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G = C42.301C23  order 128 = 27

162nd non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.301C23, C4.1752+ (1+4), (C8×D4)⋊49C2, C4⋊Q8.34C4, C8⋊9D4⋊45C2, C8⋊6D4⋊44C2, C4.21(C8○D4), C4⋊1D4.21C4, C4⋊D4.29C4, (C4×C8).29C22, C4⋊C8.367C22, (C2×C8).439C23, C42.226(C2×C4), (C2×C4).678C24, C4.4D4.22C4, C8⋊C4.98C22, C42.6C4⋊53C2, (C4×D4).303C22, C23.45(C22×C4), C42.12C4⋊55C2, C2.32(Q8○M4(2)), C22⋊C8.146C22, (C22×C4).945C23, C22.202(C23×C4), (C2×C42).785C22, (C22×C8).451C22, C2.52(C22.11C24), (C2×M4(2)).248C22, C22.26C24.29C2, C2.31(C2×C8○D4), C4⋊C4.121(C2×C4), (C2×D4).145(C2×C4), C22⋊C4.22(C2×C4), (C2×C4).83(C22×C4), (C2×Q8).127(C2×C4), (C22×C8)⋊C2⋊37C2, (C22×C4).358(C2×C4), (C2×C4○D4).98C22, SmallGroup(128,1713)

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

Derived series
C1 — C22 — C42.301C23
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C22.26C24 — C42.301C23
Lower central
C1 — C22 — C42.301C23
Upper central
C1 — C2×C4 — C42.301C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.301C23

Subgroups: 332 in 201 conjugacy classes, 126 normal (28 characteristic)
C1, C2 [×3], C2 [×5], C4 [×4], C4 [×8], C22, C22 [×15], C8 [×8], C2×C4 [×6], C2×C4 [×4], C2×C4 [×13], D4 [×12], Q8 [×2], C23, C23 [×4], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×8], C2×C8 [×4], M4(2) [×4], C22×C4 [×3], C22×C4 [×4], C2×D4 [×10], C2×Q8 [×2], C4○D4 [×4], C4×C8 [×2], C8⋊C4 [×2], C22⋊C8 [×12], C4⋊C8 [×4], C2×C42, C4×D4 [×4], C4⋊D4 [×4], C4.4D4 [×2], C4⋊1D4, C4⋊Q8, C22×C8 [×4], C2×M4(2) [×4], C2×C4○D4 [×2], (C22×C8)⋊C2 [×4], C42.12C4, C42.6C4, C8×D4 [×2], C8⋊9D4 [×4], C8⋊6D4 [×2], C22.26C24, C42.301C23

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C8○D4 [×2], C23×C4, 2+ (1+4) [×2], C22.11C24, C2×C8○D4, Q8○M4(2), C42.301C23

Generators and relations
 G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, cac-1=a-1b2, dad=a-1, ae=ea, bc=cb, bd=db, be=eb, dcd=ece=a2b2c, ede=b2d >

Smallest permutation representation
►On 64 points
Generators in S64
(1 18 51 37)(2 34 52 23)(3 20 53 39)(4 36 54 17)(5 22 55 33)(6 38 56 19)(7 24 49 35)(8 40 50 21)(9 62 32 44)(10 41 25 59)(11 64 26 46)(12 43 27 61)(13 58 28 48)(14 45 29 63)(15 60 30 42)(16 47 31 57)

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

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

Matrix representation ►G ⊆ GL6(𝔽17)

0130000
1300000
0000016
0000160
000100
001000
,
400000
040000
001000
000100
000010
000001
,
200000
020000
0016000
000100
0000160
000001
,
040000
1300000
000100
001000
0000016
0000160
,
010000
100000
000100
001000
000001
000010

G:=sub<GL(6,GF(17))| [0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16,0,0,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
 

44 conjugacy classes

class 1 2A2B2C2D···2H4A4B4C4D4E4F4G4H4I···4O8A···8H8I···8T
order12222···2444444444···48···88···8
size11114···4111122224···42···24···4

44 irreducible representations

dim111111111111244
type+++++++++
imageC1C2C2C2C2C2C2C2C4C4C4C4C8○D42+ (1+4)Q8○M4(2)
kernelC42.301C23(C22×C8)⋊C2C42.12C4C42.6C4C8×D4C8⋊9D4C8⋊6D4C22.26C24C4⋊D4C4.4D4C4⋊1D4C4⋊Q8C4C4C2
# reps141124218422822

In GAP, Magma, Sage, TeX 

C_4^2._{301}C_2^3
 
% in TeX
 
G:=Group("C4^2.301C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1713);
 
// by ID
 
G=gap.SmallGroup(128,1713);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,891,675,1018,521,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=d^2=e^2=1,c^2=b,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d=a^-1,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=e*c*e=a^2*b^2*c,e*d*e=b^2*d>;
 
// generators/relations
 

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