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G = C42.52D4order 128 = 27

34th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.52D4, C42.616C23, D4⋊C832C2, Q8⋊C836C2, C4.4(C8○D4), C4⋊D4.7C4, (C4×C8).8C22, C22⋊Q8.7C4, C42.66(C2×C4), C4.4D4.6C4, (C4×D4).7C22, C42.C2.8C4, (C4×Q8).7C22, C4⋊C8.200C22, (C22×C4).206D4, C4.135(C8⋊C22), C4⋊M4(2)⋊18C2, C42.12C413C2, C4.129(C8.C22), C23.48(C22⋊C4), (C2×C42).169C22, C2.7(C23.36D4), C2.12(C42⋊C22), C23.36C23.4C2, C4⋊C4.56(C2×C4), (C2×D4).54(C2×C4), (C2×Q8).49(C2×C4), (C2×C4).1452(C2×D4), (C2×C4).82(C22⋊C4), (C2×C4).321(C22×C4), (C22×C4).191(C2×C4), C22.171(C2×C22⋊C4), C2.21((C22×C8)⋊C2), SmallGroup(128,227)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.52D4
C1C2C22C2×C4C42C2×C42C23.36C23 — C42.52D4
C1C2C2×C4 — C42.52D4
C1C2×C4C2×C42 — C42.52D4
C1C22C22C42 — C42.52D4

Generators and relations for C42.52D4
 G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=a-1, dad-1=a-1b2, bc=cb, bd=db, dcd-1=a2b-1c3 >

Subgroups: 212 in 108 conjugacy classes, 46 normal (36 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×7], C22, C22 [×6], C8 [×4], C2×C4 [×6], C2×C4 [×10], D4 [×4], Q8 [×2], C23, C23, C42 [×4], C42, C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×4], M4(2) [×2], C22×C4 [×3], C22×C4, C2×D4, C2×D4, C2×Q8, C4×C8 [×2], C22⋊C8, C4⋊C8 [×2], C4⋊C8 [×2], C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C422C2, C2×M4(2), D4⋊C8 [×2], Q8⋊C8 [×2], C4⋊M4(2), C42.12C4, C23.36C23, C42.52D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×C22⋊C4, C8○D4 [×2], C8⋊C22, C8.C22, (C22×C8)⋊C2, C23.36D4, C42⋊C22, C42.52D4

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

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A···8H8I8J8K8L
order122222444444444444448···88888
size111148111122224448884···48888

32 irreducible representations

dim1111111111222444
type+++++++++-
imageC1C2C2C2C2C2C4C4C4C4D4D4C8○D4C8⋊C22C8.C22C42⋊C22
kernelC42.52D4D4⋊C8Q8⋊C8C4⋊M4(2)C42.12C4C23.36C23C4⋊D4C22⋊Q8C4.4D4C42.C2C42C22×C4C4C4C4C2
# reps1221112222228112

Matrix representation of C42.52D4 in GL6(𝔽17)

16150000
010000
00041615
00041515
0016100
001161313
,
1300000
0130000
0013000
0001300
0000130
0000013
,
800000
080000
0051544
005040
0013121212
0012020
,
800000
990000
0051544
0051348
0013121212
00121424

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,16,1,0,0,4,4,1,16,0,0,16,15,0,13,0,0,15,15,0,13],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,5,5,13,12,0,0,15,0,12,0,0,0,4,4,12,2,0,0,4,0,12,0],[8,9,0,0,0,0,0,9,0,0,0,0,0,0,5,5,13,12,0,0,15,13,12,14,0,0,4,4,12,2,0,0,4,8,12,4] >;

C42.52D4 in GAP, Magma, Sage, TeX

C_4^2._{52}D_4
% in TeX

G:=Group("C4^2.52D4");
// GroupNames label

G:=SmallGroup(128,227);
// by ID

G=gap.SmallGroup(128,227);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,1059,1123,570,136,172]);
// 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=a^-1,d*a*d^-1=a^-1*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
// generators/relations

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