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G = C42.432D4order 128 = 27

65th non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.432D4, (C2×C8)⋊29D4, C4.10(C4×D4), (C2×C4).75D8, C41D412C4, C42(D4⋊C4), C429C43C2, C2.2(C84D4), C2.3(C85D4), (C2×C4).73SD16, C22.46(C2×D8), C4.73(C4⋊D4), C42.266(C2×C4), C2.3(C4.4D8), C23.796(C2×D4), (C22×C4).580D4, C22.71(C2×SD16), C22.32(C41D4), (C22×C8).490C22, (C22×D4).41C22, (C22×C4).1401C23, (C2×C42).1072C22, C22.63(C4.4D4), C2.9(C24.3C22), (C2×C4×C8)⋊13C2, (C2×D4⋊C4)⋊7C2, (C2×C4).735(C2×D4), (C2×C41D4).5C2, (C2×D4).103(C2×C4), C2.22(C2×D4⋊C4), (C2×C4⋊C4).84C22, (C2×C4).592(C4○D4), (C2×C4).415(C22×C4), (C2×C4).256(C22⋊C4), C22.279(C2×C22⋊C4), SmallGroup(128,689)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.432D4
C1C2C22C23C22×C4C2×C42C2×C4×C8 — C42.432D4
C1C2C2×C4 — C42.432D4
C1C23C2×C42 — C42.432D4
C1C2C2C22×C4 — C42.432D4

Generators and relations for C42.432D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=dbd=b-1, dcd=bc-1 >

Subgroups: 564 in 218 conjugacy classes, 80 normal (16 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×12], C4 [×2], C22 [×3], C22 [×4], C22 [×20], C8 [×4], C2×C4 [×2], C2×C4 [×16], C2×C4 [×6], D4 [×28], C23, C23 [×16], C42 [×4], C4⋊C4 [×8], C2×C8 [×4], C2×C8 [×4], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×4], C2×D4 [×26], C24 [×2], C4×C8 [×2], D4⋊C4 [×8], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C41D4 [×4], C41D4 [×2], C22×C8 [×2], C22×D4 [×2], C22×D4 [×2], C429C4, C2×C4×C8, C2×D4⋊C4 [×4], C2×C41D4, C42.432D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], D8 [×4], SD16 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], D4⋊C4 [×8], C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C41D4, C2×D8 [×2], C2×SD16 [×2], C24.3C22, C2×D4⋊C4 [×2], C4.4D8 [×2], C85D4, C84D4, C42.432D4

Smallest permutation representation of C42.432D4
On 64 points
Generators in S64
(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 39 11 13)(2 40 12 14)(3 37 9 15)(4 38 10 16)(5 19 58 42)(6 20 59 43)(7 17 60 44)(8 18 57 41)(21 54 46 62)(22 55 47 63)(23 56 48 64)(24 53 45 61)(25 31 51 33)(26 32 52 34)(27 29 49 35)(28 30 50 36)
(1 64 25 60)(2 63 26 59)(3 62 27 58)(4 61 28 57)(5 9 54 49)(6 12 55 52)(7 11 56 51)(8 10 53 50)(13 23 33 44)(14 22 34 43)(15 21 35 42)(16 24 36 41)(17 39 48 31)(18 38 45 30)(19 37 46 29)(20 40 47 32)
(1 4)(2 3)(5 22)(6 21)(7 24)(8 23)(9 12)(10 11)(13 38)(14 37)(15 40)(16 39)(17 61)(18 64)(19 63)(20 62)(25 28)(26 27)(29 34)(30 33)(31 36)(32 35)(41 56)(42 55)(43 54)(44 53)(45 60)(46 59)(47 58)(48 57)(49 52)(50 51)

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4L4M4N4O4P8A···8P
order12···222224···444448···8
size11···188882···288882···2

44 irreducible representations

dim111111222222
type+++++++++
imageC1C2C2C2C2C4D4D4D4D8SD16C4○D4
kernelC42.432D4C429C4C2×C4×C8C2×D4⋊C4C2×C41D4C41D4C42C2×C8C22×C4C2×C4C2×C4C2×C4
# reps111418242884

Matrix representation of C42.432D4 in GL5(𝔽17)

10000
00100
016000
000160
000016
,
160000
01000
00100
00001
000160
,
40000
013000
00400
00033
000314
,
160000
00100
01000
000016
000160

G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,1,0],[4,0,0,0,0,0,13,0,0,0,0,0,4,0,0,0,0,0,3,3,0,0,0,3,14],[16,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,16,0] >;

C42.432D4 in GAP, Magma, Sage, TeX

C_4^2._{432}D_4
% in TeX

G:=Group("C4^2.432D4");
// GroupNames label

G:=SmallGroup(128,689);
// by ID

G=gap.SmallGroup(128,689);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,100,2019,248]);
// Polycyclic

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

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