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G = C23.13SD16order 128 = 27

3rd non-split extension by C23 of SD16 acting via SD16/C4=C22

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C23.13SD16, C4.21C4≀C2, C2.D8.2C4, (C2×C4).103D8, (C2×C8).303D4, (C2×Q16).1C4, C22⋊C16.4C2, C4.8(C23⋊C4), C8.18D4.2C2, (C22×C4).188D4, C2.5(D8.C4), C2.3(C8.17D4), C4.C42.6C2, (C22×C8).100C22, C22.58(D4⋊C4), C2.15(C22.SD16), (C2×C8).20(C2×C4), (C2×C4).220(C22⋊C4), SmallGroup(128,82)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — C23.13SD16
C1C2C4C2×C4C22×C4C22×C8C8.18D4 — C23.13SD16
C1C2C2×C4C2×C8 — C23.13SD16
C1C22C22×C4C22×C8 — C23.13SD16
C1C2C2C2C2C4C2×C4C22×C8 — C23.13SD16

Generators and relations for C23.13SD16
 G = < a,b,c,d,e | a2=b2=c2=1, d8=e2=c, eae-1=ab=ba, ac=ca, dad-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=abd3 >

4C2
2C22
2C22
2C22
2C4
8C4
8C4
2C2×C4
2C8
2C8
2C2×C4
4C8
4Q8
4C2×C4
4C2×C4
4Q8
4C8
2M4(2)
2C4⋊C4
2C2×Q8
2M4(2)
4C16
4C2×C8
4C4⋊C4
4Q16
4C2×C8
4M4(2)
4C22⋊C4
2C2×C16
2Q8⋊C4
2C22⋊Q8
2C2×M4(2)

Character table of C23.13SD16

 class 12A2B2C2D4A4B4C4D4E4F8A8B8C8D8E8F8G8H8I8J16A16B16C16D16E16F16G16H
 size 1111422221616222244888844444444
ρ111111111111111111111111111111    trivial
ρ2111111111-1-11111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ311111111111111111-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ4111111111-1-1111111-1-1-1-111111111    linear of order 2
ρ51111-1-111-1-11-1-1-1-111-i-iiii-iii-i-i-ii    linear of order 4
ρ61111-1-111-11-1-1-1-1-111-i-iii-ii-i-iiii-i    linear of order 4
ρ71111-1-111-11-1-1-1-1-111ii-i-ii-iii-i-i-ii    linear of order 4
ρ81111-1-111-1-11-1-1-1-111ii-i-i-ii-i-iiii-i    linear of order 4
ρ92222-2-222-2002222-2-2000000000000    orthogonal lifted from D4
ρ1022222222200-2-2-2-2-2-2000000000000    orthogonal lifted from D4
ρ112222-22-2-22000000000000-2-2-222-222    orthogonal lifted from D8
ρ122222-22-2-22000000000000222-2-22-2-2    orthogonal lifted from D8
ρ132-22-2002-2000-2i2i2i-2i00-1+i1-i-1-i1+i00000000    complex lifted from C4≀C2
ρ142-22-2002-20002i-2i-2i2i00-1-i1+i-1+i1-i00000000    complex lifted from C4≀C2
ρ152-22-2002-2000-2i2i2i-2i001-i-1+i1+i-1-i00000000    complex lifted from C4≀C2
ρ162-22-2002-20002i-2i-2i2i001+i-1-i1-i-1+i00000000    complex lifted from C4≀C2
ρ1722222-2-2-2-2000000000000-2--2-2--2-2--2-2--2    complex lifted from SD16
ρ1822222-2-2-2-2000000000000--2-2--2-2--2-2--2-2    complex lifted from SD16
ρ192-2-2202i00-2i00-2--2-2--2-220000ζ1611169ζ16151613ζ16316ζ1613167ζ161116ζ167165ζ169163ζ1615165    complex lifted from D8.C4
ρ202-2-220-2i002i00--2-2--2-2-220000ζ167165ζ16316ζ16151613ζ169163ζ1615165ζ1611169ζ1613167ζ161116    complex lifted from D8.C4
ρ212-2-220-2i002i00-2--2-2--22-20000ζ169163ζ1615165ζ161116ζ16151613ζ1611169ζ1613167ζ16316ζ167165    complex lifted from D8.C4
ρ222-2-2202i00-2i00--2-2--2-22-20000ζ1613167ζ161116ζ1615165ζ16316ζ167165ζ169163ζ16151613ζ1611169    complex lifted from D8.C4
ρ232-2-220-2i002i00--2-2--2-2-220000ζ16151613ζ1611169ζ167165ζ161116ζ1613167ζ16316ζ1615165ζ169163    complex lifted from D8.C4
ρ242-2-2202i00-2i00--2-2--2-22-20000ζ1615165ζ169163ζ1613167ζ1611169ζ16151613ζ161116ζ167165ζ16316    complex lifted from D8.C4
ρ252-2-2202i00-2i00-2--2-2--2-220000ζ16316ζ167165ζ1611169ζ1615165ζ169163ζ16151613ζ161116ζ1613167    complex lifted from D8.C4
ρ262-2-220-2i002i00-2--2-2--22-20000ζ161116ζ1613167ζ169163ζ167165ζ16316ζ1615165ζ1611169ζ16151613    complex lifted from D8.C4
ρ274-44-400-44000000000000000000000    orthogonal lifted from C23⋊C4
ρ2844-4-40000000-22-22222200000000000000    symplectic lifted from C8.17D4, Schur index 2
ρ2944-4-400000002222-22-2200000000000000    symplectic lifted from C8.17D4, Schur index 2

Smallest permutation representation of C23.13SD16
On 64 points
Generators in S64
(1 21)(3 23)(5 25)(7 27)(9 29)(11 31)(13 17)(15 19)(33 41)(34 50)(35 43)(36 52)(37 45)(38 54)(39 47)(40 56)(42 58)(44 60)(46 62)(48 64)(49 57)(51 59)(53 61)(55 63)
(1 29)(2 30)(3 31)(4 32)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(33 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 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 47 9 39)(2 58 10 50)(3 61 11 53)(4 40 12 48)(5 43 13 35)(6 54 14 62)(7 57 15 49)(8 36 16 44)(17 59 25 51)(18 38 26 46)(19 41 27 33)(20 52 28 60)(21 55 29 63)(22 34 30 42)(23 37 31 45)(24 64 32 56)

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

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

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

Matrix representation of C23.13SD16 in GL4(𝔽17) generated by

1000
01600
00160
00121
,
16000
01600
00160
00016
,
16000
01600
0010
0001
,
6000
0500
0039
00814
,
0100
16000
00515
001212
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,16,12,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[6,0,0,0,0,5,0,0,0,0,3,8,0,0,9,14],[0,16,0,0,1,0,0,0,0,0,5,12,0,0,15,12] >;

C23.13SD16 in GAP, Magma, Sage, TeX

C_2^3._{13}{\rm SD}_{16}
% in TeX

G:=Group("C2^3.13SD16");
// GroupNames label

G:=SmallGroup(128,82);
// by ID

G=gap.SmallGroup(128,82);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,456,422,387,520,1690,248,2804,1411,172,4037,2028,124]);
// Polycyclic

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

Export

Subgroup lattice of C23.13SD16 in TeX
Character table of C23.13SD16 in TeX

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