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G = C425C4order 64 = 26

2nd semidirect product of C42 and C4 acting via C4/C2=C2

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

Aliases: C425C4, C23.59C23, (C2×C42).3C2, (C22×C4).3C22, C2.8(C42⋊C2), C2.1(C422C2), C22.18(C4○D4), C2.C42.2C2, C22.32(C22×C4), (C2×C4).54(C2×C4), SmallGroup(64,64)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C425C4
C1C2C22C23C22×C4C2×C42 — C425C4
C1C22 — C425C4
C1C23 — C425C4
C1C23 — C425C4

Generators and relations for C425C4
 G = < a,b,c | a4=b4=c4=1, ab=ba, cac-1=ab2, cbc-1=a2b-1 >

Subgroups: 105 in 69 conjugacy classes, 41 normal (5 characteristic)
C1, C2 [×7], C4 [×10], C22, C22 [×6], C2×C4 [×6], C2×C4 [×18], C23, C42 [×4], C22×C4 [×7], C2.C42 [×6], C2×C42, C425C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C22×C4, C4○D4 [×6], C42⋊C2 [×3], C422C2 [×4], C425C4

Character table of C425C4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q4R4S4T
 size 1111111122222222222244444444
ρ11111111111111111111111111111    trivial
ρ211111111-1-11-1-1-1-1-1-1111-1-1111-1-11    linear of order 2
ρ311111111-11-1-1-1-1111-1-1-1-11-111-11-1    linear of order 2
ρ4111111111-1-1111-1-1-1-1-1-11-1-1111-1-1    linear of order 2
ρ511111111-11-1-1-1-1111-1-1-11-11-1-11-11    linear of order 2
ρ611111111-1-11-1-1-1-1-1-111111-1-1-111-1    linear of order 2
ρ7111111111-1-1111-1-1-1-1-1-1-111-1-1-111    linear of order 2
ρ811111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ91-11-11-11-1-1-111-111-11-11-1-i-iii-iii-i    linear of order 4
ρ101-11-11-11-1-11-11-11-11-11-11-ii-ii-ii-ii    linear of order 4
ρ111-11-11-11-11-1-1-11-11-111-11i-i-ii-i-iii    linear of order 4
ρ121-11-11-11-1111-11-1-11-1-11-1iiii-i-i-i-i    linear of order 4
ρ131-11-11-11-1-1-111-111-11-11-1ii-i-ii-i-ii    linear of order 4
ρ141-11-11-11-1-11-11-11-11-11-11i-ii-ii-ii-i    linear of order 4
ρ151-11-11-11-11-1-1-11-11-111-11-iii-iii-i-i    linear of order 4
ρ161-11-11-11-1111-11-1-11-1-11-1-i-i-i-iiiii    linear of order 4
ρ1722-222-2-2-202i0000-2i-2i2i00000000000    complex lifted from C4○D4
ρ1822-222-2-2-20-2i00002i2i-2i00000000000    complex lifted from C4○D4
ρ19222-2-22-2-22i00-2i-2i2i00000000000000    complex lifted from C4○D4
ρ202-2-22-222-2002i000000-2i-2i2i00000000    complex lifted from C4○D4
ρ2122-2-2-2-222002i0000002i-2i-2i00000000    complex lifted from C4○D4
ρ222-2-2-222-2202i00002i-2i-2i00000000000    complex lifted from C4○D4
ρ232-222-2-2-222i002i-2i-2i00000000000000    complex lifted from C4○D4
ρ2422-2-2-2-22200-2i000000-2i2i2i00000000    complex lifted from C4○D4
ρ252-2-22-222-200-2i0000002i2i-2i00000000    complex lifted from C4○D4
ρ262-222-2-2-22-2i00-2i2i2i00000000000000    complex lifted from C4○D4
ρ272-2-2-222-220-2i0000-2i2i2i00000000000    complex lifted from C4○D4
ρ28222-2-22-2-2-2i002i2i-2i00000000000000    complex lifted from C4○D4

Smallest permutation representation of C425C4
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 28 31 40)(2 25 32 37)(3 26 29 38)(4 27 30 39)(5 55 22 48)(6 56 23 45)(7 53 24 46)(8 54 21 47)(9 51 16 36)(10 52 13 33)(11 49 14 34)(12 50 15 35)(17 58 63 43)(18 59 64 44)(19 60 61 41)(20 57 62 42)
(1 47 11 63)(2 55 12 18)(3 45 9 61)(4 53 10 20)(5 52 44 27)(6 34 41 40)(7 50 42 25)(8 36 43 38)(13 62 30 46)(14 17 31 54)(15 64 32 48)(16 19 29 56)(21 51 58 26)(22 33 59 39)(23 49 60 28)(24 35 57 37)

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

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

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

C425C4 is a maximal subgroup of
C23.165C24  C4×C422C2  C24.547C23  C23.202C24  C23.218C24  C24.205C23  C23.225C24  C23.229C24  C23.235C24  C23.238C24  C23.262C24  C23.263C24  C24.230C23  C24.577C23  C23.395C24  C23.410C24  C23.414C24  C24.309C23  C23.418C24  C24.313C23  C23.424C24  C23.425C24  C23.432C24  C23.433C24  C23.472C24  C23.473C24  C24.341C23  C23.478C24  C4222D4  C42.183D4  C428Q8  C42.38Q8  C4229D4  C42.189D4  C42.190D4  C42.191D4  C4211Q8  C23.637C24  C24.426C23  C23.649C24  C23.656C24  C24.438C23  C23.658C24  C23.659C24  C23.662C24  C23.664C24  C23.666C24  C23.669C24  C24.445C23  C23.675C24  C23.676C24  C23.677C24  C4233D4  C42.200D4  C4212Q8  C4243D4  C4215Q8  C42⋊C12
 (C4×C4p)⋊C4: C42.6Q8  C427Dic3  C425Dic5  C425F5  C425Dic7 ...
 (C22×C4).D2p: C42.59D4  C42.60D4  C42.63D4  C3⋊(C425C4)  C52(C425C4)  C7⋊(C425C4) ...
C425C4 is a maximal quotient of
C425C8  C424C4.C2
 (C4×C4p)⋊C4: C8⋊C417C4  C427Dic3  C425Dic5  C425F5  C425Dic7 ...
 (C22×C4).D2p: C24.624C23  C24.633C23  C3⋊(C425C4)  C52(C425C4)  C7⋊(C425C4) ...

Matrix representation of C425C4 in GL5(𝔽5)

10000
03000
00300
00020
00003
,
40000
01000
00400
00030
00003
,
20000
00100
01000
00001
00040

G:=sub<GL(5,GF(5))| [1,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,2,0,0,0,0,0,3],[4,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,3,0,0,0,0,0,3],[2,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,1,0] >;

C425C4 in GAP, Magma, Sage, TeX

C_4^2\rtimes_5C_4
% in TeX

G:=Group("C4^2:5C4");
// GroupNames label

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

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

G:=PCGroup([6,-2,2,2,-2,2,2,192,121,151,362,50]);
// Polycyclic

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

Export

Character table of C425C4 in TeX

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