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G = C4313C2order 128 = 27

13rd semidirect product of C43 and C2 acting faithfully

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

Aliases: C4313C2, C4237D4, C24.124C23, C23.760C24, C41(C422C2), C22.470(C22×D4), (C22×C4).1266C23, (C2×C42).1095C22, (C22×D4).315C22, C24.C22186C2, C24.3C22.81C2, C23.65C23171C2, C2.C42.455C22, C2.60(C22.26C24), C2.114(C23.36C23), (C2×C4).689(C2×D4), (C2×C422C2)⋊30C2, (C2×C4).530(C4○D4), (C2×C4⋊C4).563C22, C2.26(C2×C422C2), C22.601(C2×C4○D4), (C2×C22⋊C4).369C22, SmallGroup(128,1592)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C4313C2
C1C2C22C23C22×C4C2×C42C43 — C4313C2
C1C23 — C4313C2
C1C23 — C4313C2
C1C23 — C4313C2

Generators and relations for C4313C2
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, ac=ca, dad=ac2, bc=cb, dbd=b-1, dcd=a2c-1 >

Subgroups: 468 in 258 conjugacy classes, 108 normal (9 characteristic)
C1, C2, C2 [×6], C2 [×2], C4 [×4], C4 [×18], C22, C22 [×6], C22 [×14], C2×C4 [×18], C2×C4 [×30], D4 [×4], C23, C23 [×14], C42 [×4], C42 [×12], C22⋊C4 [×24], C4⋊C4 [×18], C22×C4, C22×C4 [×12], C2×D4 [×6], C24 [×2], C2.C42 [×6], C2×C42, C2×C42 [×6], C2×C22⋊C4 [×12], C2×C4⋊C4 [×9], C422C2 [×8], C22×D4, C43, C24.C22 [×6], C23.65C23 [×3], C24.3C22 [×3], C2×C422C2 [×2], C4313C2
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×12], C24, C422C2 [×4], C22×D4, C2×C4○D4 [×6], C2×C422C2, C23.36C23 [×3], C22.26C24 [×3], C4313C2

Smallest permutation representation of C4313C2
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 36 26 47)(2 33 27 48)(3 34 28 45)(4 35 25 46)(5 58 23 15)(6 59 24 16)(7 60 21 13)(8 57 22 14)(9 62 54 17)(10 63 55 18)(11 64 56 19)(12 61 53 20)(29 43 52 39)(30 44 49 40)(31 41 50 37)(32 42 51 38)
(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)
(2 11)(4 9)(5 43)(6 22)(7 41)(8 24)(13 50)(14 16)(15 52)(17 35)(18 63)(19 33)(20 61)(21 37)(23 39)(25 54)(27 56)(29 58)(30 32)(31 60)(34 45)(36 47)(38 44)(40 42)(46 62)(48 64)(49 51)(57 59)

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I4A···4AB4AC···4AH
order12···2224···44···4
size11···1882···28···8

44 irreducible representations

dim11111122
type+++++++
imageC1C2C2C2C2C2D4C4○D4
kernelC4313C2C43C24.C22C23.65C23C24.3C22C2×C422C2C42C2×C4
# reps116332424

Matrix representation of C4313C2 in GL6(𝔽5)

010000
400000
001100
003400
000030
000003
,
400000
040000
004400
002100
000040
000004
,
200000
020000
002000
000200
000001
000010
,
100000
040000
004400
000100
000010
000004

G:=sub<GL(6,GF(5))| [0,4,0,0,0,0,1,0,0,0,0,0,0,0,1,3,0,0,0,0,1,4,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,2,0,0,0,0,4,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,4,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4] >;

C4313C2 in GAP, Magma, Sage, TeX

C_4^3\rtimes_{13}C_2
% in TeX

G:=Group("C4^3:13C2");
// GroupNames label

G:=SmallGroup(128,1592);
// by ID

G=gap.SmallGroup(128,1592);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,568,758,184,2019,80]);
// 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*c^2,b*c=c*b,d*b*d=b^-1,d*c*d=a^2*c^-1>;
// generators/relations

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