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G = C22×C22⋊C4order 64 = 26

Direct product of C22 and C22⋊C4

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

Aliases: C22×C22⋊C4, C244C4, C25.2C2, C23.58D4, C22.5C24, C23.68C23, C24.35C22, (C23×C4)⋊2C2, C235(C2×C4), (C2×C4)⋊3C23, C2.1(C23×C4), C2.1(C22×D4), C222(C22×C4), C22.57(C2×D4), (C22×C4)⋊15C22, SmallGroup(64,193)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C22×C22⋊C4
C1C2C22C23C24C25 — C22×C22⋊C4
C1C2 — C22×C22⋊C4
C1C24 — C22×C22⋊C4
C1C22 — C22×C22⋊C4

Generators and relations for C22×C22⋊C4
 G = < a,b,c,d,e | a2=b2=c2=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, de=ed >

Subgroups: 505 in 337 conjugacy classes, 169 normal (6 characteristic)
C1, C2, C2 [×14], C2 [×8], C4 [×8], C22, C22 [×42], C22 [×56], C2×C4 [×8], C2×C4 [×24], C23 [×43], C23 [×56], C22⋊C4 [×16], C22×C4 [×12], C22×C4 [×8], C24, C24 [×14], C24 [×8], C2×C22⋊C4 [×12], C23×C4 [×2], C25, C22×C22⋊C4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C24, C2×C22⋊C4 [×12], C23×C4, C22×D4 [×2], C22×C22⋊C4

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

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

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

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

C22×C22⋊C4 is a maximal subgroup of
C24.17Q8  C232C42  C24.50D4  C24.5Q8  C24.52D4  C25.3C4  C24.68D4  C24.78D4  C23⋊C42  C24.90D4  C23.194C24  C24.91D4  C23.203C24  C23.224C24  C23.240C24  C23.304C24  C24.94D4  C248D4  C23.311C24  C24.95D4  C23.318C24  C23.324C24  C244Q8  C23.372C24  C23.380C24  C23.382C24  C24.96D4  C23.434C24  C23.439C24  C23.461C24  C2410D4  C24.97D4  C245Q8  D4×C22×C4  C22.79C25
C22×C22⋊C4 is a maximal quotient of
C25.85C22  C23.179C24  C24.90D4  C23.191C24  C23.192C24  C24.542C23  C24.549C23  C23.223C24  C24.73(C2×C4)  D4○(C22⋊C8)  C23.C24  C23.4C24  M4(2).24C23  M4(2).25C23  C24.98D4  2+ 1+45C4  2- 1+44C4  2- 1+45C4

40 conjugacy classes

class 1 2A···2O2P···2W4A···4P
order12···22···24···4
size11···12···22···2

40 irreducible representations

dim111112
type+++++
imageC1C2C2C2C4D4
kernelC22×C22⋊C4C2×C22⋊C4C23×C4C25C24C23
# reps11221168

Matrix representation of C22×C22⋊C4 in GL5(𝔽5)

40000
04000
00400
00040
00004
,
10000
04000
00100
00010
00001
,
10000
01000
00400
00010
00004
,
10000
01000
00100
00040
00004
,
30000
04000
00400
00001
00010

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

C22×C22⋊C4 in GAP, Magma, Sage, TeX

C_2^2\times C_2^2\rtimes C_4
% in TeX

G:=Group("C2^2xC2^2:C4");
// GroupNames label

G:=SmallGroup(64,193);
// by ID

G=gap.SmallGroup(64,193);
# by ID

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

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

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