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G = C84Q8order 64 = 26

3rd semidirect product of C8 and Q8 acting via Q8/C4=C2

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

Aliases: C84Q8, C4.3M4(2), C42.13C22, C4⋊C4.9C4, C4⋊C8.10C2, C2.6(C4×Q8), (C4×C8).15C2, (C2×Q8).7C4, (C4×Q8).4C2, C4.24(C2×Q8), C8⋊C4.5C2, C2.9(C8○D4), C4.55(C4○D4), (C2×C8).103C22, (C2×C4).157C23, C2.11(C2×M4(2)), C22.49(C22×C4), (C2×C4).30(C2×C4), SmallGroup(64,127)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C84Q8
C1C2C4C2×C4C2×C8C4×C8 — C84Q8
C1C22 — C84Q8
C1C2×C4 — C84Q8
C1C2C2C2×C4 — C84Q8

Generators and relations for C84Q8
 G = < a,b,c | a8=b4=1, c2=b2, ab=ba, cac-1=a5, cbc-1=b-1 >

2C4
2C4
2C4
2C4
2C4
2C8
2Q8
2Q8
2C8
2C8

Character table of C84Q8

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E8F8G8H8I8J8K8L
 size 1111111122224444222222224444
ρ11111111111111111111111111111    trivial
ρ2111111111111-1-1-1-111111111-1-1-1-1    linear of order 2
ρ311111111-1-1-1-11-1-11-11-11-11-11-111-1    linear of order 2
ρ411111111-1-1-1-1-111-1-11-11-11-111-1-11    linear of order 2
ρ5111111111111-1-1-1-1-1-1-1-1-1-1-1-11111    linear of order 2
ρ61111111111111111-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ711111111-1-1-1-11-1-111-11-11-11-11-1-11    linear of order 2
ρ811111111-1-1-1-1-111-11-11-11-11-1-111-1    linear of order 2
ρ91111-1-1-1-11-1-111-11-1-i-iii-iii-i-i-iii    linear of order 4
ρ101111-1-1-1-11-1-111-11-1ii-i-ii-i-iiii-i-i    linear of order 4
ρ111111-1-1-1-11-1-11-11-11ii-i-ii-i-ii-i-iii    linear of order 4
ρ121111-1-1-1-11-1-11-11-11-i-iii-iii-iii-i-i    linear of order 4
ρ131111-1-1-1-1-111-1-1-111i-i-iiii-i-i-ii-ii    linear of order 4
ρ141111-1-1-1-1-111-1-1-111-iii-i-i-iiii-ii-i    linear of order 4
ρ151111-1-1-1-1-111-111-1-1-iii-i-i-iii-ii-ii    linear of order 4
ρ161111-1-1-1-1-111-111-1-1i-i-iiii-i-ii-ii-i    linear of order 4
ρ172-2-22-22-2200000000-202020-200000    symplectic lifted from Q8, Schur index 2
ρ182-2-22-22-220000000020-20-20200000    symplectic lifted from Q8, Schur index 2
ρ192-22-22i2i-2i-2i-2i2-22i0000000000000000    complex lifted from M4(2)
ρ202-22-2-2i-2i2i2i-2i-222i0000000000000000    complex lifted from M4(2)
ρ212-2-222-22-200000000-2i0-2i02i02i00000    complex lifted from C4○D4
ρ222-2-222-22-2000000002i02i0-2i0-2i00000    complex lifted from C4○D4
ρ232-22-22i2i-2i-2i2i-22-2i0000000000000000    complex lifted from M4(2)
ρ242-22-2-2i-2i2i2i2i2-2-2i0000000000000000    complex lifted from M4(2)
ρ2522-2-2-2i2i2i-2i00000000085083087080000    complex lifted from C8○D4
ρ2622-2-2-2i2i2i-2i00000000080870830850000    complex lifted from C8○D4
ρ2722-2-22i-2i-2i2i00000000087080850830000    complex lifted from C8○D4
ρ2822-2-22i-2i-2i2i00000000083085080870000    complex lifted from C8○D4

Smallest permutation representation of C84Q8
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 26 21 38)(2 27 22 39)(3 28 23 40)(4 29 24 33)(5 30 17 34)(6 31 18 35)(7 32 19 36)(8 25 20 37)(9 48 64 56)(10 41 57 49)(11 42 58 50)(12 43 59 51)(13 44 60 52)(14 45 61 53)(15 46 62 54)(16 47 63 55)
(1 45 21 53)(2 42 22 50)(3 47 23 55)(4 44 24 52)(5 41 17 49)(6 46 18 54)(7 43 19 51)(8 48 20 56)(9 37 64 25)(10 34 57 30)(11 39 58 27)(12 36 59 32)(13 33 60 29)(14 38 61 26)(15 35 62 31)(16 40 63 28)

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

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

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

C84Q8 is a maximal subgroup of
C8.17Q16  C812SD16  C89Q16  D4.M4(2)  Q8.M4(2)  C89SD16  C8⋊M4(2)  C8⋊SD16  C82SD16  C8.SD16  C8⋊Q16  C82Q16  C8.3Q16  C42.249C23  C42.251C23  C42.253C23  C42.255C23  C42.290C23  C42.291C23  C42.292C23  C42.293C23  C42.294C23  D46M4(2)  C42.302C23  Q8.4M4(2)  C42.696C23  C42.304C23  C42.305C23  C42.698C23  D48M4(2)  C42.307C23  C42.308C23  C42.309C23  C42.310C23  C42.507C23  C42.508C23  C42.509C23  C42.510C23  C42.511C23  C42.512C23  C42.513C23  C42.514C23  C42.515C23  C42.516C23  C42.517C23  C42.518C23  D84Q8  SD16⋊Q8  SD162Q8  Q164Q8  SD163Q8  D85Q8  Q165Q8  C42.72C23  C42.73C23  C42.74C23  C42.75C23  C42.531C23  C42.532C23  C42.533C23
 C4p.M4(2): C86Q16  C8.M4(2)  C2412Q8  C42.198D6  C42.210D6  C4011Q8  C42.198D10  C42.210D10 ...
 C2p.(C4×Q8): C42.286C23  C42.287C23  M4(2)⋊9Q8  Q8×M4(2)  C24⋊Q8  C42.27D6  C40⋊Q8  Dic5.5M4(2) ...
C84Q8 is a maximal quotient of
C4⋊C813C4  C4⋊C814C4  C4⋊C43C8  (C2×C8).Q8  Dic5.M4(2)  C20.M4(2)  C20.6M4(2)
 C42.D2p: C42.61Q8  C42.27Q8  C42.327D4  C42.120D4  C2412Q8  C24⋊Q8  C42.27D6  C42.198D6 ...

Matrix representation of C84Q8 in GL4(𝔽17) generated by

0100
13000
0001
00130
,
1000
0100
0009
00150
,
16000
0100
00108
00157
G:=sub<GL(4,GF(17))| [0,13,0,0,1,0,0,0,0,0,0,13,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,15,0,0,9,0],[16,0,0,0,0,1,0,0,0,0,10,15,0,0,8,7] >;

C84Q8 in GAP, Magma, Sage, TeX

C_8\rtimes_4Q_8
% in TeX

G:=Group("C8:4Q8");
// GroupNames label

G:=SmallGroup(64,127);
// by ID

G=gap.SmallGroup(64,127);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,55,650,122,88]);
// Polycyclic

G:=Group<a,b,c|a^8=b^4=1,c^2=b^2,a*b=b*a,c*a*c^-1=a^5,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C84Q8 in TeX
Character table of C84Q8 in TeX

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