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G = D4⋊C42order 128 = 27

3rd semidirect product of D4 and C42 acting via C42/C2×C4=C2

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

Aliases: D43C42, C42.93D4, (C4×D4)⋊12C4, D4⋊C49C4, C4.110(C4×D4), C4.5(C2×C42), C22.88(C4×D4), C2.2(D8⋊C4), C42.128(C2×C4), (C22×C4).671D4, C23.732(C2×D4), C22.4Q1645C2, C4.5(C42⋊C2), C2.2(SD16⋊C4), C22.54(C8⋊C22), (C2×C42).235C22, (C22×C8).380C22, (C22×C4).1309C23, C2.3(C23.37D4), C2.3(C23.36D4), (C22×D4).447C22, C22.43(C8.C22), (C4×C4⋊C4)⋊4C2, C4⋊C423(C2×C4), (C2×C8)⋊24(C2×C4), (C2×C4×D4).12C2, (C2×C8⋊C4)⋊21C2, C2.20(C4×C22⋊C4), (C2×D4).199(C2×C4), (C2×C4).1304(C2×D4), (C2×D4⋊C4).32C2, (C2×C4).540(C4○D4), (C2×C4⋊C4).748C22, (C2×C4).352(C22×C4), (C2×C4).327(C22⋊C4), C22.121(C2×C22⋊C4), SmallGroup(128,494)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — D4⋊C42
C1C2C4C2×C4C22×C4C2×C42C2×C8⋊C4 — D4⋊C42
C1C2C4 — D4⋊C42
C1C23C2×C42 — D4⋊C42
C1C2C2C22×C4 — D4⋊C42

Generators and relations for D4⋊C42
 G = < a,b,c,d | a4=b2=c4=d4=1, bab=cac-1=a-1, ad=da, cbc-1=ab, dbd-1=a2b, cd=dc >

Subgroups: 396 in 194 conjugacy classes, 84 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×2], C4 [×10], C22 [×3], C22 [×4], C22 [×16], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×24], D4 [×4], D4 [×6], C23, C23 [×10], C42 [×4], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×6], C4⋊C4 [×3], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4 [×11], C2×D4 [×6], C2×D4 [×3], C24, C2.C42, C8⋊C4 [×2], D4⋊C4 [×8], C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4 [×2], C4×D4 [×4], C4×D4 [×2], C22×C8 [×2], C23×C4, C22×D4, C22.4Q16 [×2], C4×C4⋊C4, C2×C8⋊C4, C2×D4⋊C4 [×2], C2×C4×D4, D4⋊C42
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×4], C23, C42 [×4], C22⋊C4 [×4], C22×C4 [×3], C2×D4 [×2], C4○D4 [×2], C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4 [×4], C8⋊C22 [×3], C8.C22, C4×C22⋊C4, C23.36D4, C23.37D4, SD16⋊C4 [×2], D8⋊C4 [×2], D4⋊C42

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4L4M···4X8A···8H
order12···222224···44···48···8
size11···144442···24···44···4

44 irreducible representations

dim1111111122244
type+++++++++-
imageC1C2C2C2C2C2C4C4D4D4C4○D4C8⋊C22C8.C22
kernelD4⋊C42C22.4Q16C4×C4⋊C4C2×C8⋊C4C2×D4⋊C4C2×C4×D4D4⋊C4C4×D4C42C22×C4C2×C4C22C22
# reps12112116822431

Matrix representation of D4⋊C42 in GL8(𝔽17)

160000000
016000000
001600000
000160000
000001600
00001000
000000016
00000010
,
160000000
81000000
00100000
000160000
00000100
00001000
00000001
00000010
,
82000000
109000000
00010000
001600000
000013131616
0000134161
00008844
000089413
,
10000000
01000000
001300000
000130000
0000013015
00004020
000001604
000010130

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0],[16,8,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[8,10,0,0,0,0,0,0,2,9,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,13,13,8,8,0,0,0,0,13,4,8,9,0,0,0,0,16,16,4,4,0,0,0,0,16,1,4,13],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,4,0,1,0,0,0,0,13,0,16,0,0,0,0,0,0,2,0,13,0,0,0,0,15,0,4,0] >;

D4⋊C42 in GAP, Magma, Sage, TeX

D_4\rtimes C_4^2
% in TeX

G:=Group("D4:C4^2");
// GroupNames label

G:=SmallGroup(128,494);
// by ID

G=gap.SmallGroup(128,494);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,723,100,2019,248,4037,124]);
// Polycyclic

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

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