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G = C24.422C23  order 128 = 27

262nd non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.422C23, C23.630C24, C22.4032+ (1+4), C22.3052- (1+4), (C2×Q8)⋊13D4, C2.34(Q8⋊6D4), C2.56(Q8⋊5D4), C23.86(C4○D4), C23⋊2D4.27C2, C23.Q8⋊72C2, C23.4Q8⋊52C2, C23.10D4⋊97C2, C2.54(C23⋊3D4), (C2×C42).681C22, (C23×C4).477C22, (C22×C4).199C23, C22.439(C22×D4), C23.11D4⋊101C2, C23.23D4⋊100C2, C24.3C22⋊89C2, (C22×D4).256C22, (C22×Q8).199C22, C23.67C23⋊91C2, C24.C22⋊147C2, C2.75(C22.32C24), C23.63C23⋊151C2, C2.86(C22.45C24), C2.C42.336C22, C2.23(C22.56C24), C2.86(C22.36C24), (C2×C4).124(C2×D4), (C2×C22⋊Q8)⋊45C2, (C2×C4.4D4)⋊31C2, (C2×C4).210(C4○D4), (C2×C4⋊C4).443C22, C22.492(C2×C4○D4), (C2×C22⋊C4).293C22, SmallGroup(128,1462)

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

Derived series
C1 — C23 — C24.422C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C24.C22 — C24.422C23
Lower central
C1 — C23 — C24.422C23
Upper central
C1 — C23 — C24.422C23
Jennings
C1 — C23 — C24.422C23

Subgroups: 596 in 274 conjugacy classes, 96 normal (82 characteristic)
C1, C2 [×7], C2 [×4], C4 [×16], C22 [×7], C22 [×24], C2×C4 [×8], C2×C4 [×36], D4 [×12], Q8 [×4], C23, C23 [×2], C23 [×20], C42 [×3], C22⋊C4 [×22], C4⋊C4 [×10], C22×C4 [×12], C22×C4 [×4], C2×D4 [×13], C2×Q8 [×4], C2×Q8 [×2], C24 [×3], C2.C42 [×8], C2×C42 [×2], C2×C22⋊C4 [×14], C2×C4⋊C4 [×6], C22⋊Q8 [×4], C4.4D4 [×4], C23×C4, C22×D4 [×3], C22×Q8, C23.23D4 [×2], C23.63C23, C24.C22 [×2], C24.3C22, C23.67C23, C23⋊2D4, C23.10D4 [×2], C23.Q8, C23.11D4, C23.4Q8, C2×C22⋊Q8, C2×C4.4D4, C24.422C23

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], 2+ (1+4) [×3], 2- (1+4), C23⋊3D4, C22.32C24, C22.36C24, Q8⋊5D4, Q8⋊6D4, C22.45C24, C22.56C24, C24.422C23

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=g2=cb=bc, f2=b, eae-1=gag-1=ab=ba, ac=ca, faf-1=ad=da, bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, geg-1=bce, fg=gf >

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

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

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

220000
130000
001000
000100
000001
000010
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
400000
040000
001000
000100
000010
000001
,
400000
040000
000200
002000
000030
000002
,
400000
210000
000300
002000
000020
000002
,
100000
010000
000100
004000
000001
000040

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4P4Q4R4S4T
order12···222224···44444
size11···144884···48888

32 irreducible representations

dim111111111111122244
type+++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ (1+4)2- (1+4)
kernelC24.422C23C23.23D4C23.63C23C24.C22C24.3C22C23.67C23C23⋊2D4C23.10D4C23.Q8C23.11D4C23.4Q8C2×C22⋊Q8C2×C4.4D4C2×Q8C2×C4C23C22C22
# reps121211121111144431

In GAP, Magma, Sage, TeX 

C_2^4._{422}C_2^3
 
% in TeX
 
G:=Group("C2^4.422C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1462);
 
// by ID
 
G=gap.SmallGroup(128,1462);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,758,723,100,1571,346,192]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=g^2=c*b=b*c,f^2=b,e*a*e^-1=g*a*g^-1=a*b=b*a,a*c=c*a,f*a*f^-1=a*d=d*a,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f^-1=c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,g*e*g^-1=b*c*e,f*g=g*f>;
 
// generators/relations
 

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