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G = C24.399C23  order 128 = 27

239th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.399C23, C23.592C24, C22.2722- (1+4), C22.3662+ (1+4), C4⋊C4⋊13D4, (C2×D4)⋊5Q8, C2.28(D4×Q8), C23.33(C2×Q8), C23⋊Q8⋊44C2, C2.97(D4⋊5D4), C2.49(D4⋊3Q8), C23.7Q8⋊86C2, C23.Q8⋊57C2, C23.4Q8⋊43C2, (C22×C4).872C23, (C23×C4).456C22, (C2×C42).646C22, C23.8Q8⋊104C2, C2.14(C23⋊2Q8), C22.401(C22×D4), C22.144(C22×Q8), C23.23D4.51C2, (C22×D4).229C22, (C22×Q8).183C22, C23.67C23⋊80C2, C23.78C23⋊44C2, C23.81C23⋊83C2, C2.57(C22.29C24), C24.3C22.60C2, C23.63C23⋊132C2, C2.C42.299C22, C2.72(C22.36C24), C2.13(C22.56C24), (C2×C4).96(C2×D4), (C2×C4).68(C2×Q8), (C2×C22⋊Q8)⋊39C2, (C2×C4).423(C4○D4), (C2×C4⋊C4).406C22, C22.454(C2×C4○D4), (C2×C22⋊C4).259C22, SmallGroup(128,1424)

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

Derived series
C1 — C23 — C24.399C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C23.7Q8 — C24.399C23
Lower central
C1 — C23 — C24.399C23
Upper central
C1 — C23 — C24.399C23
Jennings
C1 — C23 — C24.399C23

Subgroups: 548 in 268 conjugacy classes, 104 normal (82 characteristic)
C1, C2 [×7], C2 [×4], C4 [×18], C22 [×7], C22 [×20], C2×C4 [×10], C2×C4 [×42], D4 [×4], Q8 [×8], C23, C23 [×4], C23 [×12], C42, C22⋊C4 [×15], C4⋊C4 [×4], C4⋊C4 [×12], C22×C4 [×13], C22×C4 [×9], C2×D4 [×4], C2×D4 [×2], C2×Q8 [×9], C24 [×2], C2.C42 [×10], C2×C42, C2×C22⋊C4 [×10], C2×C4⋊C4 [×9], C22⋊Q8 [×8], C23×C4 [×2], C22×D4, C22×Q8 [×2], C23.7Q8, C23.8Q8, C23.23D4 [×2], C23.63C23, C24.3C22, C23.67C23, C23⋊Q8 [×2], C23.78C23, C23.Q8, C23.81C23, C23.4Q8, C2×C22⋊Q8 [×2], C24.399C23

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, C22×D4, C22×Q8, C2×C4○D4, 2+ (1+4) [×3], 2- (1+4), C22.29C24, C22.36C24, C23⋊2Q8, D4⋊5D4, D4×Q8, D4⋊3Q8, C22.56C24, C24.399C23

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

Smallest permutation representation
►On 64 points
Generators in S64
(5 22)(6 23)(7 24)(8 21)(13 47)(14 48)(15 45)(16 46)(17 34)(18 35)(19 36)(20 33)(29 57)(30 58)(31 59)(32 60)(37 44)(38 41)(39 42)(40 43)(49 61)(50 62)(51 63)(52 64)

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

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
010000
001000
000400
000010
000004
,
400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
020000
200000
002000
000200
000040
000001
,
100000
010000
000400
004000
000001
000010
,
010000
400000
000400
001000
000040
000001

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4N4O···4T
order12···222224···44···4
size11···144444···48···8

32 irreducible representations

dim111111111111122244
type++++++++++++++-+-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2D4Q8C4○D42+ (1+4)2- (1+4)
kernelC24.399C23C23.7Q8C23.8Q8C23.23D4C23.63C23C24.3C22C23.67C23C23⋊Q8C23.78C23C23.Q8C23.81C23C23.4Q8C2×C22⋊Q8C4⋊C4C2×D4C2×C4C22C22
# reps111211121111244431

In GAP, Magma, Sage, TeX 

C_2^4._{399}C_2^3
 
% in TeX
 
G:=Group("C2^4.399C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1424);
 
// by ID
 
G=gap.SmallGroup(128,1424);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,344,758,723,1571,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=g^2=c*b=b*c,a*b=b*a,g*a*g^-1=a*c=c*a,a*d=d*a,a*e=e*a,f*a*f=a*c*d,b*d=d*b,g*e*g^-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,f*e*f=d*e=e*d,d*f=f*d,d*g=g*d,g*f*g^-1=c*d*f>;
 
// generators/relations
 

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