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G = C429C4order 64 = 26

6th semidirect product of C42 and C4 acting via C4/C2=C2

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

Aliases: C429C4, C23.60C23, C41(C4⋊C4), (C2×C4).67D4, C2.1(C4⋊Q8), (C2×C4).14Q8, (C2×C42).9C2, C2.1(C41D4), C22.33(C2×D4), C22.11(C2×Q8), C22.33(C22×C4), (C22×C4).104C22, C2.6(C2×C4⋊C4), (C2×C4⋊C4).5C2, (C2×C4).70(C2×C4), SmallGroup(64,65)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C429C4
C1C2C22C23C22×C4C2×C42 — C429C4
C1C22 — C429C4
C1C23 — C429C4
C1C23 — C429C4

Generators and relations for C429C4
 G = < a,b,c | a4=b4=c4=1, ab=ba, cac-1=a-1, cbc-1=b-1 >

Subgroups: 129 in 93 conjugacy classes, 65 normal (6 characteristic)
C1, C2, C2 [×6], C4 [×12], C4 [×4], C22, C22 [×6], C2×C4 [×18], C2×C4 [×12], C23, C42 [×4], C4⋊C4 [×12], C22×C4 [×7], C2×C42, C2×C4⋊C4 [×6], C429C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×6], C23, C4⋊C4 [×12], C22×C4, C2×D4 [×3], C2×Q8 [×3], C2×C4⋊C4 [×3], C41D4, C4⋊Q8 [×3], C429C4

Character table of C429C4

 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
ρ182-222-2-2-222002-2-200000000000000    orthogonal lifted from D4
ρ192-222-2-2-22-200-22200000000000000    orthogonal lifted from D4
ρ202-2-2-222-220200002-2-200000000000    orthogonal lifted from D4
ρ212-2-2-222-220-20000-22200000000000    orthogonal lifted from D4
ρ2222-2-2-2-22200-2000000-22200000000    orthogonal lifted from D4
ρ2322-222-2-2-2020000-2-2200000000000    symplectic lifted from Q8, Schur index 2
ρ242-2-22-222-200-200000022-200000000    symplectic lifted from Q8, Schur index 2
ρ25222-2-22-2-2200-2-2200000000000000    symplectic lifted from Q8, Schur index 2
ρ262-2-22-222-2002000000-2-2200000000    symplectic lifted from Q8, Schur index 2
ρ2722-222-2-2-20-2000022-200000000000    symplectic lifted from Q8, Schur index 2
ρ28222-2-22-2-2-20022-200000000000000    symplectic lifted from Q8, Schur index 2

Smallest permutation representation of C429C4
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 31 38 13)(2 32 39 14)(3 29 40 15)(4 30 37 16)(5 46 24 53)(6 47 21 54)(7 48 22 55)(8 45 23 56)(9 49 27 34)(10 50 28 35)(11 51 25 36)(12 52 26 33)(17 43 61 58)(18 44 62 59)(19 41 63 60)(20 42 64 57)
(1 56 27 63)(2 55 28 62)(3 54 25 61)(4 53 26 64)(5 52 57 16)(6 51 58 15)(7 50 59 14)(8 49 60 13)(9 19 38 45)(10 18 39 48)(11 17 40 47)(12 20 37 46)(21 36 43 29)(22 35 44 32)(23 34 41 31)(24 33 42 30)

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

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

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

C429C4 is a maximal subgroup of
C42.8Q8  C423C8  C42.98D4  C42.99D4  C42.55Q8  C42.24Q8  C4≀C2⋊C4  C42.29Q8  C42.30Q8  C42.431D4  C42.432D4  C42.110D4  C42.436D4  C42.124D4  C4211D4  M4(2)⋊Q8  (C2×C4).24D8  (C2×C4).19Q16  (C2×C8).1Q8  C2.(C83Q8)  (C2×C4).27D8  (C2×C8).169D4  (C2×C8).60D4  (C2×C8).170D4  (C2×C4).28D8  (C2×C4).23Q16  C23.167C24  C4×C41D4  C4×C4⋊Q8  C24.192C23  C23.199C24  C42.160D4  C42.33Q8  D4×C4⋊C4  Q8×C4⋊C4  C23.236C24  C23.237C24  C24.230C23  C23.322C24  C23.323C24  C24.568C23  C24.268C23  C23.396C24  C23.397C24  C24.308C23  C23.400C24  C23.401C24  C23.402C24  C23.406C24  C23.407C24  C23.411C24  C23.412C24  C4218D4  C42.166D4  C42.167D4  C427Q8  C42.35Q8  C42.174D4  C42.175D4  C42.176D4  C42.36Q8  C42.37Q8  C42.180D4  C4228D4  C42.188D4  C42.39Q8  C4210Q8  C23.580C24  C23.618C24  C23.620C24  C23.621C24  C24.454C23  C23.691C24  C23.692C24  C23.693C24  C23.694C24  C23.695C24  C4235D4  C4212Q8  C4247D4  C42.440D4  C43.15C2  C4219Q8  C422C12
 C4p⋊(C4⋊C4): C42.58Q8  C42.59Q8  C42.26Q8  C4210Dic3  (C4×Dic3)⋊8C4  C428Dic5  C205(C4⋊C4)  C428F5 ...
C429C4 is a maximal quotient of
C24.625C23  C24.634C23  C429C8  C42.25Q8  C42.60Q8  C42.324D4  C42.106D4
 C4p⋊(C4⋊C4): C42.58Q8  C42.59Q8  C42.26Q8  C4210Dic3  (C4×Dic3)⋊8C4  C428Dic5  C205(C4⋊C4)  C428F5 ...

Matrix representation of C429C4 in GL5(𝔽5)

40000
03000
00200
00020
00003
,
40000
02000
00300
00040
00004
,
20000
00100
01000
00001
00040

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

C429C4 in GAP, Magma, Sage, TeX

C_4^2\rtimes_9C_4
% in TeX

G:=Group("C4^2:9C4");
// GroupNames label

G:=SmallGroup(64,65);
// by ID

G=gap.SmallGroup(64,65);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,2,192,121,55,362,86]);
// Polycyclic

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

Export

Character table of C429C4 in TeX

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