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G = C4⋊C4.98D4  order 128 = 27

53rd non-split extension by C4⋊C4 of D4 acting via D4/C2=C22

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

Aliases: C4⋊C4.98D4, (C2×C8).47D4, (C2×Q8).99D4, C4.69C22≀C2, C4.27(C4⋊D4), (C22×Q16).4C2, C4.71(C4.4D4), C4.C42.7C2, C2.19(D4.5D4), C23.277(C4○D4), (C22×C8).113C22, (C22×C4).728C23, C23.38D4.8C2, (C22×Q8).69C22, C22.235(C4⋊D4), C42⋊C2.60C22, C42.6C22.5C2, C2.20(C23.10D4), (C2×M4(2)).229C22, C22.56(C22.D4), (C2×C4).82(C4○D4), (C2×C4).1044(C2×D4), (C2×C4.10D4).11C2, SmallGroup(128,779)

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

Derived series
C1 — C22×C4 — C4⋊C4.98D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×M4(2) — C2×C4.10D4 — C4⋊C4.98D4
Lower central
C1 — C2 — C22×C4 — C4⋊C4.98D4
Upper central
C1 — C22 — C22×C4 — C4⋊C4.98D4
Jennings
C1 — C2 — C2 — C22×C4 — C4⋊C4.98D4

Generators and relations for C4⋊C4.98D4
 G = < a,b,c,d | a4=b4=1, c4=d2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=a2b-1, dbd-1=ab, dcd-1=a2c3 >

Subgroups: 248 in 128 conjugacy classes, 42 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), Q16, C22×C4, C22×C4, C2×Q8, C2×Q8, C4.10D4, Q8⋊C4, C4⋊C8, C42⋊C2, C22×C8, C2×M4(2), C2×M4(2), C2×Q16, C22×Q8, C4.C42, C2×C4.10D4, C23.38D4, C42.6C22, C22×Q16, C4⋊C4.98D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C22≀C2, C4⋊D4, C22.D4, C4.4D4, C23.10D4, D4.5D4, C4⋊C4.98D4

Character table of C4⋊C4.98D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F8G8H8I8J
 size 11112222228888884444888888
ρ111111111111111111111111111    trivial
ρ211111111111-111-11-1-1-1-11-1-1-11-1    linear of order 2
ρ31111111111-1-1-1-1-1-1111111-1-111    linear of order 2
ρ41111111111-11-1-11-1-1-1-1-11-1111-1    linear of order 2
ρ511111111111-1-11-1-1-1-1-1-1-1111-11    linear of order 2
ρ6111111111111-111-11111-1-1-1-1-1-1    linear of order 2
ρ71111111111-111-111-1-1-1-1-11-1-1-11    linear of order 2
ρ81111111111-1-11-1-111111-1-111-1-1    linear of order 2
ρ92-2-22-22-2-222200-2000000000000    orthogonal lifted from D4
ρ102-2-222-22-2-220200-200000000000    orthogonal lifted from D4
ρ112-2-22-22-2-222-2002000000000000    orthogonal lifted from D4
ρ122222-2-2-22-22000000-222-2000000    orthogonal lifted from D4
ρ132222-2-22-22-200-20020000000000    orthogonal lifted from D4
ρ142222-2-22-22-200200-20000000000    orthogonal lifted from D4
ρ152-2-222-22-2-220-200200000000000    orthogonal lifted from D4
ρ162222-2-2-22-220000002-2-22000000    orthogonal lifted from D4
ρ172-2-222-2-222-20000000000-2i0002i0    complex lifted from C4○D4
ρ182-2-22-2222-2-200000000000-2i0002i    complex lifted from C4○D4
ρ192-2-22-2222-2-2000000000002i000-2i    complex lifted from C4○D4
ρ20222222-2-2-2-2000000000000-2i2i00    complex lifted from C4○D4
ρ21222222-2-2-2-20000000000002i-2i00    complex lifted from C4○D4
ρ222-2-222-2-222-200000000002i000-2i0    complex lifted from C4○D4
ρ2344-4-40000000000002√200-2√2000000    symplectic lifted from D4.5D4, Schur index 2
ρ244-44-40000000000000-2√22√20000000    symplectic lifted from D4.5D4, Schur index 2
ρ254-44-400000000000002√2-2√20000000    symplectic lifted from D4.5D4, Schur index 2
ρ2644-4-4000000000000-2√2002√2000000    symplectic lifted from D4.5D4, Schur index 2

Smallest permutation representation of C4⋊C4.98D4
►On 64 points
Generators in S64
(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 55 53 51)(50 56 54 52)(57 59 61 63)(58 60 62 64)

