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G = C22⋊C4.7D4  order 128 = 27

5th non-split extension by C22⋊C4 of D4 acting via D4/C2=C22

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

Aliases: (C2×C8).4D4, C22⋊C4.7D4, C4.9C42⋊5C2, C4.75C22≀C2, (C2×D4).114D4, (C2×Q8).105D4, C23.35(C2×D4), D8⋊C22.5C2, C23.24D4⋊3C2, (C22×C8).76C22, C4.101(C4.4D4), C22.71(C4⋊D4), C42.6C22⋊1C2, (C22×C4).732C23, C23.C23⋊11C2, C42⋊C2.63C22, C4.25(C22.D4), (C2×M4(2)).24C22, C2.26(C23.10D4), (C2×C4).264(C2×D4), (C2×C4).83(C4○D4), (C2×C4○D4).66C22, SmallGroup(128,785)

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

Derived series
C1 — C22×C4 — C22⋊C4.7D4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C23.24D4 — C22⋊C4.7D4
Lower central
C1 — C2 — C22×C4 — C22⋊C4.7D4
Upper central
C1 — C4 — C22×C4 — C22⋊C4.7D4
Jennings
C1 — C2 — C2 — C22×C4 — C22⋊C4.7D4

Generators and relations for C22⋊C4.7D4
 G = < a,b,c,d,e | a2=b2=c4=e2=1, d4=b, cac-1=dad-1=ab=ba, ae=ea, bc=cb, bd=db, be=eb, dcd-1=bc-1, ece=ac, ede=bd3 >

Subgroups: 296 in 132 conjugacy classes, 40 normal (16 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C23⋊C4, D4⋊C4, Q8⋊C4, C4⋊C8, C42⋊C2, C42⋊C2, C22×C8, C2×M4(2), C4○D8, C8⋊C22, C8.C22, C2×C4○D4, C4.9C42, C23.C23, C23.24D4, C42.6C22, D8⋊C22, C22⋊C4.7D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C22≀C2, C4⋊D4, C22.D4, C4.4D4, C23.10D4, C22⋊C4.7D4

Character table of C22⋊C4.7D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D8E8F
 size 11222881122288888888444488
ρ111111111111111111111111111    trivial
ρ2111111111111-11-11-1-1-1-1-1-1-1-111    linear of order 2
ρ311111-1-111111-1-1-1-1-11-11111111    linear of order 2
ρ411111-1-1111111-11-11-11-1-1-1-1-111    linear of order 2
ρ5111111-1111111-1-111-1-1-11111-1-1    linear of order 2
ρ6111111-111111-1-111-1111-1-1-1-1-1-1    linear of order 2
ρ711111-111111111-1-111-11-1-1-1-1-1-1    linear of order 2
ρ811111-1111111-111-1-1-11-11111-1-1    linear of order 2
ρ922-2-220-2-2-222-202000000000000    orthogonal lifted from D4
ρ1022-22-200222-2-2000000000000-22    orthogonal lifted from D4
ρ112222200-2-2-2-2-20000020-2000000    orthogonal lifted from D4
ρ12222-2-220-2-22-22000-20000000000    orthogonal lifted from D4
ρ132222200-2-2-2-2-200000-202000000    orthogonal lifted from D4
ρ1422-2-2202-2-222-20-2000000000000    orthogonal lifted from D4
ρ15222-2-2-20-2-22-2200020000000000    orthogonal lifted from D4
ρ1622-22-200222-2-20000000000002-2    orthogonal lifted from D4
ρ1722-22-200-2-2-22200000000-2i2i2i-2i00    complex lifted from C4○D4
ρ18222-2-20022-22-2002i000-2i0000000    complex lifted from C4○D4
ρ1922-2-220022-2-22-2i0002i000000000    complex lifted from C4○D4
ρ2022-22-200-2-2-222000000002i-2i-2i2i00    complex lifted from C4○D4
ρ2122-2-220022-2-222i000-2i000000000    complex lifted from C4○D4
ρ22222-2-20022-22-200-2i0002i0000000    complex lifted from C4○D4
ρ234-400000-4i4i000000000002ζ872ζ852ζ82ζ8300    complex faithful
ρ244-400000-4i4i000000000002ζ832ζ82ζ852ζ8700    complex faithful
ρ254-4000004i-4i000000000002ζ82ζ832ζ872ζ8500    complex faithful
ρ264-4000004i-4i000000000002ζ852ζ872ζ832ζ800    complex faithful

Smallest permutation representation of C22⋊C4.7D4
►On 32 points
Generators in S32
(1 25)(2 30)(3 27)(4 32)(5 29)(6 26)(7 31)(8 28)(9 22)(10 19)(11 24)(12 21)(13 18)(14 23)(15 20)(16 17)

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

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

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

Matrix representation of C22⋊C4.7D4 ►in GL4(𝔽17) generated by

013413
401313
00013
0040
,
16000
01600
00160
00016
,
9003
05713
771015
7101510
,
6628
8951
01589
1501111
,
6628
981216
1501111
01589
G:=sub<GL(4,GF(17))| [0,4,0,0,13,0,0,0,4,13,0,4,13,13,13,0],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[9,0,7,7,0,5,7,10,0,7,10,15,3,13,15,10],[6,8,0,15,6,9,15,0,2,5,8,11,8,1,9,11],[6,9,15,0,6,8,0,15,2,12,11,8,8,16,11,9] >;
 

C22⋊C4.7D4 in GAP, Magma, Sage, TeX 

C_2^2\rtimes C_4._7D_4
 
% in TeX
 
G:=Group("C2^2:C4.7D4");
 
// GroupNames label
 
G:=SmallGroup(128,785);
 
// by ID
 
G=gap.SmallGroup(128,785);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,58,2019,1018,248,2804,172,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^4=e^2=1,d^4=b,c*a*c^-1=d*a*d^-1=a*b=b*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=b*c^-1,e*c*e=a*c,e*d*e=b*d^3>;
 
// generators/relations
 

Export

Character table of C22⋊C4.7D4 in TeX

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