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G = C4⋊C4.18D4  order 128 = 27

18th non-split extension by C4⋊C4 of D4 acting faithfully

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

Aliases: C4⋊C4.18D4, (C2×D4).20D4, (C2×Q8).20D4, (C22×C4).18D4, C23.532(C2×D4), C22⋊C8.8C22, C2.9(D4.8D4), C22.26(C4○D8), C23.48D4⋊2C2, C4⋊D4.14C22, (C22×C4).21C23, C22.SD16.3C2, C22⋊Q8.14C22, C22.142C22≀C2, C2.10(D4.7D4), C23.31D4⋊15C2, C23.46D4.4C2, C23.83C23⋊1C2, C2.12(C23.7D4), C22.16(C8.C22), C22.M4(2)⋊8C2, C2.C42.27C22, C22.33C24.3C2, (C2×C4).210(C2×D4), (C2×C4⋊C4).25C22, SmallGroup(128,347)

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

Derived series
C1 — C22×C4 — C4⋊C4.18D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C4⋊C4 — C22.33C24 — C4⋊C4.18D4
Lower central
C1 — C22 — C22×C4 — C4⋊C4.18D4
Upper central
C1 — C22 — C22×C4 — C4⋊C4.18D4
Jennings
C1 — C2 — C22 — C22×C4 — C4⋊C4.18D4

Generators and relations for C4⋊C4.18D4
 G = < a,b,c,d | a4=b4=1, c4=a2, d2=b2, bab-1=dad-1=a-1, cac-1=a-1b2, cbc-1=a-1b, dbd-1=ab, dcd-1=a2b2c3 >

Subgroups: 244 in 104 conjugacy classes, 32 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2.C42, C2.C42, C22⋊C8, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C22.M4(2), C22.SD16, C23.31D4, C23.83C23, C23.46D4, C23.48D4, C22.33C24, C4⋊C4.18D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C22≀C2, C4○D8, C8.C22, D4.7D4, D4.8D4, C23.7D4, C4⋊C4.18D4

Character table of C4⋊C4.18D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D
 size 11112284444888888888888
ρ111111111111111111111111    trivial
ρ21111111-111-1-1-1-1-1111-1-11-11    linear of order 2
ρ3111111-111111-11-11-111-1-1-1-1    linear of order 2
ρ4111111-1-111-1-11-111-11-11-11-1    linear of order 2
ρ51111111-111-11-1-1-1-11-111-11-1    linear of order 2
ρ611111111111-1111-11-1-1-1-1-1-1    linear of order 2
ρ7111111-1-111-111-11-1-1-11-11-11    linear of order 2
ρ8111111-11111-1-11-1-1-1-1-11111    linear of order 2
ρ92222-2-2-202-20000002000000    orthogonal lifted from D4
ρ102222-2-200-2200-20200000000    orthogonal lifted from D4
ρ112222220-2-2-2-2002000000000    orthogonal lifted from D4
ρ122222-2-2202-2000000-2000000    orthogonal lifted from D4
ρ1322222202-2-2200-2000000000    orthogonal lifted from D4
ρ142222-2-200-220020-200000000    orthogonal lifted from D4
ρ1522-2-2-220-2i002i00000000-√2√-2√2-√-2    complex lifted from C4○D8
ρ1622-2-2-2202i00-2i00000000-√2-√-2√2√-2    complex lifted from C4○D8
ρ1722-2-2-2202i00-2i00000000√2√-2-√2-√-2    complex lifted from C4○D8
ρ1822-2-2-220-2i002i00000000√2-√-2-√2√-2    complex lifted from C4○D8
ρ1944-4-44-400000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ204-4-4400000002i000000-2i0000    complex lifted from D4.8D4
ρ214-44-4000000000002i0-2i00000    complex lifted from C23.7D4
ρ224-44-400000000000-2i02i00000    complex lifted from C23.7D4
ρ234-4-440000000-2i0000002i0000    complex lifted from D4.8D4

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

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

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

Matrix representation of C4⋊C4.18D4 ►in GL6(𝔽17)

120000
16160000
000100
001000
001112
001616016
,
0100000
500000
001313139
0000130
0001300
000444
,
070000
5100000
001313139
000040
004000
000004
,
1300000
440000
000400
004000
000040
000004

G:=sub<GL(6,GF(17))| [1,16,0,0,0,0,2,16,0,0,0,0,0,0,0,1,1,16,0,0,1,0,1,16,0,0,0,0,1,0,0,0,0,0,2,16],[0,5,0,0,0,0,10,0,0,0,0,0,0,0,13,0,0,0,0,0,13,0,13,4,0,0,13,13,0,4,0,0,9,0,0,4],[0,5,0,0,0,0,7,10,0,0,0,0,0,0,13,0,4,0,0,0,13,0,0,0,0,0,13,4,0,0,0,0,9,0,0,4],[13,4,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4] >;
 

C4⋊C4.18D4 in GAP, Magma, Sage, TeX 

C_4\rtimes C_4._{18}D_4
 
% in TeX
 
G:=Group("C4:C4.18D4");
 
// GroupNames label
 
G:=SmallGroup(128,347);
 
// by ID
 
G=gap.SmallGroup(128,347);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,672,141,422,184,1123,570,521,136,1411]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2,d^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=a^-1*b,d*b*d^-1=a*b,d*c*d^-1=a^2*b^2*c^3>;
 
// generators/relations
 

Export

Character table of C4⋊C4.18D4 in TeX

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