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G = C42.74D4order 128 = 27

56th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.74D4, C42.155C23, C4⋊Q8.19C4, C42.96(C2×C4), C42⋊C2.5C4, (C22×C4).231D4, C8⋊C4.90C22, C4.16(C4.10D4), C23.61(C22⋊C4), C42.6C4.22C2, (C2×C42).199C22, C42.2C2212C2, C42.C2.100C22, C2.37(C42⋊C22), C23.37C23.11C2, C4⋊C4.31(C2×C4), (C2×C4).1183(C2×D4), (C22×C4).221(C2×C4), (C2×C4).149(C22×C4), C2.14(C2×C4.10D4), (C2×C4).100(C22⋊C4), C22.213(C2×C22⋊C4), SmallGroup(128,269)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.74D4
C1C2C22C2×C4C42C2×C42C23.37C23 — C42.74D4
C1C22C2×C4 — C42.74D4
C1C22C2×C42 — C42.74D4
C1C22C22C42 — C42.74D4

Generators and relations for C42.74D4
 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=a2bc3 >

Subgroups: 180 in 97 conjugacy classes, 44 normal (12 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×9], C22, C22 [×3], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×8], Q8 [×4], C23, C42 [×2], C42 [×2], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×4], C22×C4, C22×C4 [×2], C2×Q8 [×2], C8⋊C4 [×4], C22⋊C8 [×2], C4⋊C8 [×2], C2×C42, C42⋊C2 [×2], C4×Q8 [×2], C22⋊Q8 [×2], C42.C2 [×2], C4⋊Q8 [×2], C42.2C22 [×4], C42.6C4 [×2], C23.37C23, C42.74D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4.10D4 [×2], C2×C22⋊C4, C2×C4.10D4, C42⋊C22 [×2], C42.74D4

Character table of C42.74D4

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D8E8F8G8H
 size 11114222222444888888888888
ρ111111111111111111111111111    trivial
ρ211111111111111-1-1-1-1-1-11-1-1111    linear of order 2
ρ311111111111111-1-1-1-111-111-1-1-1    linear of order 2
ρ4111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ51111-111-1-111-1-11-111-1-111-11-1-11    linear of order 2
ρ61111-111-1-111-1-111-1-111-111-1-1-11    linear of order 2
ρ71111-111-1-111-1-111-1-11-11-1-1111-1    linear of order 2
ρ81111-111-1-111-1-11-111-11-1-11-111-1    linear of order 2
ρ911111-1-111-1-1-1-1111-1-1ii-i-i-i-iii    linear of order 4
ρ1011111-1-111-1-1-1-11-1-111-i-i-iii-iii    linear of order 4
ρ1111111-1-111-1-1-1-1111-1-1-i-iiiii-i-i    linear of order 4
ρ1211111-1-111-1-1-1-11-1-111iii-i-ii-i-i    linear of order 4
ρ131111-1-1-1-1-1-1-1111-11-11-ii-ii-ii-ii    linear of order 4
ρ141111-1-1-1-1-1-1-11111-11-1i-i-i-iii-ii    linear of order 4
ρ151111-1-1-1-1-1-1-1111-11-11i-ii-ii-ii-i    linear of order 4
ρ161111-1-1-1-1-1-1-11111-11-1-iiii-i-ii-i    linear of order 4
ρ172222-2-2-222222-2-2000000000000    orthogonal lifted from D4
ρ182222-22222-2-2-22-2000000000000    orthogonal lifted from D4
ρ192222222-2-2-2-22-2-2000000000000    orthogonal lifted from D4
ρ2022222-2-2-2-222-22-2000000000000    orthogonal lifted from D4
ρ2144-4-4000-4400000000000000000    symplectic lifted from C4.10D4, Schur index 2
ρ2244-4-40004-400000000000000000    symplectic lifted from C4.10D4, Schur index 2
ρ234-4-4400000-4i4i000000000000000    complex lifted from C42⋊C22
ρ244-4-44000004i-4i000000000000000    complex lifted from C42⋊C22
ρ254-44-40-4i4i0000000000000000000    complex lifted from C42⋊C22
ρ264-44-404i-4i0000000000000000000    complex lifted from C42⋊C22

Smallest permutation representation of C42.74D4
On 64 points
Generators in S64
(1 62 19 11)(2 12 20 63)(3 64 21 13)(4 14 22 57)(5 58 23 15)(6 16 24 59)(7 60 17 9)(8 10 18 61)(25 37 49 48)(26 41 50 38)(27 39 51 42)(28 43 52 40)(29 33 53 44)(30 45 54 34)(31 35 55 46)(32 47 56 36)
(1 60 23 13)(2 57 24 10)(3 62 17 15)(4 59 18 12)(5 64 19 9)(6 61 20 14)(7 58 21 11)(8 63 22 16)(25 39 53 46)(26 36 54 43)(27 33 55 48)(28 38 56 45)(29 35 49 42)(30 40 50 47)(31 37 51 44)(32 34 52 41)
(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 37 60 51 23 44 13 31)(2 26 57 36 24 54 10 43)(3 46 62 25 17 39 15 53)(4 56 59 45 18 28 12 38)(5 33 64 55 19 48 9 27)(6 30 61 40 20 50 14 47)(7 42 58 29 21 35 11 49)(8 52 63 41 22 32 16 34)

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

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

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

Matrix representation of C42.74D4 in GL8(𝔽17)

160000000
016000000
001600000
000160000
000001600
00001000
000000016
00000010
,
01000000
160000000
00010000
001600000
000013000
000001300
000000130
000000013
,
001100000
0010160000
1016000000
167000000
00000009
00000090
00008000
00000900
,
00100000
00010000
01000000
160000000
00009000
00000800
00000009
00000090

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13],[0,0,10,16,0,0,0,0,0,0,16,7,0,0,0,0,1,10,0,0,0,0,0,0,10,16,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0],[0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0] >;

C42.74D4 in GAP, Magma, Sage, TeX

C_4^2._{74}D_4
% in TeX

G:=Group("C4^2.74D4");
// GroupNames label

G:=SmallGroup(128,269);
// by ID

G=gap.SmallGroup(128,269);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1430,352,1123,1018,248,1971,102]);
// 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*c^3>;
// generators/relations

Export

Character table of C42.74D4 in TeX

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