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G = C428C4order 64 = 26

5th semidirect product of C42 and C4 acting via C4/C2=C2

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

Aliases: C428C4, C23.58C23, (C2×C4).66D4, C4.10(C4⋊C4), (C2×C4).13Q8, (C2×C42).8C2, C22.32(C2×D4), C22.10(C2×Q8), C2.1(C4.4D4), C2.1(C42.C2), C2.7(C42⋊C2), C22.17(C4○D4), C2.C42.1C2, (C22×C4).89C22, C22.31(C22×C4), C2.5(C2×C4⋊C4), (C2×C4⋊C4).4C2, (C2×C4).53(C2×C4), SmallGroup(64,63)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C428C4
C1C2C22C23C22×C4C2×C42 — C428C4
C1C22 — C428C4
C1C23 — C428C4
C1C23 — C428C4

Generators and relations for C428C4
 G = < a,b,c | a4=b4=c4=1, ab=ba, cac-1=ab2, cbc-1=a2b >

Subgroups: 113 in 77 conjugacy classes, 49 normal (9 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×8], C22 [×3], C22 [×4], C2×C4 [×10], C2×C4 [×16], C23, C42 [×4], C4⋊C4 [×4], C22×C4, C22×C4 [×6], C2.C42 [×4], C2×C42, C2×C4⋊C4 [×2], C428C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C4○D4 [×4], C2×C4⋊C4, C42⋊C2 [×2], C4.4D4 [×2], C42.C2 [×2], C428C4

Character table of C428C4

 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
ρ1722-2-2-2-2220020000002-2-200000000    orthogonal lifted from D4
ρ1822-2-2-2-22200-2000000-22200000000    orthogonal lifted from D4
ρ192-2-22-222-200-200000022-200000000    symplectic lifted from Q8, Schur index 2
ρ202-2-22-222-2002000000-2-2200000000    symplectic lifted from Q8, Schur index 2
ρ2122-222-2-2-202i0000-2i-2i2i00000000000    complex lifted from C4○D4
ρ2222-222-2-2-20-2i00002i2i-2i00000000000    complex lifted from C4○D4
ρ23222-2-22-2-22i00-2i-2i2i00000000000000    complex lifted from C4○D4
ρ242-2-2-222-2202i00002i-2i-2i00000000000    complex lifted from C4○D4
ρ252-222-2-2-222i002i-2i-2i00000000000000    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 C428C4
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 25 14 10)(2 26 15 11)(3 27 16 12)(4 28 13 9)(5 37 22 30)(6 38 23 31)(7 39 24 32)(8 40 21 29)(17 50 61 33)(18 51 62 34)(19 52 63 35)(20 49 64 36)(41 46 58 55)(42 47 59 56)(43 48 60 53)(44 45 57 54)
(1 50 6 42)(2 34 7 60)(3 52 8 44)(4 36 5 58)(9 62 30 48)(10 19 31 54)(11 64 32 46)(12 17 29 56)(13 49 22 41)(14 33 23 59)(15 51 24 43)(16 35 21 57)(18 37 53 28)(20 39 55 26)(25 63 38 45)(27 61 40 47)

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

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

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

C428C4 is a maximal subgroup of
C23.165C24  C4×C4.4D4  C4×C42.C2  C24.192C23  C24.547C23  C23.202C24  C42.161D4  C4214D4  C424Q8  C24.203C23  C23.225C24  C24.208C23  C23.231C24  C23.233C24  C23.234C24  C23.237C24  C23.238C24  C24.212C23  C23.261C24  C23.264C24  C24.254C23  C23.321C24  C24.267C23  C24.569C23  C23.388C24  C24.301C23  C24.304C23  C23.395C24  C23.405C24  C23.408C24  C23.409C24  C23.413C24  C23.414C24  C24.309C23  C23.417C24  C23.418C24  C23.419C24  C23.420C24  C23.424C24  C23.428C24  C23.429C24  C4217D4  C42.165D4  C426Q8  C24.326C23  C42.173D4  C42.174D4  C42.175D4  C42.177D4  C42.36Q8  C42.37Q8  C24.338C23  C24.339C23  C24.340C23  C42.178D4  C42.179D4  C4225D4  C429Q8  C4227D4  C42.187D4  C4230D4  C42.192D4  C42.39Q8  C4210Q8  C4211Q8  C24.405C23  C23.606C24  C23.619C24  C24.418C23  C24.430C23  C24.437C23  C23.654C24  C23.655C24  C24.440C23  C23.662C24  C23.663C24  C24.443C23  C23.666C24  C23.667C24  C23.668C24  C23.669C24  C23.672C24  C23.673C24  C23.674C24  C23.675C24  C23.681C24  C23.682C24  C23.683C24  C23.685C24  C23.689C24  C4234D4  C42.199D4  C42.201D4  C4213Q8  C42.40Q8  C42.439D4  C43.15C2  C4218Q8  C43.18C2
 (C4×C4p)⋊C4: C42.5Q8  C42.56Q8  C42.60Q8  C4211Dic3  C429Dic5  C429F5  C429Dic7 ...
 (C2×C4p).Q8: C42.26Q8  C42.437D4  (C2×C8).24Q8  (C4×Dic3)⋊9C4  C20.48(C4⋊C4)  (C4×Dic7)⋊9C4 ...
 (C22×C4).D2p: C42.7Q8  C42⋊C8  C42.58D4  C42.61D4  C42.62D4  C42.100D4  C42.101D4  C42.24Q8 ...
C428C4 is a maximal quotient of
C428C8  C42.23Q8  C42.322D4  C42.104D4
 C4p.(C4⋊C4): C8.(C4⋊C4)  C4211Dic3  (C4×Dic3)⋊9C4  C429Dic5  C20.48(C4⋊C4)  C429F5  C429Dic7  (C4×Dic7)⋊9C4 ...
 (C22×C4).D2p: C24.624C23  C24.625C23  C24.632C23  C42.55Q8  C42.56Q8  C42.24Q8  C3⋊(C428C4)  C52(C428C4) ...

Matrix representation of C428C4 in GL5(𝔽5)

10000
02000
00200
00004
00010
,
10000
00100
01000
00004
00010
,
20000
00300
02000
00020
00003

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

C428C4 in GAP, Magma, Sage, TeX

C_4^2\rtimes_8C_4
% in TeX

G:=Group("C4^2:8C4");
// GroupNames label

G:=SmallGroup(64,63);
// by ID

G=gap.SmallGroup(64,63);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,2,192,121,103,362,50]);
// 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>;
// generators/relations

Export

Character table of C428C4 in TeX

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