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G = C42.73D4order 128 = 27

55th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.73D4, C42.154C23, C4⋊Q8.18C4, C41D4.12C4, C42.95(C2×C4), (C22×C4).230D4, C42.6C439C2, C8⋊C4.89C22, C4.16(C4.D4), C23.60(C22⋊C4), (C2×C42).198C22, C42.C2213C2, C4.4D4.119C22, C2.36(C42⋊C22), C22.26C24.11C2, (C2×C4○D4).5C4, (C2×D4).26(C2×C4), (C2×Q8).26(C2×C4), (C2×C4).1182(C2×D4), C2.14(C2×C4.D4), (C2×C4).99(C22⋊C4), (C2×C4).148(C22×C4), (C22×C4).220(C2×C4), C22.212(C2×C22⋊C4), SmallGroup(128,268)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.73D4
C1C2C22C2×C4C42C2×C42C22.26C24 — C42.73D4
C1C22C2×C4 — C42.73D4
C1C22C2×C42 — C42.73D4
C1C22C22C42 — C42.73D4

Generators and relations for C42.73D4
 G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, bd=db, dcd-1=a2b-1c3 >

Subgroups: 276 in 119 conjugacy classes, 44 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×2], C4 [×7], C22, C22 [×9], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×10], Q8 [×2], C23, C23 [×2], C42 [×2], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×4], C8⋊C4 [×4], C22⋊C8 [×2], C4⋊C8 [×2], C2×C42, C4×D4 [×2], C4⋊D4 [×2], C4.4D4 [×2], C41D4, C4⋊Q8, C2×C4○D4 [×2], C42.C22 [×4], C42.6C4 [×2], C22.26C24, C42.73D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4.D4 [×2], C2×C22⋊C4, C2×C4.D4, C42⋊C22 [×2], C42.73D4

Character table of C42.73D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D8E8F8G8H
 size 11114882222224448888888888
ρ111111111111111111111111111    trivial
ρ211111-1-1111111111-1-1-1111-1-1-11    linear of order 2
ρ3111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ411111-1-1111111111-1-11-1-1-1111-1    linear of order 2
ρ51111-1-111-11-111-11-1-11-111-111-1-1    linear of order 2
ρ61111-11-11-11-111-11-11-1111-1-1-11-1    linear of order 2
ρ71111-1-111-11-111-11-1-111-1-11-1-111    linear of order 2
ρ81111-11-11-11-111-11-11-1-1-1-1111-11    linear of order 2
ρ91111-111-1-1-1-1-1-1111-1-1-ii-ii-iii-i    linear of order 4
ρ101111-1-1-1-1-1-1-1-1-111111ii-iii-i-i-i    linear of order 4
ρ111111-111-1-1-1-1-1-1111-1-1i-ii-ii-i-ii    linear of order 4
ρ121111-1-1-1-1-1-1-1-1-111111-i-ii-i-iiii    linear of order 4
ρ1311111-11-11-11-1-1-11-11-1ii-i-i-ii-ii    linear of order 4
ρ14111111-1-11-11-1-1-11-1-11-ii-i-ii-iii    linear of order 4
ρ1511111-11-11-11-1-1-11-11-1-i-iiii-ii-i    linear of order 4
ρ16111111-1-11-11-1-1-11-1-11i-iii-ii-i-i    linear of order 4
ρ1722222002-2-2-2-22-2-220000000000    orthogonal lifted from D4
ρ182222-20022-22-222-2-20000000000    orthogonal lifted from D4
ρ192222-200-22222-2-2-220000000000    orthogonal lifted from D4
ρ202222200-2-22-22-22-2-20000000000    orthogonal lifted from D4
ρ214-4-440000-404000000000000000    orthogonal lifted from C4.D4
ρ224-4-44000040-4000000000000000    orthogonal lifted from C4.D4
ρ234-44-4000-4i00004i0000000000000    complex lifted from C42⋊C22
ρ244-44-40004i0000-4i0000000000000    complex lifted from C42⋊C22
ρ2544-4-4000004i0-4i00000000000000    complex lifted from C42⋊C22
ρ2644-4-400000-4i04i00000000000000    complex lifted from C42⋊C22

Smallest permutation representation of C42.73D4
On 64 points
Generators in S64
(1 35 56 48)(2 41 49 36)(3 37 50 42)(4 43 51 38)(5 39 52 44)(6 45 53 40)(7 33 54 46)(8 47 55 34)(9 32 20 58)(10 59 21 25)(11 26 22 60)(12 61 23 27)(13 28 24 62)(14 63 17 29)(15 30 18 64)(16 57 19 31)
(1 27 52 57)(2 32 53 62)(3 29 54 59)(4 26 55 64)(5 31 56 61)(6 28 49 58)(7 25 50 63)(8 30 51 60)(9 45 24 36)(10 42 17 33)(11 47 18 38)(12 44 19 35)(13 41 20 40)(14 46 21 37)(15 43 22 34)(16 48 23 39)
(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 30 27 51 52 60 57 8)(2 54 32 59 53 3 62 29)(4 5 26 31 55 56 64 61)(6 50 28 63 49 7 58 25)(9 21 45 37 24 14 36 46)(10 40 42 13 17 41 33 20)(11 16 47 48 18 23 38 39)(12 43 44 22 19 34 35 15)

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

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

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

Matrix representation of C42.73D4 in GL8(𝔽17)

00100000
00010000
160000000
016000000
00000010
00000001
000016000
000001600
,
1615000000
01000000
0016150000
00010000
000013000
000001300
000000130
000000013
,
1312000000
44000000
00450000
0013130000
000011689
0000111188
000089611
00008866
,
1312000000
01000000
00450000
000160000
000011689
00006699
000089611
0000991111

G:=sub<GL(8,GF(17))| [0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[16,0,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13],[13,4,0,0,0,0,0,0,12,4,0,0,0,0,0,0,0,0,4,13,0,0,0,0,0,0,5,13,0,0,0,0,0,0,0,0,11,11,8,8,0,0,0,0,6,11,9,8,0,0,0,0,8,8,6,6,0,0,0,0,9,8,11,6],[13,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,5,16,0,0,0,0,0,0,0,0,11,6,8,9,0,0,0,0,6,6,9,9,0,0,0,0,8,9,6,11,0,0,0,0,9,9,11,11] >;

C42.73D4 in GAP, Magma, Sage, TeX

C_4^2._{73}D_4
% in TeX

G:=Group("C4^2.73D4");
// GroupNames label

G:=SmallGroup(128,268);
// by ID

G=gap.SmallGroup(128,268);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,352,1123,1018,248,1971,102]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2*b^2,d^2=b,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
// generators/relations

Export

Character table of C42.73D4 in TeX

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