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G = C42⋊32D4  order 128 = 27

26th semidirect product of C42 and D4 acting via D4/C2=C22

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

Aliases: C42⋊32D4, C24.54C23, C23.562C24, C22.2512- 1+4, C22.3362+ 1+4, C23⋊Q8⋊35C2, C23⋊2D4.20C2, C23.Q8⋊46C2, C23.10D4⋊67C2, (C22×C4).167C23, (C2×C42).626C22, C22.374(C22×D4), C24.3C22⋊70C2, (C22×D4).210C22, (C22×Q8).168C22, C24.C22⋊112C2, C23.78C23⋊33C2, C2.50(C22.29C24), C2.53(C22.32C24), C2.C42.276C22, C2.51(C22.26C24), C2.34(C22.31C24), C2.63(C22.36C24), (C4×C4⋊C4)⋊115C2, (C2×C4).407(C2×D4), (C2×C4.4D4)⋊23C2, (C2×C42⋊2C2)⋊16C2, (C2×C4).182(C4○D4), (C2×C4⋊C4).896C22, C22.429(C2×C4○D4), (C2×C22⋊C4).240C22, SmallGroup(128,1394)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C23 — C42⋊32D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C22⋊C4 — C24.3C22 — C42⋊32D4
Lower central
C1 — C23 — C42⋊32D4
Upper central
C1 — C23 — C42⋊32D4
Jennings
C1 — C23 — C42⋊32D4

Generators and relations for C42⋊32D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1, dad=a-1b2, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 580 in 267 conjugacy classes, 96 normal (34 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, C24, C2.C42, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4.4D4, C42⋊2C2, C22×D4, C22×D4, C22×Q8, C4×C4⋊C4, C24.C22, C24.3C22, C23⋊2D4, C23⋊Q8, C23.10D4, C23.10D4, C23.78C23, C23.Q8, C2×C4.4D4, C2×C42⋊2C2, C42⋊32D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.26C24, C22.29C24, C22.31C24, C22.32C24, C22.36C24, C42⋊32D4

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J4A4B4C4D4E···4P4Q···4U
order12···222244444···44···4
size11···188822224···48···8

32 irreducible representations

dim111111111112244
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2D4C4○D42+ 1+42- 1+4
kernelC42⋊32D4C4×C4⋊C4C24.C22C24.3C22C23⋊2D4C23⋊Q8C23.10D4C23.78C23C23.Q8C2×C4.4D4C2×C42⋊2C2C42C2×C4C22C22
# reps112211312114831

Matrix representation of C42⋊32D4 ►in GL8(𝔽5)

41000000
01000000
00100000
00010000
00000010
00000001
00004000
00000400
,
30000000
03000000
00400000
00040000
00000100
00001000
00000001
00000010
,
23000000
03000000
00330000
00020000
00001000
00000100
00000040
00000004
,
32000000
12000000
00330000
00420000
00001000
00000400
00000040
00000001

G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[2,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[3,1,0,0,0,0,0,0,2,2,0,0,0,0,0,0,0,0,3,4,0,0,0,0,0,0,3,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1] >;
 

C42⋊32D4 in GAP, Magma, Sage, TeX 

C_4^2\rtimes_{32}D_4
 
% in TeX
 
G:=Group("C4^2:32D4");
 
// GroupNames label
 
G:=SmallGroup(128,1394);
 
// by ID
 
G=gap.SmallGroup(128,1394);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,456,758,723,185,136]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a^-1*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
 
// generators/relations
 

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