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G = C24.286C23  order 128 = 27

126th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.286C23, C23.366C24, C22.1272- (1+4), C22.1722+ (1+4), C22⋊C4.129D4, C23.174(C2×D4), C2.31(D4⋊6D4), C2.49(D4⋊5D4), C23.34(C4○D4), (C23×C4).90C22, C23.8Q8⋊52C2, C23.Q8⋊17C2, C23.11D4⋊16C2, C23.34D4⋊27C2, (C22×C4).819C23, (C2×C42).509C22, C23.10D4.6C2, C22.246(C22×D4), C24.C22⋊49C2, (C22×D4).520C22, C23.63C23⋊45C2, C23.65C23⋊60C2, C2.38(C22.19C24), C2.C42.123C22, C2.17(C22.33C24), C2.24(C22.47C24), C2.16(C22.50C24), C2.34(C23.36C23), (C2×C4×D4).53C2, (C4×C22⋊C4)⋊65C2, (C2×C4).897(C2×D4), (C2×C42⋊2C2)⋊1C2, (C2×C4).116(C4○D4), (C2×C4⋊C4).246C22, C22.243(C2×C4○D4), (C2×C22⋊C4).140C22, SmallGroup(128,1198)

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

Derived series
C1 — C23 — C24.286C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C22⋊C4 — C24.286C23
Lower central
C1 — C23 — C24.286C23
Upper central
C1 — C23 — C24.286C23
Jennings
C1 — C23 — C24.286C23

Subgroups: 500 in 266 conjugacy classes, 100 normal (82 characteristic)
C1, C2 [×7], C2 [×4], C4 [×18], C22 [×7], C22 [×20], C2×C4 [×10], C2×C4 [×42], D4 [×8], C23, C23 [×4], C23 [×12], C42 [×5], C22⋊C4 [×4], C22⋊C4 [×16], C4⋊C4 [×17], C22×C4 [×13], C22×C4 [×10], C2×D4 [×6], C24 [×2], C2.C42 [×10], C2×C42 [×3], C2×C22⋊C4 [×10], C2×C4⋊C4 [×9], C4×D4 [×4], C42⋊2C2 [×4], C23×C4 [×2], C22×D4, C4×C22⋊C4, C23.34D4, C23.8Q8 [×2], C23.63C23 [×2], C24.C22, C23.65C23 [×2], C23.10D4 [×2], C23.Q8, C23.11D4, C2×C4×D4, C2×C42⋊2C2, C24.286C23

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×8], C24, C22×D4, C2×C4○D4 [×4], 2+ (1+4), 2- (1+4), C22.19C24, C23.36C23, C22.33C24, D4⋊5D4, D4⋊6D4, C22.47C24, C22.50C24, C24.286C23

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

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

(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 9)(2 10)(3 11)(4 12)(5 23)(6 24)(7 21)(8 22)(13 25)(14 26)(15 27)(16 28)(17 62)(18 63)(19 64)(20 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 59)(46 60)(47 57)(48 58)

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

(2 30)(4 32)(5 55)(6 8)(7 53)(10 34)(12 36)(14 38)(16 40)(17 58)(18 20)(19 60)(21 43)(22 24)(23 41)(26 50)(28 52)(42 44)(45 47)(46 64)(48 62)(54 56)(57 59)(61 63)

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

Matrix representation ►G ⊆ GL6(𝔽5)

400000
040000
004000
000100
000022
000013
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
400000
040000
004000
000400
000010
000001
,
100000
040000
001000
000400
000010
000034
,
010000
100000
000100
001000
000030
000003
,
100000
010000
002000
000200
000020
000013

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4V4W4X4Y4Z
order12···222224···44···44444
size11···144442···24···48888

38 irreducible representations

dim11111111111122244
type++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ (1+4)2- (1+4)
kernelC24.286C23C4×C22⋊C4C23.34D4C23.8Q8C23.63C23C24.C22C23.65C23C23.10D4C23.Q8C23.11D4C2×C4×D4C2×C42⋊2C2C22⋊C4C2×C4C23C22C22
# reps111221221111412411

In GAP, Magma, Sage, TeX 

C_2^4._{286}C_2^3
 
% in TeX
 
G:=Group("C2^4.286C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1198);
 
// by ID
 
G=gap.SmallGroup(128,1198);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,758,723,100,675,192]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=1,f^2=b,g^2=c*b=b*c,e*a*e=g*a*g^-1=a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,b*d=d*b,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^-1=d*e=e*d,d*f=f*d,d*g=g*d,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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