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G = C23.502C24order 128 = 27

219th central stem extension by C23 of C24

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

Aliases: C23.502C24, C24.353C23, C22.2832+ 1+4, (C22×C4)⋊14D4, C23.186(C2×D4), C23.62(C4○D4), C232D4.14C2, C23.8Q879C2, C23.11D453C2, C23.23D463C2, C23.10D449C2, C2.20(C233D4), (C22×C4).123C23, (C2×C42).589C22, (C23×C4).132C22, C22.332(C22×D4), C24.3C2261C2, (C22×D4).538C22, C23.65C2398C2, C24.C22100C2, C2.75(C22.19C24), C23.63C23106C2, C2.68(C22.45C24), C2.C42.232C22, C2.45(C22.26C24), C2.31(C22.53C24), C2.78(C22.47C24), (C2×C4×D4)⋊50C2, (C4×C22⋊C4)⋊94C2, (C2×C4).1199(C2×D4), (C2×C4).161(C4○D4), (C2×C4⋊C4).342C22, C22.378(C2×C4○D4), (C2×C22.D4)⋊23C2, (C2×C22⋊C4).202C22, SmallGroup(128,1334)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.502C24
C1C2C22C23C22×C4C23×C4C2×C4×D4 — C23.502C24
C1C23 — C23.502C24
C1C23 — C23.502C24
C1C23 — C23.502C24

Generators and relations for C23.502C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=c, g2=db=bd, eae-1=ab=ba, ac=ca, faf=ad=da, ag=ga, bc=cb, geg-1=be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 596 in 291 conjugacy classes, 100 normal (82 characteristic)
C1, C2 [×7], C2 [×5], C4 [×17], C22 [×7], C22 [×27], C2×C4 [×10], C2×C4 [×39], D4 [×16], C23, C23 [×4], C23 [×19], C42 [×4], C22⋊C4 [×24], C4⋊C4 [×11], C22×C4 [×12], C22×C4 [×4], C22×C4 [×6], C2×D4 [×15], C24 [×3], C2.C42 [×8], C2×C42 [×3], C2×C22⋊C4 [×13], C2×C4⋊C4 [×6], C4×D4 [×4], C22.D4 [×4], C23×C4 [×2], C22×D4 [×3], C4×C22⋊C4, C23.8Q8, C23.23D4, C23.63C23, C24.C22 [×3], C23.65C23, C24.3C22, C232D4, C23.10D4 [×2], C23.11D4, C2×C4×D4, C2×C22.D4, C23.502C24
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], C22.19C24, C22.26C24, C233D4, C22.45C24, C22.47C24 [×2], C22.53C24, C23.502C24

Smallest permutation representation of C23.502C24
On 64 points
Generators in S64
(1 55)(2 52)(3 53)(4 50)(5 41)(6 35)(7 43)(8 33)(9 59)(10 46)(11 57)(12 48)(13 51)(14 56)(15 49)(16 54)(17 38)(18 29)(19 40)(20 31)(21 36)(22 44)(23 34)(24 42)(25 47)(26 58)(27 45)(28 60)(30 62)(32 64)(37 63)(39 61)
(1 13)(2 14)(3 15)(4 16)(5 23)(6 24)(7 21)(8 22)(9 27)(10 28)(11 25)(12 26)(17 64)(18 61)(19 62)(20 63)(29 39)(30 40)(31 37)(32 38)(33 44)(34 41)(35 42)(36 43)(45 59)(46 60)(47 57)(48 58)(49 53)(50 54)(51 55)(52 56)
(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 25)(2 26)(3 27)(4 28)(5 39)(6 40)(7 37)(8 38)(9 15)(10 16)(11 13)(12 14)(17 33)(18 34)(19 35)(20 36)(21 31)(22 32)(23 29)(24 30)(41 61)(42 62)(43 63)(44 64)(45 53)(46 54)(47 55)(48 56)(49 59)(50 60)(51 57)(52 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 2)(3 4)(5 24)(6 23)(7 22)(8 21)(9 10)(11 12)(13 14)(15 16)(17 36)(18 35)(19 34)(20 33)(25 26)(27 28)(29 40)(30 39)(31 38)(32 37)(41 62)(42 61)(43 64)(44 63)(45 50)(46 49)(47 52)(48 51)(53 60)(54 59)(55 58)(56 57)
(1 29 11 5)(2 40 12 24)(3 31 9 7)(4 38 10 22)(6 14 30 26)(8 16 32 28)(13 39 25 23)(15 37 27 21)(17 46 44 50)(18 57 41 55)(19 48 42 52)(20 59 43 53)(33 54 64 60)(34 51 61 47)(35 56 62 58)(36 49 63 45)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L4A···4H4I···4V4W4X4Y
order12···2222224···44···4444
size11···1444482···24···4888

38 irreducible representations

dim11111111111112224
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ 1+4
kernelC23.502C24C4×C22⋊C4C23.8Q8C23.23D4C23.63C23C24.C22C23.65C23C24.3C22C232D4C23.10D4C23.11D4C2×C4×D4C2×C22.D4C22×C4C2×C4C23C22
# reps111113111211141242

Matrix representation of C23.502C24 in GL6(𝔽5)

220000
130000
001000
000100
000024
000033
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
400000
040000
001000
000100
000010
000001
,
400000
040000
000400
001000
000040
000011
,
400000
210000
000100
001000
000040
000004
,
200000
020000
001000
000100
000012
000044

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

C23.502C24 in GAP, Magma, Sage, TeX

C_2^3._{502}C_2^4
% in TeX

G:=Group("C2^3.502C2^4");
// GroupNames label

G:=SmallGroup(128,1334);
// by ID

G=gap.SmallGroup(128,1334);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,758,723,675,304]);
// Polycyclic

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