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G = C23.2D8  order 128 = 27

2nd non-split extension by C23 of D8 acting via D8/C2=D4

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

Aliases: C23.2D8, C4.6C4≀C2, C4.Q8⋊2C4, (C2×Q16)⋊2C4, (C2×C8).20D4, (C2×C4).2SD16, C4.2(C23⋊C4), C8.D4.1C2, C23.C8.4C2, (C22×C4).28D4, C4.9C42.1C2, C2.5(C22.SD16), (C2×M4(2)).2C22, C22.14(D4⋊C4), (C2×C8).2(C2×C4), (C2×C4).53(C22⋊C4), SmallGroup(128,72)

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

Derived series
C1 — C2×C8 — C23.2D8
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×M4(2) — C8.D4 — C23.2D8
Lower central
C1 — C2 — C2×C4 — C2×C8 — C23.2D8
Upper central
C1 — C2 — C2×C4 — C2×M4(2) — C23.2D8
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C2×C4 — C2×M4(2) — C23.2D8

Generators and relations for C23.2D8
 G = < a,b,c,d,e | a2=b2=c2=1, d8=c, e2=a, dad-1=ab=ba, ac=ca, ae=ea, dbd-1=ebe-1=bc=cb, cd=dc, ce=ec, ede-1=ad7 >

C1
C2
2C2
4C2
C22
C4
C4
2C22
2C4
4C22
8C4
8C4
8C4
8C4
C2×C4
C23
C2×C4
2C2×C4
2C8
2C8
2C2×C4
4C2×C4
4Q8
4C2×C4
4C2×C4
4Q8
4C2×C4
C2×C8
C22×C4
C2×C8
2C42
2C4⋊C4
2C2×Q8
2C42
4C16
4M4(2)
4C22⋊C4
4Q16
4C22⋊C4
4C4⋊C4
4C4⋊C4
C2×M4(2)
C4.Q8
C2×Q16
2M5(2)
2Q8⋊C4
2C22⋊Q8
2C42⋊C2
C4.9C42
C23.C8
C8.D4
C23.2D8

Character table of C23.2D8

 class 12A2B2C4A4B4C4D4E4F4G4H4I8A8B8C16A16B16C16D
 size 1124224888816164488888
ρ111111111111111111111    trivial
ρ21111111-1-1-1-111111-1-1-1-1    linear of order 2
ρ311111111111-1-1111-1-1-1-1    linear of order 2
ρ41111111-1-1-1-1-1-11111111    linear of order 2
ρ5111-111-1ii-i-i1-1-1-11i-i-ii    linear of order 4
ρ6111-111-1-i-iii1-1-1-11-iii-i    linear of order 4
ρ7111-111-1ii-i-i-11-1-11-iii-i    linear of order 4
ρ8111-111-1-i-iii-11-1-11i-i-ii    linear of order 4
ρ9222-222-200000022-20000    orthogonal lifted from D4
ρ102222222000000-2-2-20000    orthogonal lifted from D4
ρ112222-2-2-2000000000-√2-√2√2√2    orthogonal lifted from D8
ρ122222-2-2-2000000000√2√2-√2-√2    orthogonal lifted from D8
ρ1322-202-201+i-1-i1-i-1+i00-2i2i00000    complex lifted from C4≀C2
ρ1422-202-201-i-1+i1+i-1-i002i-2i00000    complex lifted from C4≀C2
ρ15222-2-2-22000000000-√-2√-2-√-2√-2    complex lifted from SD16
ρ16222-2-2-22000000000√-2-√-2√-2-√-2    complex lifted from SD16
ρ1722-202-20-1+i1-i-1-i1+i002i-2i00000    complex lifted from C4≀C2
ρ1822-202-20-1-i1+i-1+i1-i00-2i2i00000    complex lifted from C4≀C2
ρ1944-40-4400000000000000    orthogonal lifted from C23⋊C4
ρ208-8000000000000000000    symplectic faithful, Schur index 2

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

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

(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)

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

Matrix representation of C23.2D8 ►in GL8(𝔽17)

10000000
01000000
00100000
00010000
000016000
000001600
000000160
000000016
,
00010000
001600000
016000000
10000000
00000001
000000160
000001600
00001000
,
160000000
016000000
001600000
000160000
000016000
000001600
000000160
000000016
,
8512812995
128912812129
1289129558
9125912995
81212985128
58812128912
12995128912
81212991259
,
01000000
10000000
00010000
00100000
000000016
000000160
00000100
00001000

G:=sub<GL(8,GF(17))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[8,12,12,9,8,5,12,8,5,8,8,12,12,8,9,12,12,9,9,5,12,8,9,12,8,12,12,9,9,12,5,9,12,8,9,12,8,12,12,9,9,12,5,9,5,8,8,12,9,12,5,9,12,9,9,5,5,9,8,5,8,12,12,9],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0] >;
 

C23.2D8 in GAP, Magma, Sage, TeX 

C_2^3._2D_8
 
% in TeX
 
G:=Group("C2^3.2D8");
 
// GroupNames label
 
G:=SmallGroup(128,72);
 
// by ID
 
G=gap.SmallGroup(128,72);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,456,422,387,520,1690,521,1411,172,4037,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^8=c,e^2=a,d*a*d^-1=a*b=b*a,a*c=c*a,a*e=e*a,d*b*d^-1=e*b*e^-1=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e^-1=a*d^7>;
 
// generators/relations
 

Export

Subgroup lattice of C23.2D8 in TeX
Character table of C23.2D8 in TeX

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