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G = C42:5C4order 64 = 26

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

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

Aliases: C42:5C4, C23.59C23, (C2xC42).3C2, (C22xC4).3C22, C2.8(C42:C2), C2.1(C42:2C2), C22.18(C4oD4), C2.C42.2C2, C22.32(C22xC4), (C2xC4).54(C2xC4), SmallGroup(64,64)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42:5C4
C1C2C22C23C22xC4C2xC42 — C42:5C4
C1C22 — C42:5C4
C1C23 — C42:5C4
C1C23 — C42:5C4

Generators and relations for C42:5C4
 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, C4, C22, C22, C2xC4, C2xC4, C23, C42, C22xC4, C2.C42, C2xC42, C42:5C4
Quotients: C1, C2, C4, C22, C2xC4, C23, C22xC4, C4oD4, C42:C2, C42:2C2, C42:5C4

Character table of C42:5C4

 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 C4oD4
ρ1822-222-2-2-20-2i00002i2i-2i00000000000    complex lifted from C4oD4
ρ19222-2-22-2-22i00-2i-2i2i00000000000000    complex lifted from C4oD4
ρ202-2-22-222-2002i000000-2i-2i2i00000000    complex lifted from C4oD4
ρ2122-2-2-2-222002i0000002i-2i-2i00000000    complex lifted from C4oD4
ρ222-2-2-222-2202i00002i-2i-2i00000000000    complex lifted from C4oD4
ρ232-222-2-2-222i002i-2i-2i00000000000000    complex lifted from C4oD4
ρ2422-2-2-2-22200-2i000000-2i2i2i00000000    complex lifted from C4oD4
ρ252-2-22-222-200-2i0000002i2i-2i00000000    complex lifted from C4oD4
ρ262-222-2-2-22-2i00-2i2i2i00000000000000    complex lifted from C4oD4
ρ272-2-2-222-220-2i0000-2i2i2i00000000000    complex lifted from C4oD4
ρ28222-2-22-2-2-2i002i2i-2i00000000000000    complex lifted from C4oD4

Smallest permutation representation of C42:5C4
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)]])

C42:5C4 is a maximal subgroup of
C23.165C24  C4xC42:2C2  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  C42:22D4  C42.183D4  C42:8Q8  C42.38Q8  C42:29D4  C42.189D4  C42.190D4  C42.191D4  C42:11Q8  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  C42:33D4  C42.200D4  C42:12Q8  C42:43D4  C42:15Q8  C42:C12
 (C4xC4p):C4: C42.6Q8  C42:7Dic3  C42:5Dic5  C42:5F5  C42:5Dic7 ...
 (C22xC4).D2p: C42.59D4  C42.60D4  C42.63D4  C3:(C42:5C4)  C5:2(C42:5C4)  C7:(C42:5C4) ...
C42:5C4 is a maximal quotient of
C42:5C8  C42:4C4.C2
 (C4xC4p):C4: C8:C4:17C4  C42:7Dic3  C42:5Dic5  C42:5F5  C42:5Dic7 ...
 (C22xC4).D2p: C24.624C23  C24.633C23  C3:(C42:5C4)  C5:2(C42:5C4)  C7:(C42:5C4) ...

Matrix representation of C42:5C4 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] >;

C42:5C4 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 C42:5C4 in TeX

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