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G = C8.5Q8order 64 = 26

4th non-split extension by C8 of Q8 acting via Q8/C4=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C8.5Q8, C42.84C22, (C4×C8).9C2, C4.6(C2×Q8), (C2×C4).59D4, C2.7(C4⋊Q8), C2.D8.6C2, C4.Q8.7C2, C2.19(C4○D8), C4⋊C4.23C22, (C2×C8).94C22, C42.C2.4C2, (C2×C4).124C23, C22.120(C2×D4), SmallGroup(64,180)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C8.5Q8
C1C2C4C2×C4C2×C8C4×C8 — C8.5Q8
C1C2C2×C4 — C8.5Q8
C1C22C42 — C8.5Q8
C1C2C2C2×C4 — C8.5Q8

Generators and relations for C8.5Q8
 G = < a,b,c | a8=b4=1, c2=a4b2, ab=ba, cac-1=a3, cbc-1=a4b-1 >

2C4
2C4
4C4
4C4
4C4
4C4
2C2×C4
2C2×C4
2C2×C4
2C2×C4
2C4⋊C4
2C4⋊C4
2C4⋊C4
2C4⋊C4

Character table of C8.5Q8

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F8G8H
 size 1111222222888822222222
ρ11111111111111111111111    trivial
ρ21111-11-11-1-1-11-11-111-1-11-11    linear of order 2
ρ31111111111-111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ41111-11-11-1-111-1-11-1-111-11-1    linear of order 2
ρ511111111111-1-11-1-1-1-1-1-1-1-1    linear of order 2
ρ61111-11-11-1-1-1-1111-1-111-11-1    linear of order 2
ρ71111111111-1-1-1-111111111    linear of order 2
ρ81111-11-11-1-11-11-1-111-1-11-11    linear of order 2
ρ922222-2-2-22-2000000000000    orthogonal lifted from D4
ρ102222-2-22-2-22000000000000    orthogonal lifted from D4
ρ112-22-20-202000000200-2-2020    symplectic lifted from Q8, Schur index 2
ρ122-22-2020-200000002-200-202    symplectic lifted from Q8, Schur index 2
ρ132-22-2020-20000000-220020-2    symplectic lifted from Q8, Schur index 2
ρ142-22-20-202000000-200220-20    symplectic lifted from Q8, Schur index 2
ρ152-2-222i000-2i00000-2--2--22-2-22-2    complex lifted from C4○D8
ρ1622-2-200-2i002i0000-2--2-2-22--22-2    complex lifted from C4○D8
ρ172-2-22-2i0002i000002--2--2-22-2-2-2    complex lifted from C4○D8
ρ1822-2-2002i00-2i00002--2-22-2--2-2-2    complex lifted from C4○D8
ρ1922-2-2002i00-2i0000-2-2--2-22-22--2    complex lifted from C4○D8
ρ2022-2-200-2i002i00002-2--22-2-2-2--2    complex lifted from C4○D8
ρ212-2-222i000-2i000002-2-2-22--2-2--2    complex lifted from C4○D8
ρ222-2-22-2i0002i00000-2-2-22-2--22--2    complex lifted from C4○D8

Smallest permutation representation of C8.5Q8
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 33 19 30)(2 34 20 31)(3 35 21 32)(4 36 22 25)(5 37 23 26)(6 38 24 27)(7 39 17 28)(8 40 18 29)(9 46 58 55)(10 47 59 56)(11 48 60 49)(12 41 61 50)(13 42 62 51)(14 43 63 52)(15 44 64 53)(16 45 57 54)
(1 51 23 46)(2 54 24 41)(3 49 17 44)(4 52 18 47)(5 55 19 42)(6 50 20 45)(7 53 21 48)(8 56 22 43)(9 37 62 30)(10 40 63 25)(11 35 64 28)(12 38 57 31)(13 33 58 26)(14 36 59 29)(15 39 60 32)(16 34 61 27)

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

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

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

C8.5Q8 is a maximal subgroup of
C8.1Q16  M4(2)⋊3Q8  M4(2)⋊4Q8  C42.386C23  C42.388C23  C42.389C23  C42.423C23  C42.424C23  C42.485C23  C42.486C23  C42.488C23  C42.492C23  C42.493C23  C42.496C23  D86Q8  SD164Q8  Q166Q8  SD163Q8  D85Q8  Q165Q8
 C8p.Q8: C16.5Q8  C24.13Q8  C8.8Dic6  C8.6Dic6  C40.13Q8  C8.8Dic10  C8.6Dic10  C56.13Q8 ...
 C42.D2p: C8.16Q16  D8.Q8  Q16.Q8  D8.2Q8  C8.22SD16  C8.13SD16  C8.14SD16  C16⋊Q8 ...
C8.5Q8 is a maximal quotient of
C42.60Q8  C87(C4⋊C4)  C85(C4⋊C4)
 C42.D2p: C42.437D4  C24.13Q8  C42.215D6  C40.13Q8  C42.215D10  C56.13Q8  C42.215D14 ...
 C4⋊C4.D2p: C2.(C8⋊Q8)  (C2×C8).24Q8  C8.8Dic6  C8.6Dic6  C8.8Dic10  C8.6Dic10  C56.8Q8  C56.4Q8 ...

Matrix representation of C8.5Q8 in GL4(𝔽17) generated by

1000
0100
00150
0009
,
11300
91600
0040
0004
,
131600
0400
0001
0010
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,15,0,0,0,0,9],[1,9,0,0,13,16,0,0,0,0,4,0,0,0,0,4],[13,0,0,0,16,4,0,0,0,0,0,1,0,0,1,0] >;

C8.5Q8 in GAP, Magma, Sage, TeX

C_8._5Q_8
% in TeX

G:=Group("C8.5Q8");
// GroupNames label

G:=SmallGroup(64,180);
// by ID

G=gap.SmallGroup(64,180);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,48,121,247,362,86,1444,88]);
// Polycyclic

G:=Group<a,b,c|a^8=b^4=1,c^2=a^4*b^2,a*b=b*a,c*a*c^-1=a^3,c*b*c^-1=a^4*b^-1>;
// generators/relations

Export

Subgroup lattice of C8.5Q8 in TeX
Character table of C8.5Q8 in TeX

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