(10 27 12 25)(17 24 19 22)(18 21 20 23)
(1 20)(2 17)(3 18)(4 19)(5 27)(6 28)(7 25)(8 26)(9 15)(10 16)(11 13)(12 14)(21 29)(22 30)(23 31)(24 32)

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

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

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

C22×C4○D4 is a maximal subgroup of
C24.51(C2×C4)  C24.165C23  (C23×C4).C4  C24.65D4  C24.66D4  C23.179C24  C24.542C23  C24.549C23  C23.223C24  C23.288C24  C23.304C24  C24.243C23  C24.244C23  C24.262C23  C24.263C23  C24.264C23  C24.360C23  C24.361C23  C24.73(C2×C4)  D4○(C22⋊C8)  C24.98D4  C24.103D4  C24.104D4  C24.105D4  C24.106D4  C22.14C25  C22.38C25  C22.74C25  C22.76C25  C22.77C25  C22.78C25  C4○D4⋊A4
C22×C4○D4 is a maximal quotient of
D4×C22×C4  Q8×C22×C4  C22.33C25  C22.44C25  C22.48C25  C22.49C25  C22.50C25  C22.64C25  C22.69C25  C22.70C25  C22.71C25  C22.72C25  C22.79C25  C22.80C25  C22.81C25  C22.82C25  C22.83C25  C22.84C25  C22.94C25  C22.95C25  C22.96C25  C22.97C25  C22.98C25  C22.99C25  C22.100C25  C22.101C25  C22.102C25  C22.103C25  C22.104C25  C22.105C25  C22.106C25  C22.107C25  C22.108C25  C23.144C24  C22.110C25  C22.111C25  C23.146C24  C22.113C25

40 conjugacy classes

class 1 2A···2G2H···2S4A···4H4I···4T
order12···22···24···44···4
size11···12···21···12···2

40 irreducible representations

dim111112
type+++++
imageC1C2C2C2C2C4○D4
kernelC22×C4○D4C23×C4C22×D4C22×Q8C2×C4○D4C22
# reps1331248

Matrix representation of C22×C4○D4 in GL4(𝔽5) generated by

4000
0400
0040
0004
,
4000
0100
0040
0004
,
4000
0400
0030
0003
,
4000
0100
0001
0040
,
4000
0400
0004
0040
G:=sub<GL(4,GF(5))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,3,0,0,0,0,3],[4,0,0,0,0,1,0,0,0,0,0,4,0,0,1,0],[4,0,0,0,0,4,0,0,0,0,0,4,0,0,4,0] >;

C22×C4○D4 in GAP, Magma, Sage, TeX

C_2^2\times C_4\circ D_4
% in TeX

G:=Group("C2^2xC4oD4");
// GroupNames label

G:=SmallGroup(64,263);
// by ID

G=gap.SmallGroup(64,263);
# by ID

G:=PCGroup([6,-2,2,2,2,2,-2,409,158]);
// Polycyclic

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

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