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G = C23.5D8order 128 = 27

5th non-split extension by C23 of D8 acting via D8/C2=D4

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

Aliases: C23.5D8, C24.13D4, C23⋊C84C2, C4⋊C4.10D4, (C2×D4).15D4, C22⋊C82C22, (C22×C4).48D4, C22.15(C2×D8), C2.8(C22⋊D8), C233D4.2C2, C22.D81C2, C23.524(C2×D4), C22.SD167C2, C4⋊D4.9C22, C2.9(D4.9D4), C23.11D41C2, (C22×C4).13C23, C22.134C22≀C2, C22.42(C8⋊C22), C2.C425C22, C2.4(C23.7D4), (C2×C4⋊C4)⋊1C22, (C2×C4).202(C2×D4), (C2×C22⋊C4).96C22, SmallGroup(128,339)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — C23.5D8
C1C2C22C23C22×C4C2×C22⋊C4C233D4 — C23.5D8
C1C22C22×C4 — C23.5D8
C1C22C22×C4 — C23.5D8
C1C2C22C22×C4 — C23.5D8

Generators and relations for C23.5D8
 G = < a,b,c,d,e | a2=b2=c2=d8=1, e2=c, dad-1=eae-1=ab=ba, ac=ca, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=cd-1 >

Subgroups: 412 in 140 conjugacy classes, 34 normal (18 characteristic)
C1, C2 [×3], C2 [×6], C4 [×7], C22 [×3], C22 [×19], C8 [×2], C2×C4 [×2], C2×C4 [×13], D4 [×11], C23, C23 [×2], C23 [×10], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], C22×C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×9], C24, C24, C2.C42, C2.C42, C22⋊C8 [×2], D4⋊C4 [×2], C2.D8 [×2], C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C22≀C2 [×2], C4⋊D4 [×2], C4⋊D4, C22.D4 [×2], C22×D4, C23⋊C8, C22.SD16 [×2], C23.11D4, C22.D8 [×2], C233D4, C23.5D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×2], C2×D4 [×3], C22≀C2, C2×D8, C8⋊C22, C22⋊D8, D4.9D4, C23.7D4, C23.5D8

Character table of C23.5D8

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I8A8B8C8D
 size 11112244884488888888888
ρ111111111111111111111111    trivial
ρ211111111-1-11111-11-111-1-1-1-1    linear of order 2
ρ311111111-1-111-11-1-1-1-1-11111    linear of order 2
ρ4111111111111-111-11-1-1-1-1-1-1    linear of order 2
ρ5111111-1-11-111-1-1-1111-11-11-1    linear of order 2
ρ6111111-1-1-1111-1-111-11-1-11-11    linear of order 2
ρ7111111-1-1-11111-11-1-1-111-11-1    linear of order 2
ρ8111111-1-11-1111-1-1-11-11-11-11    linear of order 2
ρ92222-2-200022-200-200000000    orthogonal lifted from D4
ρ102222-2-200-20-2200002000000    orthogonal lifted from D4
ρ11222222-2-200-2-202000000000    orthogonal lifted from D4
ρ122222-2-2000-22-200200000000    orthogonal lifted from D4
ρ132222-2-20020-220000-2000000    orthogonal lifted from D4
ρ142222222200-2-20-2000000000    orthogonal lifted from D4
ρ1522-2-2-22-22000000000002-2-22    orthogonal lifted from D8
ρ1622-2-2-222-20000000000022-2-2    orthogonal lifted from D8
ρ1722-2-2-22-2200000000000-222-2    orthogonal lifted from D8
ρ1822-2-2-222-200000000000-2-222    orthogonal lifted from D8
ρ1944-4-44-400000000000000000    orthogonal lifted from C8⋊C22
ρ204-4-4400000000-2i000002i0000    complex lifted from D4.9D4
ρ214-4-44000000002i00000-2i0000    complex lifted from D4.9D4
ρ224-44-4000000000002i0-2i00000    complex lifted from C23.7D4
ρ234-44-400000000000-2i02i00000    complex lifted from C23.7D4

