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G = C23.49D8  order 128 = 27

20th non-split extension by C23 of D8 acting via D8/D4=C2

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

Aliases: C23.49D8, C22.8SD32, C16⋊4C4⋊9C2, (C2×C8).74D4, (C2×C4).42D8, C8⋊7D4.6C2, C2.D16⋊13C2, C22⋊C16⋊12C2, C8.70(C4○D4), C2.12(C2×SD32), (C2×C16).45C22, (C2×C8).533C23, (C2×D8).10C22, (C22×C4).353D4, C22.119(C2×D8), C2.17(C16⋊C22), C2.D8.18C22, C4.15(C8.C22), (C22×C8).130C22, C4.40(C22.D4), C2.13(C22.D8), (C2×C2.D8)⋊17C2, (C2×C4).801(C2×D4), SmallGroup(128,965)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C8 — C23.49D8
Chief series
C1 — C2 — C4 — C8 — C2×C8 — C2.D8 — C2×C2.D8 — C23.49D8
Lower central
C1 — C2 — C4 — C2×C8 — C23.49D8
Upper central
C1 — C22 — C22×C4 — C22×C8 — C23.49D8
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C23.49D8

Generators and relations for C23.49D8
 G = < a,b,c,d,e | a2=b2=c2=e2=1, d8=c, dad-1=eae=ab=ba, ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=bd7 >

Subgroups: 212 in 75 conjugacy classes, 32 normal (20 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, D4, C23, C23, C16, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C22×C4, C2×D4, D4⋊C4, C2.D8, C2.D8, C2.D8, C2×C16, C2×C4⋊C4, C4⋊D4, C22×C8, C2×D8, C22⋊C16, C2.D16, C16⋊4C4, C2×C2.D8, C8⋊7D4, C23.49D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C4○D4, SD32, C22.D4, C2×D8, C8.C22, C22.D8, C2×SD32, C16⋊C22, C23.49D8

Character table of C23.49D8

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 1111221622488881622224444444444
ρ111111111111111111111111111111    trivial
ρ21111-1-1111-1-11-11-11111-1-1111-1-11-1-1    linear of order 2
ρ31111-1-1-111-1-11-1111111-1-1-1-1-111-111    linear of order 2
ρ4111111-11111111-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ51111-1-1111-11-11-1-11111-1-1-1-1-111-111    linear of order 2
ρ61111111111-1-1-1-11111111-1-1-1-1-1-1-1-1    linear of order 2
ρ7111111-1111-1-1-1-1-111111111111111    linear of order 2
ρ81111-1-1-111-11-11-111111-1-1111-1-11-1-1    linear of order 2
ρ92222-2-2022-200000-2-2-2-22200000000    orthogonal lifted from D4
ρ10222222022200000-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ112222-2-20-2-2200000000000-√2√2-√2-√2√2√2√2-√2    orthogonal lifted from D8
ρ122222220-2-2-200000000000-√2√2-√2√2-√2√2-√2√2    orthogonal lifted from D8
ρ132222-2-20-2-2200000000000√2-√2√2√2-√2-√2-√2√2    orthogonal lifted from D8
ρ142222220-2-2-200000000000√2-√2√2-√2√2-√2√2-√2    orthogonal lifted from D8
ρ152-2-220002-200-2i02i02-2-220000000000    complex lifted from C4○D4
ρ162-2-220002-202i0-2i00-222-20000000000    complex lifted from C4○D4
ρ172-2-220002-20-2i02i00-222-20000000000    complex lifted from C4○D4
ρ182-2-220002-2002i0-2i02-2-220000000000    complex lifted from C4○D4
ρ192-22-2-22000000000√2-√2√2-√2-√2√2ζ1615+ζ169ζ165+ζ163ζ167+ζ16ζ165+ζ163ζ1615+ζ169ζ1613+ζ1611ζ167+ζ16ζ1613+ζ1611    complex lifted from SD32
ρ202-22-2-22000000000√2-√2√2-√2-√2√2ζ167+ζ16ζ1613+ζ1611ζ1615+ζ169ζ1613+ζ1611ζ167+ζ16ζ165+ζ163ζ1615+ζ169ζ165+ζ163    complex lifted from SD32
ρ212-22-2-22000000000-√2√2-√2√2√2-√2ζ165+ζ163ζ167+ζ16ζ1613+ζ1611ζ167+ζ16ζ165+ζ163ζ1615+ζ169ζ1613+ζ1611ζ1615+ζ169    complex lifted from SD32
ρ222-22-2-22000000000-√2√2-√2√2√2-√2ζ1613+ζ1611ζ1615+ζ169ζ165+ζ163ζ1615+ζ169ζ1613+ζ1611ζ167+ζ16ζ165+ζ163ζ167+ζ16    complex lifted from SD32
ρ232-22-22-2000000000√2-√2√2-√2√2-√2ζ167+ζ16ζ1613+ζ1611ζ1615+ζ169ζ165+ζ163ζ1615+ζ169ζ165+ζ163ζ167+ζ16ζ1613+ζ1611    complex lifted from SD32
ρ242-22-22-2000000000√2-√2√2-√2√2-√2ζ1615+ζ169ζ165+ζ163ζ167+ζ16ζ1613+ζ1611ζ167+ζ16ζ1613+ζ1611ζ1615+ζ169ζ165+ζ163    complex lifted from SD32
ρ252-22-22-2000000000-√2√2-√2√2-√2√2ζ1613+ζ1611ζ1615+ζ169ζ165+ζ163ζ167+ζ16ζ165+ζ163ζ167+ζ16ζ1613+ζ1611ζ1615+ζ169    complex lifted from SD32
ρ262-22-22-2000000000-√2√2-√2√2-√2√2ζ165+ζ163ζ167+ζ16ζ1613+ζ1611ζ1615+ζ169ζ1613+ζ1611ζ1615+ζ169ζ165+ζ163ζ167+ζ16    complex lifted from SD32
ρ2744-4-400000000000-2√2-2√22√22√20000000000    orthogonal lifted from C16⋊C22
ρ2844-4-4000000000002√22√2-2√2-2√20000000000    orthogonal lifted from C16⋊C22
ρ294-4-44000-4400000000000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C23.49D8
►On 64 points
Generators in S64
(1 23)(2 60)(3 25)(4 62)(5 27)(6 64)(7 29)(8 50)(9 31)(10 52)(11 17)(12 54)(13 19)(14 56)(15 21)(16 58)(18 37)(20 39)(22 41)(24 43)(26 45)(28 47)(30 33)(32 35)(34 51)(36 53)(38 55)(40 57)(42 59)(44 61)(46 63)(48 49)

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

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

Matrix representation of C23.49D8 ►in GL4(𝔽17) generated by

1000
0100
00139
0044
,
1000
0100
00160
00016
,
16000
01600
0010
0001
,
16700
101600
0012
00016
,
0100
1000
0010
001616
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,13,4,0,0,9,4],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[16,10,0,0,7,16,0,0,0,0,1,0,0,0,2,16],[0,1,0,0,1,0,0,0,0,0,1,16,0,0,0,16] >;
 

C23.49D8 in GAP, Magma, Sage, TeX 

C_2^3._{49}D_8
 
% in TeX
 
G:=Group("C2^3.49D8");
 
// GroupNames label
 
G:=SmallGroup(128,965);
 
// by ID
 
G=gap.SmallGroup(128,965);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,422,394,1684,438,242,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=e^2=1,d^8=c,d*a*d^-1=e*a*e=a*b=b*a,a*c=c*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b*d^7>;
 
// generators/relations
 

Export

Character table of C23.49D8 in TeX

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