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G = C24.390C23  order 128 = 27

230th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.390C23, C23.583C24, C22.3572+ (1+4), C22.2662- (1+4), C4⋊C4⋊12D4, (C2×D4)⋊4Q8, C2.47(D42), C2.27(D4×Q8), C23.32(C2×Q8), C23.Q8⋊54C2, (C22×C4).869C23, (C2×C42).640C22, (C23×C4).451C22, C2.13(C23⋊2Q8), C22.392(C22×D4), C22.143(C22×Q8), C23.23D4.49C2, (C22×D4).222C22, (C22×Q8).178C22, C23.78C23⋊40C2, C24.3C22.59C2, C23.65C23⋊117C2, C2.C42.292C22, C2.10(C22.57C24), C2.42(C22.31C24), (C2×C4).91(C2×D4), (C2×C4).67(C2×Q8), (C2×C22⋊Q8)⋊37C2, (C2×C4⋊C4).399C22, (C2×C22⋊C4).252C22, SmallGroup(128,1415)

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

Derived series
C1 — C23 — C24.390C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C22⋊Q8 — C24.390C23
Lower central
C1 — C23 — C24.390C23
Upper central
C1 — C23 — C24.390C23
Jennings
C1 — C23 — C24.390C23

Subgroups: 580 in 296 conjugacy classes, 112 normal (14 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×20], C22 [×3], C22 [×4], C22 [×20], C2×C4 [×14], C2×C4 [×40], D4 [×4], Q8 [×8], C23, C23 [×4], C23 [×12], C42, C22⋊C4 [×18], C4⋊C4 [×8], C4⋊C4 [×19], C22×C4, C22×C4 [×12], C22×C4 [×10], C2×D4 [×4], C2×D4 [×2], C2×Q8 [×10], C24 [×2], C2.C42 [×6], C2×C42, C2×C22⋊C4 [×10], C2×C4⋊C4, C2×C4⋊C4 [×12], C22⋊Q8 [×16], C23×C4 [×2], C22×D4, C22×Q8 [×2], C23.23D4 [×2], C23.65C23 [×2], C24.3C22, C23.78C23 [×2], C23.Q8 [×4], C2×C22⋊Q8 [×4], C24.390C23

Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], Q8 [×4], C23 [×15], C2×D4 [×12], C2×Q8 [×6], C24, C22×D4 [×2], C22×Q8, 2+ (1+4) [×2], 2- (1+4) [×2], C22.31C24 [×2], C23⋊2Q8, D42, D4×Q8 [×2], C22.57C24, C24.390C23

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=b, 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 9)(6 10)(7 11)(8 12)(17 47)(18 48)(19 45)(20 46)(21 34)(22 35)(23 36)(24 33)(29 57)(30 58)(31 59)(32 60)(37 42)(38 43)(39 44)(40 41)(49 63)(50 64)(51 61)(52 62)

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
010000
001000
000400
000010
000004
,
400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
010000
400000
004000
000100
000040
000004
,
100000
010000
000200
003000
000004
000040
,
300000
020000
004000
000100
000004
000010

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[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],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,4,0] >;
 

32 conjugacy classes

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

32 irreducible representations

dim11111112244
type++++++++-+-
imageC1C2C2C2C2C2C2D4Q82+ (1+4)2- (1+4)
kernelC24.390C23C23.23D4C23.65C23C24.3C22C23.78C23C23.Q8C2×C22⋊Q8C4⋊C4C2×D4C22C22
# reps12212448422

In GAP, Magma, Sage, TeX 

C_2^4._{390}C_2^3
 
% in TeX
 
G:=Group("C2^4.390C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1415);
 
// by ID
 
G=gap.SmallGroup(128,1415);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,120,758,723,1571,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=b,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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