(1 19 63 55)(2 52 64 24)(3 21 57 49)(4 54 58 18)(5 23 59 51)(6 56 60 20)(7 17 61 53)(8 50 62 22)(9 38 29 46)(10 43 30 35)(11 40 31 48)(12 45 32 37)(13 34 25 42)(14 47 26 39)(15 36 27 44)(16 41 28 33)

(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)

(1 38 5 34)(2 37 6 33)(3 36 7 40)(4 35 8 39)(9 53 13 49)(10 52 14 56)(11 51 15 55)(12 50 16 54)(17 25 21 29)(18 32 22 28)(19 31 23 27)(20 30 24 26)(41 64 45 60)(42 63 46 59)(43 62 47 58)(44 61 48 57)
 
G:=sub<Sym(64)| (1,3,5,7)(2,4,6,8)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,55,53,51)(50,56,54,52)(57,59,61,63)(58,60,62,64), (1,19,63,55)(2,52,64,24)(3,21,57,49)(4,54,58,18)(5,23,59,51)(6,56,60,20)(7,17,61,53)(8,50,62,22)(9,38,29,46)(10,43,30,35)(11,40,31,48)(12,45,32,37)(13,34,25,42)(14,47,26,39)(15,36,27,44)(16,41,28,33), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,38,5,34)(2,37,6,33)(3,36,7,40)(4,35,8,39)(9,53,13,49)(10,52,14,56)(11,51,15,55)(12,50,16,54)(17,25,21,29)(18,32,22,28)(19,31,23,27)(20,30,24,26)(41,64,45,60)(42,63,46,59)(43,62,47,58)(44,61,48,57)>;
 
G:=Group( (1,3,5,7)(2,4,6,8)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,55,53,51)(50,56,54,52)(57,59,61,63)(58,60,62,64), (1,19,63,55)(2,52,64,24)(3,21,57,49)(4,54,58,18)(5,23,59,51)(6,56,60,20)(7,17,61,53)(8,50,62,22)(9,38,29,46)(10,43,30,35)(11,40,31,48)(12,45,32,37)(13,34,25,42)(14,47,26,39)(15,36,27,44)(16,41,28,33), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,38,5,34)(2,37,6,33)(3,36,7,40)(4,35,8,39)(9,53,13,49)(10,52,14,56)(11,51,15,55)(12,50,16,54)(17,25,21,29)(18,32,22,28)(19,31,23,27)(20,30,24,26)(41,64,45,60)(42,63,46,59)(43,62,47,58)(44,61,48,57) );
 
G=PermutationGroup([[(1,3,5,7),(2,4,6,8),(9,15,13,11),(10,16,14,12),(17,23,21,19),(18,24,22,20),(25,31,29,27),(26,32,30,28),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,55,53,51),(50,56,54,52),(57,59,61,63),(58,60,62,64)], [(1,19,63,55),(2,52,64,24),(3,21,57,49),(4,54,58,18),(5,23,59,51),(6,56,60,20),(7,17,61,53),(8,50,62,22),(9,38,29,46),(10,43,30,35),(11,40,31,48),(12,45,32,37),(13,34,25,42),(14,47,26,39),(15,36,27,44),(16,41,28,33)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,38,5,34),(2,37,6,33),(3,36,7,40),(4,35,8,39),(9,53,13,49),(10,52,14,56),(11,51,15,55),(12,50,16,54),(17,25,21,29),(18,32,22,28),(19,31,23,27),(20,30,24,26),(41,64,45,60),(42,63,46,59),(43,62,47,58),(44,61,48,57)]])
 

Matrix representation of C4⋊C4.98D4 ►in GL6(𝔽17)

100000
010000
0016200
0016100
0000115
0000116
,
0130000
1300000
000006
0000146
0011600
0014000
,
16110000
610000
000600
0014600
000006
0000146
,
010000
100000
006200
0071100
000062
0000711

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,16,0,0,0,0,2,1,0,0,0,0,0,0,1,1,0,0,0,0,15,16],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,0,11,14,0,0,0,0,6,0,0,0,0,14,0,0,0,0,6,6,0,0],[16,6,0,0,0,0,11,1,0,0,0,0,0,0,0,14,0,0,0,0,6,6,0,0,0,0,0,0,0,14,0,0,0,0,6,6],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,6,7,0,0,0,0,2,11,0,0,0,0,0,0,6,7,0,0,0,0,2,11] >;
 

C4⋊C4.98D4 in GAP, Magma, Sage, TeX 

C_4\rtimes C_4._{98}D_4
 
% in TeX
 
G:=Group("C4:C4.98D4");
 
// GroupNames label
 
G:=SmallGroup(128,779);
 
// by ID
 
G=gap.SmallGroup(128,779);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,448,141,456,422,387,58,2019,1018,248,718,1027]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b^-1,d*b*d^-1=a*b,d*c*d^-1=a^2*c^3>;
 
// generators/relations
 

Export

Character table of C4⋊C4.98D4 in TeX

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