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G = C4⋊C4.19D4  order 128 = 27

19th non-split extension by C4⋊C4 of D4 acting faithfully

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

Aliases: C4⋊C4.19D4, (C2×D4).21D4, (C2×C4).8SD16, (C22×C4).54D4, C23.533(C2×D4), C2.9(C22⋊SD16), C22.SD16⋊16C2, C4⋊D4.15C22, (C22×C4).22C23, C22.34(C2×SD16), C2.10(D4.8D4), C22.143C22≀C2, C23.46D4⋊26C2, C22⋊C8.118C22, C22.45(C8⋊C22), C23.83C23⋊2C2, C2.13(C23.7D4), C22.M4(2)⋊11C2, C22.31C24.3C2, C2.C42.28C22, (C2×C4).211(C2×D4), (C2×C4⋊C4).26C22, SmallGroup(128,348)

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

Derived series
C1 — C22×C4 — C4⋊C4.19D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C4⋊C4 — C22.31C24 — C4⋊C4.19D4
Lower central
C1 — C22 — C22×C4 — C4⋊C4.19D4
Upper central
C1 — C22 — C22×C4 — C4⋊C4.19D4
Jennings
C1 — C2 — C22 — C22×C4 — C4⋊C4.19D4

Generators and relations for C4⋊C4.19D4
 G = < a,b,c,d | a4=b4=c4=1, d2=a2b2, bab-1=dad-1=a-1, cac-1=a-1b2, cbc-1=dbd-1=a-1b-1, dcd-1=a2c-1 >

Subgroups: 316 in 123 conjugacy classes, 34 normal (18 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C2.C42, C2.C42, C22⋊C8, D4⋊C4, C4.Q8, C2×C4⋊C4, C4⋊D4, C4⋊D4, C22⋊Q8, C2×C4○D4, C22.M4(2), C22.SD16, C23.83C23, C23.46D4, C22.31C24, C4⋊C4.19D4
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C22≀C2, C2×SD16, C8⋊C22, C22⋊SD16, D4.8D4, C23.7D4, C4⋊C4.19D4

Character table of C4⋊C4.19D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D
 size 11112288444488888888888
ρ111111111111111111111111    trivial
ρ21111111-11-1-11-11-11-11-1-11-11    linear of order 2
ρ3111111111111-111-11-1-1-1-1-1-1    linear of order 2
ρ41111111-11-1-1111-1-1-1-111-11-1    linear of order 2
ρ5111111-1-111111-1-11111-1-1-1-1    linear of order 2
ρ6111111-111-1-11-1-111-11-11-11-1    linear of order 2
ρ7111111-1-11111-1-1-1-11-1-11111    linear of order 2
ρ8111111-111-1-111-11-1-1-11-11-11    linear of order 2
ρ92222-2-2-20-200202000000000    orthogonal lifted from D4
ρ102222-2-202200-200-200000000    orthogonal lifted from D4
ρ1122222200-2-2-2-200002000000    orthogonal lifted from D4
ρ122222-2-20-2200-200200000000    orthogonal lifted from D4
ρ1322222200-222-20000-2000000    orthogonal lifted from D4
ρ142222-2-220-20020-2000000000    orthogonal lifted from D4
ρ1522-2-2-22000-2200000000-√-2√-2√-2-√-2    complex lifted from SD16
ρ1622-2-2-220002-200000000-√-2-√-2√-2√-2    complex lifted from SD16
ρ1722-2-2-22000-2200000000√-2-√-2-√-2√-2    complex lifted from SD16
ρ1822-2-2-220002-200000000√-2√-2-√-2-√-2    complex lifted from SD16
ρ1944-4-44-400000000000000000    orthogonal lifted from C8⋊C22
ρ204-4-4400000000-2i000002i0000    complex lifted from D4.8D4
ρ214-4-44000000002i00000-2i0000    complex lifted from D4.8D4
ρ224-44-4000000000002i0-2i00000    complex lifted from C23.7D4
ρ234-44-400000000000-2i02i00000    complex lifted from C23.7D4

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

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

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

Matrix representation of C4⋊C4.19D4 ►in GL6(𝔽17)

16100000
1010000
0013000
000400
000040
0000013
,
16100000
010000
0000160
0000016
001000
000100
,
0160000
100000
001000
0001600
000040
000004
,
0160000
100000
000100
001000
0000013
000040

G:=sub<GL(6,GF(17))| [16,10,0,0,0,0,10,1,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,13],[16,0,0,0,0,0,10,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,16,0,0,0,0,0,0,16,0,0],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,13,0] >;
 

C4⋊C4.19D4 in GAP, Magma, Sage, TeX 

C_4\rtimes C_4._{19}D_4
 
% in TeX
 
G:=Group("C4:C4.19D4");
 
// GroupNames label
 
G:=SmallGroup(128,348);
 
// by ID
 
G=gap.SmallGroup(128,348);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,672,141,422,352,1123,570,521,136,1411]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2*b^2,b*a*b^-1=d*a*d^-1=a^-1,c*a*c^-1=a^-1*b^2,c*b*c^-1=d*b*d^-1=a^-1*b^-1,d*c*d^-1=a^2*c^-1>;
 
// generators/relations
 

Export

Character table of C4⋊C4.19D4 in TeX

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