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G = C24.326C23  order 128 = 27

166th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.326C23, C23.453C24, C22.2382+ (1+4), C22.1842- (1+4), C22⋊C4.76D4, C23.52(C2×D4), C42⋊8C4⋊42C2, C2.53(D4⋊6D4), C2.73(D4⋊5D4), C23.8Q8⋊65C2, C23.4Q8⋊22C2, C23.7Q8⋊68C2, C23.152(C4○D4), C23.11D4⋊43C2, (C2×C42).558C22, (C23×C4).398C22, (C22×C4).539C23, C22.304(C22×D4), C24.C22⋊80C2, C23.10D4.19C2, C23.23D4.34C2, (C22×D4).168C22, C23.63C23⋊84C2, C23.65C23⋊86C2, C23.81C23⋊38C2, C2.46(C22.45C24), C2.C42.190C22, C2.33(C22.26C24), C2.24(C22.33C24), C2.61(C22.46C24), C2.77(C23.36C23), (C4×C22⋊C4)⋊84C2, (C2×C4).906(C2×D4), (C2×C42⋊2C2)⋊9C2, (C2×C4).386(C4○D4), (C2×C4⋊C4).306C22, C22.329(C2×C4○D4), (C2×C22⋊C4).505C22, (C2×C22.D4).17C2, SmallGroup(128,1285)

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

Derived series
C1 — C23 — C24.326C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C4×C22⋊C4 — C24.326C23
Lower central
C1 — C23 — C24.326C23
Upper central
C1 — C23 — C24.326C23
Jennings
C1 — C23 — C24.326C23

Subgroups: 500 in 262 conjugacy classes, 100 normal (82 characteristic)
C1, C2 [×7], C2 [×4], C4 [×18], C22 [×7], C22 [×20], C2×C4 [×10], C2×C4 [×42], D4 [×4], C23, C23 [×4], C23 [×12], C42 [×5], C22⋊C4 [×4], C22⋊C4 [×18], C4⋊C4 [×16], C22×C4 [×13], C22×C4 [×10], C2×D4 [×5], C24 [×2], C2.C42 [×10], C2×C42 [×3], C2×C22⋊C4 [×10], C2×C4⋊C4 [×9], C22.D4 [×4], C42⋊2C2 [×4], C23×C4 [×2], C22×D4, C4×C22⋊C4 [×2], C23.7Q8, C42⋊8C4, C23.8Q8, C23.23D4, C23.63C23, C24.C22, C23.65C23, C23.10D4, C23.11D4, C23.81C23, C23.4Q8, C2×C22.D4, C2×C42⋊2C2, C24.326C23

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), C23.36C23, C22.26C24, C22.33C24, D4⋊5D4, D4⋊6D4, C22.45C24, C22.46C24, C24.326C23

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

Smallest permutation representation
►On 64 points
Generators in S64
(2 9)(4 11)(5 42)(6 60)(7 44)(8 58)(14 26)(16 28)(17 54)(18 46)(19 56)(20 48)(21 57)(22 41)(23 59)(24 43)(30 39)(32 37)(34 51)(36 49)(45 62)(47 64)(53 61)(55 63)

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

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

(1 12)(2 9)(3 10)(4 11)(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 38)(30 39)(31 40)(32 37)(33 50)(34 51)(35 52)(36 49)(41 58)(42 59)(43 60)(44 57)(45 54)(46 55)(47 56)(48 53)

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
010000
001000
002400
000010
000014
,
400000
040000
004000
000400
000010
000001
,
100000
010000
004000
000400
000040
000004
,
100000
010000
001000
000100
000040
000004
,
010000
100000
003200
001200
000034
000032
,
200000
030000
002000
004300
000013
000014
,
400000
010000
002000
004300
000030
000003

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

38 conjugacy classes

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

38 irreducible representations

dim11111111111111122244
type+++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ (1+4)2- (1+4)
kernelC24.326C23C4×C22⋊C4C23.7Q8C42⋊8C4C23.8Q8C23.23D4C23.63C23C24.C22C23.65C23C23.10D4C23.11D4C23.81C23C23.4Q8C2×C22.D4C2×C42⋊2C2C22⋊C4C2×C4C23C22C22
# reps121111111111111412411

In GAP, Magma, Sage, TeX 

C_2^4._{326}C_2^3
 
% in TeX
 
G:=Group("C2^4.326C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1285);
 
// by ID
 
G=gap.SmallGroup(128,1285);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,568,758,723,675,136]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=1,f^2=d*b=b*d,g^2=c,a*b=b*a,e*a*e=a*c=c*a,f*a*f^-1=a*d=d*a,a*g=g*a,b*c=c*b,f*e*f^-1=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,d*e=e*d,d*f=f*d,d*g=g*d,f*g=g*f>;
 
// generators/relations
 

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