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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

 Derived series C1 — C2 — C22×C4○D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C22×C4○D4
 Lower central C1 — C2 — C22×C4○D4
 Upper central C1 — C22×C4 — C22×C4○D4
 Jennings C1 — C2 — 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)])

40 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2S 4A ··· 4H 4I ··· 4T order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

40 irreducible representations

 dim 1 1 1 1 1 2 type + + + + + image C1 C2 C2 C2 C2 C4○D4 kernel C22×C4○D4 C23×C4 C22×D4 C22×Q8 C2×C4○D4 C22 # reps 1 3 3 1 24 8

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

 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 1 0 0 4 0
,
 4 0 0 0 0 4 0 0 0 0 0 4 0 0 4 0
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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