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G = C42⋊29D4  order 128 = 27

23rd semidirect product of C42 and D4 acting via D4/C2=C22

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

Aliases: C42⋊29D4, C23.531C24, C24.370C23, C22.3082+ 1+4, C42⋊5C4⋊24C2, C23⋊2D4⋊26C2, C23⋊Q8⋊29C2, C23.65(C4○D4), C23.11D4⋊60C2, C23.23D4⋊70C2, (C2×C42).608C22, (C22×C4).141C23, (C23×C4).137C22, C22.356(C22×D4), (C22×D4).541C22, (C22×Q8).156C22, C2.40(C22.32C24), C2.82(C22.19C24), C2.40(C22.29C24), C2.C42.256C22, (C2×C4×D4)⋊53C2, (C2×C4).390(C2×D4), (C2×C4.4D4)⋊21C2, (C2×C4⋊C4).892C22, C22.403(C2×C4○D4), (C2×C22⋊C4).220C22, SmallGroup(128,1363)

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

Derived series
C1 — C23 — C42⋊29D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C22⋊C4 — C23.23D4 — C42⋊29D4
Lower central
C1 — C23 — C42⋊29D4
Upper central
C1 — C23 — C42⋊29D4
Jennings
C1 — C23 — C42⋊29D4

Generators and relations for C42⋊29D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1b2, dad=a-1, cbc-1=a2b, bd=db, dcd=c-1 >

Subgroups: 724 in 314 conjugacy classes, 96 normal (18 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C4.4D4, C23×C4, C22×D4, C22×D4, C22×Q8, C42⋊5C4, C23.23D4, C23⋊2D4, C23⋊Q8, C23.11D4, C2×C4×D4, C2×C4.4D4, C42⋊29D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, C22.19C24, C22.29C24, C22.32C24, C42⋊29D4

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A4B4C4D4E···4L4M···4R
order12···222222244444···44···4
size11···144448822224···48···8

32 irreducible representations

dim11111111224
type++++++++++
imageC1C2C2C2C2C2C2C2D4C4○D42+ 1+4
kernelC42⋊29D4C42⋊5C4C23.23D4C23⋊2D4C23⋊Q8C23.11D4C2×C4×D4C2×C4.4D4C42C23C22
# reps11442211484

Matrix representation of C42⋊29D4 ►in GL8(𝔽5)

01000000
10000000
00120000
00040000
00000010
00000001
00004000
00000400
,
20000000
02000000
00300000
00030000
00001200
00000400
00000012
00000004
,
01000000
40000000
00400000
00110000
00000010
00000044
00001000
00004400
,
01000000
10000000
00400000
00040000
00000010
00000001
00001000
00000100

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

C42⋊29D4 in GAP, Magma, Sage, TeX 

C_4^2\rtimes_{29}D_4
 
% in TeX
 
G:=Group("C4^2:29D4");
 
// GroupNames label
 
G:=SmallGroup(128,1363);
 
// by ID
 
G=gap.SmallGroup(128,1363);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,758,723,185,136]);
 
// 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*b^2,d*a*d=a^-1,c*b*c^-1=a^2*b,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

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