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G = C42.73C23order 128 = 27

73rd non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.73C23, C4.1492+ 1+4, C4⋊C4.185D4, C84Q820C2, Q83Q813C2, C42Q1644C2, C4⋊Q1624C2, C8.20(C4○D4), (C2×Q8).253D4, C2.72(Q8○D8), (C4×SD16).8C2, C8.2D4.3C2, Q16⋊C430C2, C4⋊C4.449C23, C4⋊C8.151C22, (C2×C4).590C24, (C4×C8).208C22, (C2×C8).222C23, D4.D4.2C2, C4⋊Q8.217C22, Q8.D4.4C2, C8⋊C4.77C22, C2.44(Q86D4), (C4×D4).223C22, (C2×D4).284C23, C4.54(C8.C22), (C2×Q16).94C22, (C2×Q8).269C23, (C4×Q8).213C22, C4.Q8.141C22, Q8⋊C4.95C22, (C2×SD16).76C22, C4.4D4.90C22, C22.850(C22×D4), D4⋊C4.177C22, C22.50C24.11C2, C4.168(C2×C4○D4), (C2×C4).654(C2×D4), C2.92(C2×C8.C22), SmallGroup(128,2130)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.73C23
C1C2C4C2×C4C42C4×D4C22.50C24 — C42.73C23
C1C2C2×C4 — C42.73C23
C1C22C4×Q8 — C42.73C23
C1C2C2C2×C4 — C42.73C23

Generators and relations for C42.73C23
 G = < a,b,c,d,e | a4=b4=1, c2=b2, d2=e2=a2, ab=ba, cac-1=eae-1=a-1, dad-1=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ce=ec, de=ed >

Subgroups: 304 in 175 conjugacy classes, 88 normal (38 characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×2], C4 [×12], C22, C22 [×3], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×9], D4 [×2], Q8 [×12], C23, C42, C42 [×2], C42 [×5], C22⋊C4 [×5], C4⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×13], C2×C8 [×2], C2×C8 [×2], SD16 [×4], Q16 [×8], C22×C4, C2×D4, C2×Q8 [×2], C2×Q8 [×4], C4×C8, C8⋊C4 [×2], D4⋊C4, Q8⋊C4, Q8⋊C4 [×4], C4⋊C8, C4⋊C8 [×2], C4.Q8, C42⋊C2, C4×D4, C4×Q8 [×2], C4×Q8 [×4], C4×Q8, C22⋊Q8, C4.4D4 [×2], C42.C2 [×3], C422C2 [×2], C4⋊Q8 [×2], C4⋊Q8 [×2], C2×SD16, C2×SD16 [×2], C2×Q16 [×6], C4×SD16, Q16⋊C4 [×2], C84Q8, D4.D4, C42Q16, C42Q16 [×2], Q8.D4 [×2], C4⋊Q16, C8.2D4 [×2], C22.50C24, Q83Q8, C42.73C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×2], C24, C8.C22 [×2], C22×D4, C2×C4○D4, 2+ 1+4, Q86D4, C2×C8.C22, Q8○D8, C42.73C23

