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G = C42⋊13D4  order 128 = 27

7th semidirect product of C42 and D4 acting via D4/C2=C22

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

Aliases: C42⋊13D4, C23.206C24, C24.197C23, C22.452+ 1+4, C22.292- 1+4, C4⋊D4⋊16C4, C4.130(C4×D4), C23.8Q8⋊7C2, C23.23D4⋊6C2, C23.10(C22×C4), C22.97(C23×C4), C23.7Q8⋊16C2, C22.94(C22×D4), (C23×C4).292C22, C24.C22⋊6C2, (C22×C4).471C23, (C2×C42).413C22, C24.3C22⋊14C2, C2.5(C22.29C24), (C22×D4).105C22, C23.65C23⋊14C2, C2.14(C22.11C24), C2.C42.42C22, C2.3(C22.34C24), C2.5(C22.36C24), C2.4(C22.31C24), C2.13(C23.33C23), (C2×C4×D4)⋊6C2, (C4×C4⋊C4)⋊26C2, C4⋊C4⋊28(C2×C4), C2.23(C2×C4×D4), (C2×D4)⋊14(C2×C4), C22⋊C4⋊10(C2×C4), (C22×C4)⋊24(C2×C4), (C2×C4).678(C2×D4), (C2×C4⋊D4).15C2, (C2×C4).27(C22×C4), C22.91(C2×C4○D4), (C2×C42⋊C2)⋊10C2, (C2×C4).647(C4○D4), (C2×C4⋊C4).807C22, (C2×C22⋊C4).27C22, SmallGroup(128,1056)

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

Derived series
C1 — C22 — C42⋊13D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C42⋊C2 — C42⋊13D4
Lower central
C1 — C22 — C42⋊13D4
Upper central
C1 — C23 — C42⋊13D4
Jennings
C1 — C23 — C42⋊13D4

Generators and relations for C42⋊13D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, ac=ca, dad=ab2, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 636 in 334 conjugacy classes, 148 normal (42 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4⋊D4, C23×C4, C23×C4, C22×D4, C22×D4, C4×C4⋊C4, C23.7Q8, C23.8Q8, C23.23D4, C24.C22, C23.65C23, C24.3C22, C24.3C22, C2×C42⋊C2, C2×C4×D4, C2×C4⋊D4, C42⋊13D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C4×D4, C23×C4, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C4×D4, C22.11C24, C23.33C23, C22.29C24, C22.31C24, C22.34C24, C22.36C24, C42⋊13D4

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

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

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

44 conjugacy classes

class 1 2A···2G2H···2M4A···4L4M···4AD
order12···22···24···44···4
size11···14···42···24···4

44 irreducible representations

dim1111111111112244
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C4D4C4○D42+ 1+42- 1+4
kernelC42⋊13D4C4×C4⋊C4C23.7Q8C23.8Q8C23.23D4C24.C22C23.65C23C24.3C22C2×C42⋊C2C2×C4×D4C2×C4⋊D4C4⋊D4C42C2×C4C22C22
# reps11122213111164431

Matrix representation of C42⋊13D4 ►in GL8(𝔽5)

20000000
02000000
00100000
00010000
00000010
00004113
00004000
00004104
,
40000000
04000000
00400000
00040000
00000100
00004000
00004113
00004014
,
23000000
03000000
00440000
00210000
00000400
00004000
00001442
00000001
,
40000000
31000000
00400000
00210000
00000100
00001000
00001442
00001401

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

C42⋊13D4 in GAP, Magma, Sage, TeX 

C_4^2\rtimes_{13}D_4
 
% in TeX
 
G:=Group("C4^2:13D4");
 
// GroupNames label
 
G:=SmallGroup(128,1056);
 
// by ID
 
G=gap.SmallGroup(128,1056);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,568,758,219,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a*b^2,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

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