Smallest permutation representation of C23.5D8
On 32 points
Generators in S32
(2 16)(3 20)(4 30)(6 12)(7 24)(8 26)(9 29)(10 21)(13 25)(14 17)(19 28)(23 32)
(1 15)(2 28)(3 9)(4 30)(5 11)(6 32)(7 13)(8 26)(10 21)(12 23)(14 17)(16 19)(18 27)(20 29)(22 31)(24 25)
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 17)(9 29)(10 30)(11 31)(12 32)(13 25)(14 26)(15 27)(16 28)
(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)
(1 17 18 8)(2 7 19 24)(3 23 20 6)(4 5 21 22)(9 12 29 32)(10 31 30 11)(13 16 25 28)(14 27 26 15)

G:=sub<Sym(32)| (2,16)(3,20)(4,30)(6,12)(7,24)(8,26)(9,29)(10,21)(13,25)(14,17)(19,28)(23,32), (1,15)(2,28)(3,9)(4,30)(5,11)(6,32)(7,13)(8,26)(10,21)(12,23)(14,17)(16,19)(18,27)(20,29)(22,31)(24,25), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,17)(9,29)(10,30)(11,31)(12,32)(13,25)(14,26)(15,27)(16,28), (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), (1,17,18,8)(2,7,19,24)(3,23,20,6)(4,5,21,22)(9,12,29,32)(10,31,30,11)(13,16,25,28)(14,27,26,15)>;

G:=Group( (2,16)(3,20)(4,30)(6,12)(7,24)(8,26)(9,29)(10,21)(13,25)(14,17)(19,28)(23,32), (1,15)(2,28)(3,9)(4,30)(5,11)(6,32)(7,13)(8,26)(10,21)(12,23)(14,17)(16,19)(18,27)(20,29)(22,31)(24,25), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,17)(9,29)(10,30)(11,31)(12,32)(13,25)(14,26)(15,27)(16,28), (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), (1,17,18,8)(2,7,19,24)(3,23,20,6)(4,5,21,22)(9,12,29,32)(10,31,30,11)(13,16,25,28)(14,27,26,15) );

G=PermutationGroup([(2,16),(3,20),(4,30),(6,12),(7,24),(8,26),(9,29),(10,21),(13,25),(14,17),(19,28),(23,32)], [(1,15),(2,28),(3,9),(4,30),(5,11),(6,32),(7,13),(8,26),(10,21),(12,23),(14,17),(16,19),(18,27),(20,29),(22,31),(24,25)], [(1,18),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,17),(9,29),(10,30),(11,31),(12,32),(13,25),(14,26),(15,27),(16,28)], [(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)], [(1,17,18,8),(2,7,19,24),(3,23,20,6),(4,5,21,22),(9,12,29,32),(10,31,30,11),(13,16,25,28),(14,27,26,15)])

Matrix representation of C23.5D8 in GL6(𝔽17)

1600000
0160000
001000
000100
00160160
00160016
,
100000
010000
0011500
0001600
000101
000110
,
100000
010000
0016000
0001600
0000160
0000016
,
1430000
14140000
00413413
0041300
000004
00130130
,
1430000
330000
00413413
0000413
00134130
00130130

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,16,16,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,15,16,1,1,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[14,14,0,0,0,0,3,14,0,0,0,0,0,0,4,4,0,13,0,0,13,13,0,0,0,0,4,0,0,13,0,0,13,0,4,0],[14,3,0,0,0,0,3,3,0,0,0,0,0,0,4,0,13,13,0,0,13,0,4,0,0,0,4,4,13,13,0,0,13,13,0,0] >;

C23.5D8 in GAP, Magma, Sage, TeX

C_2^3._5D_8
% in TeX

G:=Group("C2^3.5D8");
// GroupNames label

G:=SmallGroup(128,339);
// by ID

G=gap.SmallGroup(128,339);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,422,1123,570,521,136,1411]);
// Polycyclic

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

Export

Character table of C23.5D8 in TeX

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