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G = C42.425D4order 128 = 27

58th non-split extension by C42 of D4 acting via D4/C22=C2

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C42.425D4, C23.31M4(2), (C22×C4)⋊8C8, C42(C22⋊C8), C221(C4⋊C8), (C23×C4).36C4, (C2×C42).36C4, C23.32(C2×C8), (C22×C4).77Q8, C23.66(C4⋊C4), C24.115(C2×C4), (C22×C4).543D4, (C2×C4).60M4(2), (C22×C8).8C22, C4.179(C4⋊D4), C4.108(C22⋊Q8), (C22×C42).16C2, C22.34(C22×C8), C2.3(C4⋊M4(2)), C2.4(C42.6C4), C2.4(C24.4C4), (C2×C42).995C22, (C23×C4).633C22, C22.7C422C2, C23.263(C22×C4), C22.45(C2×M4(2)), C2.4(C23.7Q8), C2.5(C42.12C4), (C22×C4).1617C23, C22.54(C42⋊C2), (C2×C4⋊C8)⋊9C2, C2.8(C2×C4⋊C8), (C2×C4).84(C2×C8), (C2×C4).82(C4⋊C4), C2.17(C2×C22⋊C8), C22.63(C2×C4⋊C4), (C2×C4).335(C2×Q8), (C2×C4).1513(C2×D4), (C2×C22⋊C8).13C2, (C2×C4).925(C4○D4), (C22×C4).479(C2×C4), (C2×C4).354(C22⋊C4), C22.124(C2×C22⋊C4), SmallGroup(128,529)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42.425D4
C1C2C4C2×C4C22×C4C23×C4C22×C42 — C42.425D4
C1C22 — C42.425D4
C1C22×C4 — C42.425D4
C1C2C2C22×C4 — C42.425D4

Generators and relations for C42.425D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=b, ab=ba, cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=b-1c3 >

Subgroups: 316 in 200 conjugacy classes, 92 normal (34 characteristic)
C1, C2 [×7], C2 [×4], C4 [×8], C4 [×8], C22 [×7], C22 [×4], C22 [×12], C8 [×4], C2×C4 [×2], C2×C4 [×18], C2×C4 [×32], C23, C23 [×6], C23 [×4], C42 [×4], C42 [×6], C2×C8 [×12], C22×C4 [×6], C22×C4 [×8], C22×C4 [×14], C24, C22⋊C8 [×4], C4⋊C8 [×4], C2×C42 [×4], C2×C42 [×4], C22×C8 [×4], C23×C4 [×3], C22.7C42 [×2], C2×C22⋊C8 [×2], C2×C4⋊C8 [×2], C22×C42, C42.425D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C22⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×6], M4(2) [×6], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C22⋊C8 [×4], C4⋊C8 [×4], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], C22×C8, C2×M4(2) [×3], C23.7Q8, C2×C22⋊C8, C24.4C4, C2×C4⋊C8, C4⋊M4(2), C42.12C4, C42.6C4, C42.425D4

Smallest permutation representation of C42.425D4
On 64 points
Generators in S64
(1 63 55 47)(2 48 56 64)(3 57 49 41)(4 42 50 58)(5 59 51 43)(6 44 52 60)(7 61 53 45)(8 46 54 62)(9 37 21 31)(10 32 22 38)(11 39 23 25)(12 26 24 40)(13 33 17 27)(14 28 18 34)(15 35 19 29)(16 30 20 36)
(1 37 5 33)(2 38 6 34)(3 39 7 35)(4 40 8 36)(9 43 13 47)(10 44 14 48)(11 45 15 41)(12 46 16 42)(17 63 21 59)(18 64 22 60)(19 57 23 61)(20 58 24 62)(25 53 29 49)(26 54 30 50)(27 55 31 51)(28 56 32 52)
(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 37 50 5 26 33 54)(2 49 38 25 6 53 34 29)(3 32 39 52 7 28 35 56)(4 51 40 27 8 55 36 31)(9 58 43 24 13 62 47 20)(10 23 44 61 14 19 48 57)(11 60 45 18 15 64 41 22)(12 17 46 63 16 21 42 59)

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

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

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

56 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4AB8A···8P
order12···222224···44···48···8
size11···122221···12···24···4

56 irreducible representations

dim11111111222222
type+++++++-
imageC1C2C2C2C2C4C4C8D4D4Q8M4(2)C4○D4M4(2)
kernelC42.425D4C22.7C42C2×C22⋊C8C2×C4⋊C8C22×C42C2×C42C23×C4C22×C4C42C22×C4C22×C4C2×C4C2×C4C23
# reps122214416422844

Matrix representation of C42.425D4 in GL5(𝔽17)

10000
01000
00100
000130
00004
,
130000
01000
00100
000160
000016
,
90000
00100
016000
000016
000160
,
80000
00100
01000
00001
000160

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

C42.425D4 in GAP, Magma, Sage, TeX

C_4^2._{425}D_4
% in TeX

G:=Group("C4^2.425D4");
// GroupNames label

G:=SmallGroup(128,529);
// by ID

G=gap.SmallGroup(128,529);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,124]);
// Polycyclic

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

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