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G = C24.269C23order 128 = 27

109th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.269C23, C23.343C24, C22.1122- 1+4, C22.1522+ 1+4, C2.17D42, C4⋊C439D4, (C2×D4).286D4, C23.164(C2×D4), C2.24(D46D4), C23.4Q85C2, C2.20(Q85D4), (C23×C4).75C22, C23.8Q838C2, C23.231(C4○D4), C23.10D423C2, C23.23D438C2, C23.11D411C2, (C2×C42).486C22, (C22×C4).511C23, C22.223(C22×D4), C24.C2237C2, (C22×D4).130C22, C223(C22.D4), C23.81C2312C2, C23.63C2334C2, C2.25(C22.19C24), C2.C42.100C22, C2.18(C22.47C24), C2.12(C22.33C24), (C2×C4×D4)⋊30C2, (C22×C4⋊C4)⋊20C2, (C2×C4).325(C2×D4), (C2×C4⋊D4).28C2, (C2×C4).362(C4○D4), (C2×C4⋊C4).225C22, C22.220(C2×C4○D4), (C2×C22.D4)⋊10C2, C2.21(C2×C22.D4), (C2×C22⋊C4).125C22, SmallGroup(128,1175)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.269C23
C1C2C22C23C24C23×C4C22×C4⋊C4 — C24.269C23
C1C23 — C24.269C23
C1C23 — C24.269C23
C1C23 — C24.269C23

Generators and relations for C24.269C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=g2=1, e2=f2=b, ab=ba, ac=ca, ede-1=gdg=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 660 in 337 conjugacy classes, 112 normal (82 characteristic)
C1, C2 [×7], C2 [×7], C4 [×17], C22 [×7], C22 [×4], C22 [×29], C2×C4 [×10], C2×C4 [×51], D4 [×16], C23, C23 [×8], C23 [×17], C42 [×2], C22⋊C4 [×21], C4⋊C4 [×4], C4⋊C4 [×13], C22×C4 [×12], C22×C4 [×21], C2×D4 [×4], C2×D4 [×15], C24 [×3], C2.C42 [×8], C2×C42, C2×C22⋊C4 [×11], C2×C4⋊C4 [×8], C2×C4⋊C4 [×4], C4×D4 [×4], C4⋊D4 [×4], C22.D4 [×4], C23×C4 [×4], C22×D4 [×3], C23.8Q8 [×2], C23.23D4 [×3], C23.63C23, C24.C22, C23.10D4, C23.11D4, C23.81C23, C23.4Q8, C22×C4⋊C4, C2×C4×D4, C2×C4⋊D4, C2×C22.D4, C24.269C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×6], C24, C22.D4 [×4], C22×D4 [×2], C2×C4○D4 [×3], 2+ 1+4, 2- 1+4, C2×C22.D4, C22.19C24, C22.33C24, D42, D46D4, Q85D4, C22.47C24, C24.269C23

Smallest permutation representation of C24.269C23
On 64 points
Generators in S64
(1 39)(2 40)(3 37)(4 38)(5 17)(6 18)(7 19)(8 20)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(21 49)(22 50)(23 51)(24 52)(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 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 51)(2 52)(3 49)(4 50)(5 36)(6 33)(7 34)(8 35)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)
(1 15)(2 48)(3 13)(4 46)(5 9)(6 42)(7 11)(8 44)(10 18)(12 20)(14 38)(16 40)(17 41)(19 43)(21 29)(22 58)(23 31)(24 60)(25 61)(26 33)(27 63)(28 35)(30 50)(32 52)(34 55)(36 53)(37 45)(39 47)(49 57)(51 59)(54 62)(56 64)
(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 13 3 15)(2 58 4 60)(5 41 7 43)(6 26 8 28)(9 19 11 17)(10 64 12 62)(14 50 16 52)(18 54 20 56)(21 31 23 29)(22 48 24 46)(25 34 27 36)(30 38 32 40)(33 42 35 44)(37 47 39 45)(49 59 51 57)(53 63 55 61)
(1 11)(2 12)(3 9)(4 10)(5 45)(6 46)(7 47)(8 48)(13 17)(14 18)(15 19)(16 20)(21 25)(22 26)(23 27)(24 28)(29 36)(30 33)(31 34)(32 35)(37 41)(38 42)(39 43)(40 44)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N4A4B4C4D4E···4T4U4V4W
order12···2222222244444···4444
size11···1222244822224···4888

38 irreducible representations

dim1111111111111222244
type++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D4C4○D42+ 1+42- 1+4
kernelC24.269C23C23.8Q8C23.23D4C23.63C23C24.C22C23.10D4C23.11D4C23.81C23C23.4Q8C22×C4⋊C4C2×C4×D4C2×C4⋊D4C2×C22.D4C4⋊C4C2×D4C2×C4C23C22C22
# reps1231111111111444811

Matrix representation of C24.269C23 in GL6(𝔽5)

400000
040000
001000
000100
000040
000004
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
400000
010000
004000
000400
000002
000030
,
040000
400000
001300
000400
000003
000030
,
400000
040000
004000
004100
000002
000020
,
010000
100000
001000
000100
000001
000010

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

C24.269C23 in GAP, Magma, Sage, TeX

C_2^4._{269}C_2^3
% in TeX

G:=Group("C2^4.269C2^3");
// GroupNames label

G:=SmallGroup(128,1175);
// by ID

G=gap.SmallGroup(128,1175);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,232,758,723,184,675]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=g^2=1,e^2=f^2=b,a*b=b*a,a*c=c*a,e*d*e^-1=g*d*g=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,f*d*f^-1=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f^-1=c*e=e*c,c*f=f*c,c*g=g*c,e*g=g*e,f*g=g*f>;
// generators/relations

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