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G = C424C4order 64 = 26

1st semidirect product of C42 and C4 acting via C4/C2=C2

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

Aliases: C424C4, C4.7C42, C23.52C23, C2.2(C2×C42), (C2×C42).2C2, C4(C2.C42), C2.1(C42⋊C2), C22.12(C4○D4), C22.14(C22×C4), (C22×C4).84C22, C2.C42.11C2, (C2×C4).51(C2×C4), SmallGroup(64,57)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C424C4
C1C2C22C23C22×C4C2×C42 — C424C4
C1C2 — C424C4
C1C22×C4 — C424C4
C1C23 — C424C4

Generators and relations for C424C4
 G = < a,b,c | a4=b4=c4=1, ab=ba, cac-1=ab2, bc=cb >

Subgroups: 113 in 89 conjugacy classes, 65 normal (5 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×12], C22 [×7], C2×C4 [×18], C2×C4 [×12], C23, C42 [×12], C22×C4, C22×C4 [×6], C2.C42 [×4], C2×C42 [×3], C424C4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, C42 [×4], C22×C4 [×3], C4○D4 [×4], C2×C42, C42⋊C2 [×6], C424C4

Smallest permutation representation of C424C4
Regular action on 64 points
Generators in S64
(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)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)
(1 7 20 54)(2 8 17 55)(3 5 18 56)(4 6 19 53)(9 31 47 38)(10 32 48 39)(11 29 45 40)(12 30 46 37)(13 43 59 27)(14 44 60 28)(15 41 57 25)(16 42 58 26)(21 52 34 62)(22 49 35 63)(23 50 36 64)(24 51 33 61)
(1 44 49 9)(2 25 50 48)(3 42 51 11)(4 27 52 46)(5 58 33 29)(6 13 34 37)(7 60 35 31)(8 15 36 39)(10 17 41 64)(12 19 43 62)(14 22 38 54)(16 24 40 56)(18 26 61 45)(20 28 63 47)(21 30 53 59)(23 32 55 57)

G:=sub<Sym(64)| (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)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,7,20,54)(2,8,17,55)(3,5,18,56)(4,6,19,53)(9,31,47,38)(10,32,48,39)(11,29,45,40)(12,30,46,37)(13,43,59,27)(14,44,60,28)(15,41,57,25)(16,42,58,26)(21,52,34,62)(22,49,35,63)(23,50,36,64)(24,51,33,61), (1,44,49,9)(2,25,50,48)(3,42,51,11)(4,27,52,46)(5,58,33,29)(6,13,34,37)(7,60,35,31)(8,15,36,39)(10,17,41,64)(12,19,43,62)(14,22,38,54)(16,24,40,56)(18,26,61,45)(20,28,63,47)(21,30,53,59)(23,32,55,57)>;

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)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,7,20,54)(2,8,17,55)(3,5,18,56)(4,6,19,53)(9,31,47,38)(10,32,48,39)(11,29,45,40)(12,30,46,37)(13,43,59,27)(14,44,60,28)(15,41,57,25)(16,42,58,26)(21,52,34,62)(22,49,35,63)(23,50,36,64)(24,51,33,61), (1,44,49,9)(2,25,50,48)(3,42,51,11)(4,27,52,46)(5,58,33,29)(6,13,34,37)(7,60,35,31)(8,15,36,39)(10,17,41,64)(12,19,43,62)(14,22,38,54)(16,24,40,56)(18,26,61,45)(20,28,63,47)(21,30,53,59)(23,32,55,57) );

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),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64)], [(1,7,20,54),(2,8,17,55),(3,5,18,56),(4,6,19,53),(9,31,47,38),(10,32,48,39),(11,29,45,40),(12,30,46,37),(13,43,59,27),(14,44,60,28),(15,41,57,25),(16,42,58,26),(21,52,34,62),(22,49,35,63),(23,50,36,64),(24,51,33,61)], [(1,44,49,9),(2,25,50,48),(3,42,51,11),(4,27,52,46),(5,58,33,29),(6,13,34,37),(7,60,35,31),(8,15,36,39),(10,17,41,64),(12,19,43,62),(14,22,38,54),(16,24,40,56),(18,26,61,45),(20,28,63,47),(21,30,53,59),(23,32,55,57)])

C424C4 is a maximal subgroup of
C421C8  C42.9Q8  C42.375D4  C42.55D4  C42.56D4  (C4×C8)⋊12C4  C42.45Q8  C42.23Q8  C424C4.C2  C42.25Q8  C428D4  C42⋊Q8  C23.165C24  C4242D4  C4214Q8  C432C2  C23.201C24  C23.202C24  C42.160D4  C42.161D4  C4214D4  C42.33Q8  C424Q8  C23.225C24  C24.208C23  C23.235C24  C23.238C24  C23.253C24  C24.221C23  C23.255C24  C4216D4  C42.163D4  C425Q8  C23.301C24  C42.34Q8  C23.426C24  C24.315C23  C23.428C24  C23.429C24  C23.430C24  C23.431C24  C23.432C24  C23.433C24  C4224D4  C42.184D4  C428Q8  C42.38Q8  C4226D4  C42.185D4  C429Q8  C42.193D4  C42.194D4  C42.195D4  C23.544C24  C23.545C24  C42.39Q8  C4231D4  C42.196D4  C4210Q8  C424C4⋊C3
 C4p.C42: C2.C43  C426Dic3  C424Dic5  C424F5  C424Dic7 ...
 C2p.(C2×C42): D4.C42  C4×C42⋊C2  D44C42  Q84C42  Dic3.5C42  Dic5.15C42  Dic7.5C42 ...
C424C4 is a maximal quotient of
C424C8  C43.C2  (C4×C8)⋊12C4
 C4p.C42: C8.16C42  C426Dic3  C424Dic5  C424F5  C424Dic7 ...
 (C22×C4).D2p: C4×C2.C42  C24.624C23  Dic3.5C42  Dic5.15C42  Dic7.5C42 ...

40 conjugacy classes

class 1 2A···2G4A···4H4I···4AF
order12···24···44···4
size11···11···12···2

40 irreducible representations

dim11112
type+++
imageC1C2C2C4C4○D4
kernelC424C4C2.C42C2×C42C42C22
# reps143248

Matrix representation of C424C4 in GL4(𝔽5) generated by

3000
0300
0010
0004
,
1000
0400
0020
0002
,
1000
0200
0001
0010
G:=sub<GL(4,GF(5))| [3,0,0,0,0,3,0,0,0,0,1,0,0,0,0,4],[1,0,0,0,0,4,0,0,0,0,2,0,0,0,0,2],[1,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0] >;

C424C4 in GAP, Magma, Sage, TeX

C_4^2\rtimes_4C_4
% in TeX

G:=Group("C4^2:4C4");
// GroupNames label

G:=SmallGroup(64,57);
// by ID

G=gap.SmallGroup(64,57);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,2,96,121,199,50]);
// Polycyclic

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

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