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G = C23.12SD16order 128 = 27

2nd non-split extension by C23 of SD16 acting via SD16/C4=C22

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

Aliases: C23.12SD16, C4.20C4≀C2, (C2×D8).1C4, C2.D8.1C4, C22⋊C164C2, (C2×C8).302D4, C87D4.2C2, (C2×C4).102D8, C4.7(C23⋊C4), (C22×C4).187D4, C4.C4211C2, C2.4(D8.C4), C2.3(M5(2)⋊C2), (C22×C8).99C22, C22.57(D4⋊C4), C2.14(C22.SD16), (C2×C8).19(C2×C4), (C2×C4).219(C22⋊C4), SmallGroup(128,81)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — C23.12SD16
C1C2C4C2×C4C22×C4C22×C8C87D4 — C23.12SD16
C1C2C2×C4C2×C8 — C23.12SD16
C1C22C22×C4C22×C8 — C23.12SD16
C1C2C2C2C2C4C2×C4C22×C8 — C23.12SD16

Generators and relations for C23.12SD16
 G = < a,b,c,d,e | a2=b2=c2=e2=1, d8=c, eae=ab=ba, ac=ca, dad-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=abcd3 >

4C2
16C2
2C22
2C22
2C4
2C22
8C22
8C22
8C4
8C22
2C2×C4
2C8
2C2×C4
2C8
4C23
4D4
4D4
4C8
4C2×C4
4C8
8D4
8D4
2M4(2)
2C2×D4
2C4⋊C4
2M4(2)
4C16
4C22⋊C4
4M4(2)
4D8
4C2×C8
4C2×D4
4C2×C8
2C2×C16
2D4⋊C4
2C4⋊D4
2C2×M4(2)

Character table of C23.12SD16

 class 12A2B2C2D2E4A4B4C4D4E8A8B8C8D8E8F8G8H8I8J16A16B16C16D16E16F16G16H
 size 1111416222216222244888844444444
ρ111111111111111111111111111111    trivial
ρ211111-11111-11111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ311111111111111111-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ411111-11111-1111111-1-1-1-111111111    linear of order 2
ρ51111-1-1-111-11-1-1-1-111ii-i-ii-iii-i-i-ii    linear of order 4
ρ61111-1-1-111-11-1-1-1-111-i-iii-ii-i-iiii-i    linear of order 4
ρ71111-11-111-1-1-1-1-1-111-i-iiii-iii-i-i-ii    linear of order 4
ρ81111-11-111-1-1-1-1-1-111ii-i-i-ii-i-iiii-i    linear of order 4
ρ92222-20-222-202222-2-2000000000000    orthogonal lifted from D4
ρ1022222022220-2-2-2-2-2-2000000000000    orthogonal lifted from D4
ρ112222-202-2-2200000000000222-2-22-2-2    orthogonal lifted from D8
ρ122222-202-2-2200000000000-2-2-222-222    orthogonal lifted from D8
ρ132-22-2000-2200-2i2i2i-2i00-1+i1-i-1-i1+i00000000    complex lifted from C4≀C2
ρ142-22-2000-22002i-2i-2i2i00-1-i1+i-1+i1-i00000000    complex lifted from C4≀C2
ρ152-22-2000-2200-2i2i2i-2i001-i-1+i1+i-1-i00000000    complex lifted from C4≀C2
ρ162-22-2000-22002i-2i-2i2i001+i-1-i1-i-1+i00000000    complex lifted from C4≀C2
ρ17222220-2-2-2-200000000000--2-2--2-2--2-2--2-2    complex lifted from SD16
ρ18222220-2-2-2-200000000000-2--2-2--2-2--2-2--2    complex lifted from SD16
ρ192-2-2200-2i002i0-2--2-2--22-20000ζ1615165ζ161116ζ1613167ζ1611169ζ167165ζ169163ζ16151613ζ16316    complex lifted from D8.C4
ρ202-2-2200-2i002i0-2--2-2--22-20000ζ1613167ζ169163ζ1615165ζ16316ζ16151613ζ161116ζ167165ζ1611169    complex lifted from D8.C4
ρ212-2-2200-2i002i0--2-2--2-2-220000ζ16316ζ16151613ζ1611169ζ1615165ζ161116ζ167165ζ169163ζ1613167    complex lifted from D8.C4
ρ222-2-22002i00-2i0-2--2-2--2-220000ζ16151613ζ16316ζ167165ζ161116ζ1615165ζ1611169ζ1613167ζ169163    complex lifted from D8.C4
ρ232-2-22002i00-2i0-2--2-2--2-220000ζ167165ζ1611169ζ16151613ζ169163ζ1613167ζ16316ζ1615165ζ161116    complex lifted from D8.C4
ρ242-2-22002i00-2i0--2-2--2-22-20000ζ161116ζ1615165ζ169163ζ167165ζ1611169ζ1613167ζ16316ζ16151613    complex lifted from D8.C4
ρ252-2-22002i00-2i0--2-2--2-22-20000ζ169163ζ1613167ζ161116ζ16151613ζ16316ζ1615165ζ1611169ζ167165    complex lifted from D8.C4
ρ262-2-2200-2i002i0--2-2--2-2-220000ζ1611169ζ167165ζ16316ζ1613167ζ169163ζ16151613ζ161116ζ1615165    complex lifted from D8.C4
ρ274-44-40004-400000000000000000000    orthogonal lifted from C23⋊C4
ρ2844-4-40000000-22-22222200000000000000    orthogonal lifted from M5(2)⋊C2
ρ2944-4-400000002222-22-2200000000000000    orthogonal lifted from M5(2)⋊C2

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

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

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

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

Matrix representation of C23.12SD16 in GL4(𝔽17) generated by

13800
13400
001616
0001
,
16000
01600
00160
00016
,
16000
01600
0010
0001
,
51100
31600
00166
0021
,
1000
11600
0010
001516
G:=sub<GL(4,GF(17))| [13,13,0,0,8,4,0,0,0,0,16,0,0,0,16,1],[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],[5,3,0,0,11,16,0,0,0,0,16,2,0,0,6,1],[1,1,0,0,0,16,0,0,0,0,1,15,0,0,0,16] >;

C23.12SD16 in GAP, Magma, Sage, TeX

C_2^3._{12}{\rm SD}_{16}
% in TeX

G:=Group("C2^3.12SD16");
// GroupNames label

G:=SmallGroup(128,81);
// by ID

G=gap.SmallGroup(128,81);
# by ID

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

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

Export

Subgroup lattice of C23.12SD16 in TeX
Character table of C23.12SD16 in TeX

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