Character table of C42.73C23

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q4R8A8B8C8D8E8F
 size 11118222244444444488888444488
ρ111111111111111111111111111111    trivial
ρ21111111-1-1-1-11-1-1-11-1-11-11-11-111-1-11    linear of order 2
ρ3111111111-1-1-111-1-1-11-111-1-1-1-1-1-111    linear of order 2
ρ41111111-1-111-1-1-11-11-1-1-111-11-1-11-11    linear of order 2
ρ51111-111111-1-11-1-1-11-11-1-111-1-1-1-111    linear of order 2
ρ61111-111-1-1-11-1-111-1-1111-1-111-1-11-11    linear of order 2
ρ71111-11111-1111-111-1-1-1-1-1-1-1111111    linear of order 2
ρ81111-111-1-11-11-11-1111-11-11-1-111-1-11    linear of order 2
ρ91111111-1-1-11-1-1-11-1-1-1-11-111-111-11-1    linear of order 2
ρ101111111111-1-111-1-111-1-1-1-111111-1-1    linear of order 2
ρ111111111-1-11-11-1-1-111-111-1-1-11-1-111-1    linear of order 2
ρ12111111111-1111111-111-1-11-1-1-1-1-1-1-1    linear of order 2
ρ131111-111-1-1-1-11-11-11-11-1-11111-1-111-1    linear of order 2
ρ141111-111111111-1111-1-111-11-1-1-1-1-1-1    linear of order 2
ρ151111-111-1-111-1-111-1111-11-1-1-111-11-1    linear of order 2
ρ161111-11111-1-1-11-1-1-1-1-11111-11111-1-1    linear of order 2
ρ1722220-2-222022-20-2-20000000000000    orthogonal lifted from D4
ρ1822220-2-2-2-20-22202-20000000000000    orthogonal lifted from D4
ρ1922220-2-2-2-202-220-220000000000000    orthogonal lifted from D4
ρ2022220-2-2220-2-2-20220000000000000    orthogonal lifted from D4
ρ212-22-202-200-2i000-2i002i2i000000-22000    complex lifted from C4○D4
ρ222-22-202-2002i0002i00-2i-2i000000-22000    complex lifted from C4○D4
ρ232-22-202-2002i000-2i00-2i2i0000002-2000    complex lifted from C4○D4
ρ242-22-202-200-2i0002i002i-2i0000002-2000    complex lifted from C4○D4
ρ254-44-40-440000000000000000000000    orthogonal lifted from 2+ 1+4
ρ264-4-440004-400000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ274-4-44000-4400000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2844-4-400000000000000000002200-2200    symplectic lifted from Q8○D8, Schur index 2
ρ2944-4-40000000000000000000-22002200    symplectic lifted from Q8○D8, Schur index 2

Smallest permutation representation of C42.73C23
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 51 11 25)(2 52 12 26)(3 49 9 27)(4 50 10 28)(5 62 54 58)(6 63 55 59)(7 64 56 60)(8 61 53 57)(13 37 35 31)(14 38 36 32)(15 39 33 29)(16 40 34 30)(17 48 44 23)(18 45 41 24)(19 46 42 21)(20 47 43 22)
(1 59 11 63)(2 58 12 62)(3 57 9 61)(4 60 10 64)(5 52 54 26)(6 51 55 25)(7 50 56 28)(8 49 53 27)(13 22 35 47)(14 21 36 46)(15 24 33 45)(16 23 34 48)(17 40 44 30)(18 39 41 29)(19 38 42 32)(20 37 43 31)
(1 37 3 39)(2 32 4 30)(5 42 7 44)(6 20 8 18)(9 29 11 31)(10 40 12 38)(13 49 15 51)(14 28 16 26)(17 54 19 56)(21 60 23 58)(22 61 24 63)(25 35 27 33)(34 52 36 50)(41 55 43 53)(45 59 47 57)(46 64 48 62)
(1 39 3 37)(2 38 4 40)(5 21 7 23)(6 24 8 22)(9 31 11 29)(10 30 12 32)(13 25 15 27)(14 28 16 26)(17 62 19 64)(18 61 20 63)(33 49 35 51)(34 52 36 50)(41 57 43 59)(42 60 44 58)(45 53 47 55)(46 56 48 54)

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

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

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

Matrix representation of C42.73C23 in GL8(𝔽17)

00100000
00010000
160000000
016000000
000012050
000001205
00005050
00000505
,
160000000
016000000
001600000
000160000
00000100
000016000
00000001
000000160
,
0014140000
001430000
1414000000
143000000
000000314
0000001414
000014300
00003300
,
016000000
10000000
000160000
00100000
000000160
00000001
00001000
000001600
,
01000000
160000000
000160000
00100000
00000010
00000001
000016000
000001600

G:=sub<GL(8,GF(17))| [0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,5,0,0,0,0,0,0,12,0,5,0,0,0,0,5,0,5,0,0,0,0,0,0,5,0,5],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[0,0,14,14,0,0,0,0,0,0,14,3,0,0,0,0,14,14,0,0,0,0,0,0,14,3,0,0,0,0,0,0,0,0,0,0,0,0,14,3,0,0,0,0,0,0,3,3,0,0,0,0,3,14,0,0,0,0,0,0,14,14,0,0],[0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;

C42.73C23 in GAP, Magma, Sage, TeX

C_4^2._{73}C_2^3
% in TeX

G:=Group("C4^2.73C2^3");
// GroupNames label

G:=SmallGroup(128,2130);
// by ID

G=gap.SmallGroup(128,2130);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,723,100,346,80,4037,1027,124]);
// Polycyclic

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

Export

Character table of C42.73C23 in TeX

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