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G = C23.50D8order 128 = 27

21st non-split extension by C23 of D8 acting via D8/D4=C2

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

Aliases: C23.50D8, C22.6SD32, (C2×C8).75D4, (C2×C4).43D8, C164C411C2, C8.72(C4○D4), C22⋊C16.8C2, C2.13(C2×SD32), C2.Q3212C2, (C2×C8).535C23, (C2×C16).47C22, C8.18D4.5C2, C22.121(C2×D8), (C22×C4).355D4, C2.D8.20C22, C2.18(Q32⋊C2), C4.17(C8.C22), (C2×Q16).10C22, (C22×C8).131C22, C4.42(C22.D4), C2.15(C22.D8), (C2×C4).803(C2×D4), (C2×C2.D8).25C2, SmallGroup(128,967)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — C23.50D8
C1C2C4C8C2×C8C2.D8C2×C2.D8 — C23.50D8
C1C2C4C2×C8 — C23.50D8
C1C22C22×C4C22×C8 — C23.50D8
C1C2C2C2C2C4C4C2×C8 — C23.50D8

Generators and relations for C23.50D8
 G = < a,b,c,d,e | a2=b2=c2=1, d8=e2=c, dad-1=eae-1=ab=ba, ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd7 >

Subgroups: 164 in 70 conjugacy classes, 32 normal (20 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C8 [×2], C8, C2×C4 [×2], C2×C4 [×8], Q8 [×2], C23, C16 [×2], C22⋊C4, C4⋊C4 [×5], C2×C8 [×2], C2×C8 [×2], Q16 [×2], C22×C4, C22×C4, C2×Q8, Q8⋊C4, C2.D8, C2.D8 [×2], C2.D8, C2×C16 [×2], C2×C4⋊C4, C22⋊Q8, C22×C8, C2×Q16, C22⋊C16, C2.Q32 [×2], C164C4 [×2], C2×C2.D8, C8.18D4, C23.50D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D8 [×2], C2×D4, C4○D4 [×2], SD32 [×2], C22.D4, C2×D8, C8.C22, C22.D8, C2×SD32, Q32⋊C2, C23.50D8

Character table of C23.50D8

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 1111222248888161622224444444444
ρ111111111111111111111111111111    trivial
ρ21111111111111-1-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31111-1-111-111-1-1-111111-1-11-1-1-1111-1    linear of order 2
ρ41111-1-111-111-1-11-11111-1-1-1111-1-1-11    linear of order 2
ρ5111111111-1-1-1-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ6111111111-1-1-1-1-1-111111111111111    linear of order 2
ρ71111-1-111-1-1-111-111111-1-1-1111-1-1-11    linear of order 2
ρ81111-1-111-1-1-1111-11111-1-11-1-1-1111-1    linear of order 2
ρ92222-2-222-2000000-2-2-2-22200000000    orthogonal lifted from D4
ρ10222222222000000-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ11222222-2-2-200000000000022-22-22-2-2    orthogonal lifted from D8
ρ12222222-2-2-2000000000000-2-22-22-222    orthogonal lifted from D8
ρ132222-2-2-2-22000000000000-22-222-22-2    orthogonal lifted from D8
ρ142222-2-2-2-220000000000002-22-2-22-22    orthogonal lifted from D8
ρ152-2-22002-2000-2i2i00-22-220000000000    complex lifted from C4○D4
ρ162-2-22002-202i-2i00002-22-20000000000    complex lifted from C4○D4
ρ172-2-22002-20-2i2i00002-22-20000000000    complex lifted from C4○D4
ρ182-2-22002-20002i-2i00-22-220000000000    complex lifted from C4○D4
ρ192-22-22-200000000022-2-2-22ζ16131611ζ165163ζ16716ζ16131611ζ16716ζ165163ζ1615169ζ1615169    complex lifted from SD32
ρ202-22-22-200000000022-2-2-22ζ165163ζ16131611ζ1615169ζ165163ζ1615169ζ16131611ζ16716ζ16716    complex lifted from SD32
ρ212-22-2-2200000000022-2-22-2ζ16131611ζ16131611ζ1615169ζ165163ζ16716ζ165163ζ1615169ζ16716    complex lifted from SD32
ρ222-22-2-2200000000022-2-22-2ζ165163ζ165163ζ16716ζ16131611ζ1615169ζ16131611ζ16716ζ1615169    complex lifted from SD32
ρ232-22-22-2000000000-2-2222-2ζ1615169ζ16716ζ16131611ζ1615169ζ16131611ζ16716ζ165163ζ165163    complex lifted from SD32
ρ242-22-22-2000000000-2-2222-2ζ16716ζ1615169ζ165163ζ16716ζ165163ζ1615169ζ16131611ζ16131611    complex lifted from SD32
ρ252-22-2-22000000000-2-222-22ζ16716ζ16716ζ16131611ζ1615169ζ165163ζ1615169ζ16131611ζ165163    complex lifted from SD32
ρ262-22-2-22000000000-2-222-22ζ1615169ζ1615169ζ165163ζ16716ζ16131611ζ16716ζ165163ζ16131611    complex lifted from SD32
ρ274-4-4400-44000000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2844-4-40000000000022-22-22220000000000    symplectic lifted from Q32⋊C2, Schur index 2
ρ2944-4-400000000000-222222-220000000000    symplectic lifted from Q32⋊C2, Schur index 2

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

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

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

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

Matrix representation of C23.50D8 in GL4(𝔽17) generated by

1000
0100
0010
00116
,
1000
0100
00160
00016
,
16000
01600
0010
0001
,
6000
01400
0049
00413
,
01400
6000
00115
00016
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,16],[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],[6,0,0,0,0,14,0,0,0,0,4,4,0,0,9,13],[0,6,0,0,14,0,0,0,0,0,1,0,0,0,15,16] >;

C23.50D8 in GAP, Magma, Sage, TeX

C_2^3._{50}D_8
% in TeX

G:=Group("C2^3.50D8");
// GroupNames label

G:=SmallGroup(128,967);
// by ID

G=gap.SmallGroup(128,967);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,422,58,1684,438,242,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^8=e^2=c,d*a*d^-1=e*a*e^-1=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^-1=b*d^7>;
// generators/relations

Export

Character table of C23.50D8 in TeX

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