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G = C42.128D4order 128 = 27

110th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.128D4, M4(2).7Q8, C4.34(C4⋊Q8), C22.15(C4×Q8), C4.85(C22⋊Q8), C4.53(C4.4D4), (C4×M4(2)).28C2, C4⋊M4(2).36C2, C23.202(C22×C4), (C2×C42).340C22, (C22×C4).707C23, C22.C42.13C2, (C2×M4(2)).219C22, C2.20(C23.67C23), C2.29(M4(2).8C22), (C2×C4⋊C4).26C4, (C2×C4).14(C2×Q8), (C2×C4).71(C4○D4), (C2×C4).1369(C2×D4), (C22×C4).28(C2×C4), (C2×C42.C2).8C2, (C2×C4⋊C4).100C22, (C2×C4).210(C22⋊C4), C22.307(C2×C22⋊C4), SmallGroup(128,730)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.128D4
C1C2C4C2×C4C22×C4C2×C42C4×M4(2) — C42.128D4
C1C2C23 — C42.128D4
C1C22C2×C42 — C42.128D4
C1C2C2C22×C4 — C42.128D4

Generators and relations for C42.128D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2b2, ab=ba, ac=ca, dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=a2b-1c3 >

Subgroups: 196 in 114 conjugacy classes, 56 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×2], C8 [×6], C2×C4 [×2], C2×C4 [×8], C2×C4 [×10], C23, C42 [×2], C42 [×2], C4⋊C4 [×12], C2×C8 [×4], M4(2) [×4], M4(2) [×6], C22×C4, C22×C4 [×6], C4×C8, C8⋊C4, C4⋊C8 [×2], C2×C42, C2×C4⋊C4 [×6], C42.C2 [×4], C2×M4(2) [×4], C22.C42 [×4], C4×M4(2), C4⋊M4(2), C2×C42.C2, C42.128D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], C2×C22⋊C4, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4, C4⋊Q8, C23.67C23, M4(2).8C22 [×2], C42.128D4

Smallest permutation representation of C42.128D4
On 64 points
Generators in S64
(1 34 31 10)(2 35 32 11)(3 36 25 12)(4 37 26 13)(5 38 27 14)(6 39 28 15)(7 40 29 16)(8 33 30 9)(17 49 48 58)(18 50 41 59)(19 51 42 60)(20 52 43 61)(21 53 44 62)(22 54 45 63)(23 55 46 64)(24 56 47 57)
(1 7 5 3)(2 4 6 8)(9 11 13 15)(10 16 14 12)(17 23 21 19)(18 20 22 24)(25 31 29 27)(26 28 30 32)(33 35 37 39)(34 40 38 36)(41 43 45 47)(42 48 46 44)(49 55 53 51)(50 52 54 56)(57 59 61 63)(58 64 62 60)
(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 53 27 58)(2 59 28 54)(3 51 29 64)(4 57 30 52)(5 49 31 62)(6 63 32 50)(7 55 25 60)(8 61 26 56)(9 24 37 43)(10 48 38 21)(11 22 39 41)(12 46 40 19)(13 20 33 47)(14 44 34 17)(15 18 35 45)(16 42 36 23)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A···4H4I4J4K4L4M4N8A···8H8I8J8K8L
order1222224···44444448···88888
size1111222···24488884···48888

32 irreducible representations

dim1111112224
type++++++-
imageC1C2C2C2C2C4D4Q8C4○D4M4(2).8C22
kernelC42.128D4C22.C42C4×M4(2)C4⋊M4(2)C2×C42.C2C2×C4⋊C4C42M4(2)C2×C4C2
# reps1411184444

Matrix representation of C42.128D4 in GL6(𝔽17)

1300000
040000
000400
0013000
0041149
00516413
,
1600000
0160000
000100
0016000
001610162
001314161
,
100000
010000
000010
0017115
0001600
0037410
,
0130000
1300000
0013000
000400
0000130
00130134

G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,13,4,5,0,0,4,0,11,16,0,0,0,0,4,4,0,0,0,0,9,13],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,16,13,0,0,1,0,10,14,0,0,0,0,16,16,0,0,0,0,2,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,3,0,0,0,7,16,7,0,0,1,1,0,4,0,0,0,15,0,10],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,13,0,0,13,0,0,0,4,0,0,0,0,0,0,13,13,0,0,0,0,0,4] >;

C42.128D4 in GAP, Magma, Sage, TeX

C_4^2._{128}D_4
% in TeX

G:=Group("C4^2.128D4");
// GroupNames label

G:=SmallGroup(128,730);
// by ID

G=gap.SmallGroup(128,730);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,400,422,436,2019,1018,2028,124]);
// Polycyclic

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

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