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G = C24.332C23order 128 = 27

172nd non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.332C23, C23.460C24, C22.1872- 1+4, C22.2452+ 1+4, C4⋊C429D4, C23⋊Q818C2, C2.80(D45D4), C2.41(Q85D4), C23.53(C4○D4), C23.11D445C2, (C2×C42).563C22, (C22×C4).542C23, (C23×C4).401C22, C22.311(C22×D4), C24.C2286C2, C23.10D4.21C2, C23.23D4.35C2, (C22×D4).174C22, (C22×Q8).138C22, C23.67C2364C2, C23.63C2387C2, C23.81C2341C2, C24.3C22.47C2, C2.49(C22.45C24), C2.C42.197C22, C2.40(C22.26C24), C2.39(C22.50C24), C2.27(C22.33C24), C2.82(C23.36C23), (C4×C4⋊C4)⋊95C2, (C2×C4).82(C2×D4), (C4×C22⋊C4)⋊90C2, (C2×C22⋊Q8)⋊23C2, (C2×C422C2)⋊12C2, (C2×C4).391(C4○D4), (C2×C4⋊C4).874C22, C22.336(C2×C4○D4), (C2×C22⋊C4).183C22, SmallGroup(128,1292)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.332C23
C1C2C22C23C22×C4C23×C4C4×C22⋊C4 — C24.332C23
C1C23 — C24.332C23
C1C23 — C24.332C23
C1C23 — C24.332C23

Generators and relations for C24.332C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=b, f2=dc=cd, g2=cb=bc, ab=ba, eae-1=ac=ca, faf-1=ad=da, ag=ga, bd=db, fef-1=geg-1=be=eb, bf=fb, bg=gb, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 500 in 259 conjugacy classes, 100 normal (82 characteristic)
C1, C2 [×7], C2 [×3], C4 [×19], C22 [×7], C22 [×17], C2×C4 [×12], C2×C4 [×37], D4 [×4], Q8 [×4], C23, C23 [×2], C23 [×13], C42 [×6], C22⋊C4 [×21], C4⋊C4 [×4], C4⋊C4 [×12], C22×C4 [×13], C22×C4 [×5], C2×D4 [×5], C2×Q8 [×5], C24 [×2], C2.C42 [×10], C2×C42 [×4], C2×C22⋊C4 [×11], C2×C4⋊C4 [×7], C22⋊Q8 [×4], C422C2 [×4], C23×C4, C22×D4, C22×Q8, C4×C22⋊C4, C4×C4⋊C4, C23.23D4, C23.63C23, C24.C22 [×3], C24.3C22, C23.67C23, C23⋊Q8, C23.10D4, C23.11D4, C23.81C23, C2×C22⋊Q8, C2×C422C2, C24.332C23
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, D45D4, Q85D4, C22.45C24, C22.50C24, C24.332C23

Smallest permutation representation of C24.332C23
On 64 points
Generators in S64
(1 9)(2 48)(3 11)(4 46)(5 30)(6 44)(7 32)(8 42)(10 23)(12 21)(13 35)(14 17)(15 33)(16 19)(18 57)(20 59)(22 47)(24 45)(25 56)(26 62)(27 54)(28 64)(29 50)(31 52)(34 58)(36 60)(37 55)(38 61)(39 53)(40 63)(41 49)(43 51)
(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 22)(2 23)(3 24)(4 21)(5 51)(6 52)(7 49)(8 50)(9 47)(10 48)(11 45)(12 46)(13 59)(14 60)(15 57)(16 58)(17 36)(18 33)(19 34)(20 35)(25 38)(26 39)(27 40)(28 37)(29 42)(30 43)(31 44)(32 41)(53 62)(54 63)(55 64)(56 61)
(1 63)(2 64)(3 61)(4 62)(5 36)(6 33)(7 34)(8 35)(9 40)(10 37)(11 38)(12 39)(13 42)(14 43)(15 44)(16 41)(17 51)(18 52)(19 49)(20 50)(21 53)(22 54)(23 55)(24 56)(25 45)(26 46)(27 47)(28 48)(29 59)(30 60)(31 57)(32 58)
(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 10 54 28)(2 9 55 27)(3 12 56 26)(4 11 53 25)(5 44 17 57)(6 43 18 60)(7 42 19 59)(8 41 20 58)(13 49 29 34)(14 52 30 33)(15 51 31 36)(16 50 32 35)(21 45 62 38)(22 48 63 37)(23 47 64 40)(24 46 61 39)
(1 20 24 33)(2 19 21 36)(3 18 22 35)(4 17 23 34)(5 64 49 53)(6 63 50 56)(7 62 51 55)(8 61 52 54)(9 59 45 15)(10 58 46 14)(11 57 47 13)(12 60 48 16)(25 44 40 29)(26 43 37 32)(27 42 38 31)(28 41 39 30)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J4A···4H4I···4X4Y4Z4AA
order12···22224···44···4444
size11···14482···24···4888

38 irreducible representations

dim1111111111111122244
type++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ 1+42- 1+4
kernelC24.332C23C4×C22⋊C4C4×C4⋊C4C23.23D4C23.63C23C24.C22C24.3C22C23.67C23C23⋊Q8C23.10D4C23.11D4C23.81C23C2×C22⋊Q8C2×C422C2C4⋊C4C2×C4C23C22C22
# reps11111311111111412411

Matrix representation of C24.332C23 in GL6(𝔽5)

010000
100000
004000
000100
000042
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000040
000004
,
400000
040000
004000
000400
000010
000001
,
100000
010000
000400
004000
000020
000023
,
040000
100000
000100
001000
000021
000003
,
400000
040000
003000
000300
000013
000004

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

C24.332C23 in GAP, Magma, Sage, TeX

C_2^4._{332}C_2^3
% in TeX

G:=Group("C2^4.332C2^3");
// GroupNames label

G:=SmallGroup(128,1292);
// by ID

G=gap.SmallGroup(128,1292);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,680,758,723,675,192]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=b,f^2=d*c=c*d,g^2=c*b=b*c,a*b=b*a,e*a*e^-1=a*c=c*a,f*a*f^-1=a*d=d*a,a*g=g*a,b*d=d*b,f*e*f^-1=g*e*g^-1=b*e=e*b,b*f=f*b,b*g=g*b,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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