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G = C23.374C24order 128 = 27

91st central stem extension by C23 of C24

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

Aliases: C23.374C24, C24.291C23, C22.1792+ 1+4, C22.1322- 1+4, C22⋊C4.132D4, C23.178(C2×D4), C2.35(D46D4), C2.56(D45D4), (C2×C42).33C22, C23.Q820C2, C23.8Q855C2, C23.11D420C2, (C22×C4).518C23, (C23×C4).362C22, C23.10D4.9C2, C22.254(C22×D4), C24.C2255C2, C23.23D4.21C2, (C22×D4).140C22, (C22×Q8).112C22, C23.63C2352C2, C23.78C2313C2, C23.67C2346C2, C2.46(C22.19C24), C2.C42.130C22, C2.18(C22.50C24), C2.40(C23.36C23), C2.10(C22.49C24), C2.25(C22.36C24), (C4×C4⋊C4)⋊63C2, (C2×C4).901(C2×D4), (C2×C42⋊C2)⋊24C2, (C2×C4).369(C4○D4), (C2×C4⋊C4).252C22, (C2×C4.4D4).22C2, C22.251(C2×C4○D4), (C2×C22⋊C4).458C22, SmallGroup(128,1206)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.374C24
C1C2C22C23C22×C4C2×C42C4×C4⋊C4 — C23.374C24
C1C23 — C23.374C24
C1C23 — C23.374C24
C1C23 — C23.374C24

Generators and relations for C23.374C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=cb=bc, f2=b, g2=c, eae-1=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 >

Subgroups: 500 in 263 conjugacy classes, 100 normal (82 characteristic)
C1, C2 [×7], C2 [×3], C4 [×19], C22 [×7], C22 [×17], C2×C4 [×12], C2×C4 [×41], D4 [×4], Q8 [×4], C23, C23 [×2], C23 [×13], C42 [×8], C22⋊C4 [×4], C22⋊C4 [×18], C4⋊C4 [×12], C22×C4 [×13], C22×C4 [×6], C2×D4 [×5], C2×Q8 [×5], C24 [×2], C2.C42 [×10], C2×C42 [×4], C2×C22⋊C4 [×11], C2×C4⋊C4 [×7], C42⋊C2 [×4], C4.4D4 [×4], C23×C4, C22×D4, C22×Q8, C4×C4⋊C4, C23.8Q8, C23.23D4, C23.63C23, C24.C22 [×4], C23.67C23, C23.10D4, C23.78C23, C23.Q8, C23.11D4, C2×C42⋊C2, C2×C4.4D4, C23.374C24
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, C22.36C24, D45D4, D46D4, C22.49C24, C22.50C24, C23.374C24

Smallest permutation representation of C23.374C24
On 64 points
Generators in S64
(1 41)(2 24)(3 43)(4 22)(5 28)(6 55)(7 26)(8 53)(9 23)(10 42)(11 21)(12 44)(13 17)(14 50)(15 19)(16 52)(18 46)(20 48)(25 38)(27 40)(29 36)(30 64)(31 34)(32 62)(33 60)(35 58)(37 54)(39 56)(45 49)(47 51)(57 61)(59 63)
(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 17 9 49)(2 64 10 33)(3 19 11 51)(4 62 12 35)(5 14 37 46)(6 57 38 31)(7 16 39 48)(8 59 40 29)(13 43 45 21)(15 41 47 23)(18 56 50 26)(20 54 52 28)(22 60 44 30)(24 58 42 32)(25 63 55 36)(27 61 53 34)
(1 13 11 47)(2 14 12 48)(3 15 9 45)(4 16 10 46)(5 62 39 33)(6 63 40 34)(7 64 37 35)(8 61 38 36)(17 43 51 23)(18 44 52 24)(19 41 49 21)(20 42 50 22)(25 59 53 31)(26 60 54 32)(27 57 55 29)(28 58 56 30)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim11111111111112244
type+++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2D4C4○D42+ 1+42- 1+4
kernelC23.374C24C4×C4⋊C4C23.8Q8C23.23D4C23.63C23C24.C22C23.67C23C23.10D4C23.78C23C23.Q8C23.11D4C2×C42⋊C2C2×C4.4D4C22⋊C4C2×C4C22C22
# reps111114111111141611

Matrix representation of C23.374C24 in GL6(𝔽5)

030000
200000
000300
002000
000040
000001
,
100000
010000
001000
000100
000040
000004
,
400000
040000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
040000
100000
003000
000300
000003
000020
,
040000
400000
000100
001000
000001
000040
,
300000
030000
003000
000300
000001
000040

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

C23.374C24 in GAP, Magma, Sage, TeX

C_2^3._{374}C_2^4
% in TeX

G:=Group("C2^3.374C2^4");
// GroupNames label

G:=SmallGroup(128,1206);
// by ID

G=gap.SmallGroup(128,1206);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,344,758,723,100,675,136]);
// Polycyclic

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