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G = C23.146C24  order 128 = 27

46th non-split extension by C23 of C24 acting via C24/C23=C2

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

Aliases: C23.146C24, C42.103C23, C22.112C25, C4.432- (1+4), C22.82- (1+4), (D4×Q8)⋊25C2, Q8○3(C4⋊D4), Q8⋊3Q8⋊23C2, C4⋊C4.304C23, (C2×C4).102C24, Q8.43(C4○D4), C4⋊Q8.223C22, (C2×D4).484C23, (C4×D4).243C22, C22⋊C4.35C23, (C2×Q8).460C23, (C4×Q8).230C22, C4⋊D4.243C22, (C2×C42).959C22, C42⋊2C2.4C22, C22⋊Q8.233C22, C2.34(C2×2- (1+4)), (C22×C4).1212C23, C4.4D4.177C22, C42.C2.158C22, (C22×Q8).365C22, C22.35C24⋊15C2, C23.37C23⋊47C2, C23.32C23⋊18C2, C22.46C24⋊25C2, C42⋊C2.238C22, C22.50C24⋊30C2, C22.D4.34C22, C23.36C23.32C2, (C2×C4×Q8)⋊63C2, C4.285(C2×C4○D4), C2.68(C22×C4○D4), (C2×C4⋊C4).965C22, SmallGroup(128,2255)

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

Derived series
C1 — C22 — C23.146C24
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C2×C4×Q8 — C23.146C24
Lower central
C1 — C22 — C23.146C24
Upper central
C1 — C22 — C23.146C24
Jennings
C1 — C22 — C23.146C24

Subgroups: 604 in 496 conjugacy classes, 392 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×24], C22, C22 [×2], C22 [×8], C2×C4 [×2], C2×C4 [×26], C2×C4 [×26], D4 [×6], Q8 [×4], Q8 [×24], C23, C23 [×2], C42 [×30], C22⋊C4 [×26], C4⋊C4, C4⋊C4 [×69], C22×C4, C22×C4 [×12], C2×D4, C2×D4 [×2], C2×Q8, C2×Q8 [×12], C2×Q8 [×12], C2×C42 [×3], C2×C4⋊C4 [×3], C42⋊C2 [×18], C4×D4 [×9], C4×Q8 [×2], C4×Q8 [×29], C4⋊D4, C22⋊Q8 [×21], C22.D4 [×6], C4.4D4 [×3], C42.C2 [×27], C42⋊2C2 [×18], C4⋊Q8 [×12], C22×Q8, C22×Q8 [×2], C2×C4×Q8, C23.32C23 [×2], C23.36C23 [×3], C23.37C23 [×3], C22.35C24 [×6], D4×Q8, D4×Q8 [×2], C22.46C24 [×6], C22.50C24 [×3], Q8⋊3Q8, Q8⋊3Q8 [×3], C23.146C24

Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], C4○D4 [×4], C24 [×31], C2×C4○D4 [×6], 2- (1+4) [×4], C25, C22×C4○D4, C2×2- (1+4) [×2], C23.146C24

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=e2=f2=b, g2=cb=bc, ab=ba, dad-1=ac=ca, ae=ea, af=fa, ag=ga, ede-1=gdg-1=bd=db, fef-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, df=fd, eg=ge, fg=gf >

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

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

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

430000
010000
001000
000100
000010
000001
,
100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
100000
440000
003000
000200
002020
000303
,
400000
040000
001220
002102
002343
003234
,
100000
010000
002000
000200
003130
001303
,
300000
030000
000100
004000
002001
000340

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4P4Q···4AJ
order122222224···44···4
size111122442···24···4

44 irreducible representations

dim1111111111244
type++++++++++--
imageC1C2C2C2C2C2C2C2C2C2C4○D42- (1+4)2- (1+4)
kernelC23.146C24C2×C4×Q8C23.32C23C23.36C23C23.37C23C22.35C24D4×Q8C22.46C24C22.50C24Q8⋊3Q8Q8C4C22
# reps1123363634822

In GAP, Magma, Sage, TeX 

C_2^3._{146}C_2^4
 
% in TeX
 
G:=Group("C2^3.146C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,2255);
 
// by ID
 
G=gap.SmallGroup(128,2255);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,232,1430,570,136,1684,242]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=e^2=f^2=b,g^2=c*b=b*c,a*b=b*a,d*a*d^-1=a*c=c*a,a*e=e*a,a*f=f*a,a*g=g*a,e*d*e^-1=g*d*g^-1=b*d=d*b,f*e*f^-1=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*f=f*d,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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