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G = C43⋊9C2  order 128 = 27

9th semidirect product of C43 and C2 acting faithfully

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

Aliases: C43⋊9C2, C42⋊36D4, C24.185C23, C23.175C24, (C4×D4)⋊18C4, C4.181(C4×D4), C4⋊1(C42⋊C2), C42.277(C2×C4), C23.75(C22×C4), C22.66(C23×C4), C22.73(C22×D4), C4○(C24.3C22), (C23×C4).281C22, C4○(C23.65C23), C4○2(C24.C22), (C2×C42).1004C22, (C22×C4).1238C23, (C22×D4).465C22, C24.C22⋊189C2, C24.3C22.83C2, C2.2(C22.26C24), C23.65C23⋊172C2, C2.C42.510C22, C2.3(C23.36C23), (C4×C4⋊C4)⋊16C2, C2.12(C2×C4×D4), (C2×C4×D4).26C2, C2.6(C4×C4○D4), C4⋊C4.198(C2×C4), (C4×C22⋊C4)⋊25C2, (C2×C4).674(C2×D4), (C2×C42⋊C2)⋊6C2, (C2×D4).209(C2×C4), C22⋊C4.52(C2×C4), C22.67(C2×C4○D4), (C2×C4).635(C4○D4), (C2×C4⋊C4).790C22, (C22×C4).294(C2×C4), (C2×C4).208(C22×C4), C2.16(C2×C42⋊C2), (C2×C22⋊C4).416C22, SmallGroup(128,1025)

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

Derived series
C1 — C22 — C43⋊9C2
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C43 — C43⋊9C2
Lower central
C1 — C22 — C43⋊9C2
Upper central
C1 — C22×C4 — C43⋊9C2
Jennings
C1 — C23 — C43⋊9C2

Subgroups: 524 in 326 conjugacy classes, 160 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×8], C4 [×18], C22 [×3], C22 [×4], C22 [×20], C2×C4 [×26], C2×C4 [×42], D4 [×8], C23, C23 [×4], C23 [×12], C42 [×8], C42 [×14], C22⋊C4 [×8], C22⋊C4 [×16], C4⋊C4 [×4], C4⋊C4 [×12], C22×C4 [×3], C22×C4 [×18], C22×C4 [×8], C2×D4 [×4], C2×D4 [×4], C24 [×2], C2.C42 [×6], C2×C42 [×3], C2×C42 [×6], C2×C22⋊C4 [×10], C2×C4⋊C4, C2×C4⋊C4 [×6], C42⋊C2 [×8], C4×D4 [×8], C23×C4 [×2], C22×D4, C43, C4×C22⋊C4 [×2], C4×C4⋊C4, C24.C22 [×4], C23.65C23 [×2], C24.3C22 [×2], C2×C42⋊C2 [×2], C2×C4×D4, C43⋊9C2

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×10], C24, C42⋊C2 [×4], C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4 [×5], C2×C42⋊C2, C2×C4×D4, C4×C4○D4, C23.36C23 [×2], C22.26C24 [×2], C43⋊9C2

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

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

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

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

Matrix representation ►G ⊆ GL5(𝔽5)

30000
03000
00300
00001
00010
,
10000
00100
04000
00010
00001
,
40000
04000
00400
00020
00002
,
40000
00100
01000
00010
00004

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

56 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4AF4AG···4AR
order12···222224···44···44···4
size11···144441···12···24···4

56 irreducible representations

dim111111111122
type++++++++++
imageC1C2C2C2C2C2C2C2C2C4D4C4○D4
kernelC43⋊9C2C43C4×C22⋊C4C4×C4⋊C4C24.C22C23.65C23C24.3C22C2×C42⋊C2C2×C4×D4C4×D4C42C2×C4
# reps11214222116420

In GAP, Magma, Sage, TeX 

C_4^3\rtimes_9C_2
 
% in TeX
 
G:=Group("C4^3:9C2");
 
// GroupNames label
 
G:=SmallGroup(128,1025);
 
// by ID
 
G=gap.SmallGroup(128,1025);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,184,80]);
 
// 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,c*d=d*c>;
 
// generators/relations
 

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