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G = C23.215C24order 128 = 27

68th central extension by C23 of C24

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

Aliases: C23.215C24, C24.202C23, C22.532+ 1+4, C4⋊D418C4, C23.23D49C2, (C2×C42).14C22, C23.14(C22×C4), C23.34D413C2, (C22×C4).480C23, (C23×C4).294C22, C22.106(C23×C4), C24.C229C2, C24.3C2216C2, C23.63C239C2, C2.6(C22.32C24), (C22×D4).107C22, C2.20(C22.11C24), C2.C42.470C22, C2.6(C22.34C24), C4⋊C412(C2×C4), (C2×D4)⋊17(C2×C4), C2.18(C4×C4○D4), C22⋊C428(C2×C4), (C4×C22⋊C4)⋊33C2, (C22×C4)⋊26(C2×C4), (C2×C4⋊D4).17C2, (C2×C4).35(C22×C4), (C2×C4).517(C4○D4), (C2×C4⋊C4).184C22, C22.100(C2×C4○D4), (C2×C22⋊C4).430C22, SmallGroup(128,1065)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C23.215C24
C1C2C22C23C22×C4C23×C4C4×C22⋊C4 — C23.215C24
C1C22 — C23.215C24
C1C23 — C23.215C24
C1C23 — C23.215C24

Generators and relations for C23.215C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=d, f2=db=bd, g2=c, faf-1=ab=ba, eae-1=ac=ca, 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 >

Subgroups: 604 in 298 conjugacy classes, 136 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×18], C22 [×3], C22 [×4], C22 [×30], C2×C4 [×12], C2×C4 [×42], D4 [×12], C23, C23 [×6], C23 [×18], C42 [×4], C22⋊C4 [×8], C22⋊C4 [×14], C4⋊C4 [×4], C4⋊C4 [×2], C22×C4 [×2], C22×C4 [×14], C22×C4 [×10], C2×D4 [×12], C2×D4 [×6], C24, C24 [×2], C2.C42 [×10], C2×C42 [×2], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×10], C2×C4⋊C4, C2×C4⋊C4 [×2], C4⋊D4 [×8], C23×C4, C23×C4 [×2], C22×D4, C22×D4 [×2], C4×C22⋊C4, C4×C22⋊C4 [×2], C23.34D4, C23.23D4 [×4], C23.63C23 [×2], C24.C22 [×2], C24.3C22 [×2], C2×C4⋊D4, C23.215C24
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×4], C24, C23×C4, C2×C4○D4 [×2], 2+ 1+4 [×4], C4×C4○D4, C22.11C24 [×2], C22.32C24 [×2], C22.34C24 [×2], C23.215C24

Smallest permutation representation of C23.215C24
On 64 points
Generators in S64
(1 10)(2 39)(3 12)(4 37)(5 46)(6 20)(7 48)(8 18)(9 64)(11 62)(13 56)(14 29)(15 54)(16 31)(17 33)(19 35)(21 60)(22 27)(23 58)(24 25)(26 51)(28 49)(30 43)(32 41)(34 45)(36 47)(38 61)(40 63)(42 53)(44 55)(50 59)(52 57)
(1 28)(2 25)(3 26)(4 27)(5 54)(6 55)(7 56)(8 53)(9 52)(10 49)(11 50)(12 51)(13 48)(14 45)(15 46)(16 47)(17 41)(18 42)(19 43)(20 44)(21 40)(22 37)(23 38)(24 39)(29 34)(30 35)(31 36)(32 33)(57 64)(58 61)(59 62)(60 63)
(1 61)(2 62)(3 63)(4 64)(5 35)(6 36)(7 33)(8 34)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 48)(18 45)(19 46)(20 47)(21 51)(22 52)(23 49)(24 50)(25 59)(26 60)(27 57)(28 58)(29 53)(30 54)(31 55)(32 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 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 41 26 19)(2 18 27 44)(3 43 28 17)(4 20 25 42)(5 23 56 40)(6 39 53 22)(7 21 54 38)(8 37 55 24)(9 31 50 34)(10 33 51 30)(11 29 52 36)(12 35 49 32)(13 60 46 61)(14 64 47 59)(15 58 48 63)(16 62 45 57)
(1 34 61 8)(2 30 62 54)(3 36 63 6)(4 32 64 56)(5 25 35 59)(7 27 33 57)(9 13 37 41)(10 45 38 18)(11 15 39 43)(12 47 40 20)(14 23 42 49)(16 21 44 51)(17 52 48 22)(19 50 46 24)(26 31 60 55)(28 29 58 53)

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

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

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

44 conjugacy classes

class 1 2A···2G2H···2M4A···4L4M···4AD
order12···22···24···44···4
size11···14···42···24···4

44 irreducible representations

dim11111111124
type+++++++++
imageC1C2C2C2C2C2C2C2C4C4○D42+ 1+4
kernelC23.215C24C4×C22⋊C4C23.34D4C23.23D4C23.63C23C24.C22C24.3C22C2×C4⋊D4C4⋊D4C2×C4C22
# reps131422211684

Matrix representation of C23.215C24 in GL8(𝔽5)

01000000
10000000
00010000
00100000
00004000
00000400
00001010
00000101
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
10000000
01000000
00400000
00040000
00004000
00000400
00000040
00000004
,
02000000
30000000
00040000
00100000
00000403
00004030
00000101
00001010
,
10000000
01000000
00300000
00030000
00004030
00000403
00000010
00000001
,
20000000
02000000
00200000
00020000
00000100
00004000
00000001
00000040

G:=sub<GL(8,GF(5))| [0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,1,0,0,0,0,0,0,4,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,1,0,0,0,0,4,0,1,0,0,0,0,0,0,3,0,1,0,0,0,0,3,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,3,0,1,0,0,0,0,0,0,3,0,1],[2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0] >;

C23.215C24 in GAP, Magma, Sage, TeX

C_2^3._{215}C_2^4
% in TeX

G:=Group("C2^3.215C2^4");
// GroupNames label

G:=SmallGroup(128,1065);
// by ID

G=gap.SmallGroup(128,1065);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,568,758,219,675,136]);
// Polycyclic

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