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G = C24.221C23order 128 = 27

61st non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.221C23, C23.254C24, C22.852+ 1+4, C4229(C2×C4), C4.4D425C4, C424C416C2, (C23×C4).59C22, C23.23(C22×C4), C22.145(C23×C4), (C2×C42).444C22, (C22×C4).773C23, (C22×Q8).93C22, C24.C2223C2, C23.23D4.10C2, (C22×D4).110C22, C23.67C2324C2, C2.7(C22.45C24), C2.35(C22.11C24), C2.C42.481C22, C2.5(C22.49C24), (C2×Q8)⋊16(C2×C4), C2.41(C4×C4○D4), (C4×C22⋊C4)⋊45C2, C22⋊C432(C2×C4), (C2×D4).131(C2×C4), (C2×C4).726(C4○D4), (C2×C4⋊C4).193C22, (C2×C4).290(C22×C4), (C2×C4.4D4).18C2, C22.139(C2×C4○D4), (C2×C22⋊C4).445C22, SmallGroup(128,1104)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C24.221C23
C1C2C22C23C22×C4C2×C42C424C4 — C24.221C23
C1C22 — C24.221C23
C1C23 — C24.221C23
C1C23 — C24.221C23

Generators and relations for C24.221C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=d, f2=b, g2=c, eae-1=gag-1=ab=ba, faf-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, eg=ge, fg=gf >

Subgroups: 508 in 280 conjugacy classes, 140 normal (12 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×22], C22 [×3], C22 [×4], C22 [×20], C2×C4 [×18], C2×C4 [×38], D4 [×4], Q8 [×4], C23, C23 [×4], C23 [×12], C42 [×4], C42 [×10], C22⋊C4 [×16], C22⋊C4 [×10], C4⋊C4 [×2], C22×C4, C22×C4 [×12], C22×C4 [×10], C2×D4 [×4], C2×D4 [×2], C2×Q8 [×4], C2×Q8 [×2], C24 [×2], C2.C42 [×12], C2×C42, C2×C42 [×6], C2×C22⋊C4 [×10], C2×C4⋊C4 [×2], C4.4D4 [×8], C23×C4 [×2], C22×D4, C22×Q8, C424C4 [×2], C4×C22⋊C4 [×4], C23.23D4 [×2], C24.C22 [×4], C23.67C23 [×2], C2×C4.4D4, C24.221C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×8], C24, C23×C4, C2×C4○D4 [×4], 2+ 1+4 [×2], C4×C4○D4 [×2], C22.11C24, C22.45C24 [×2], C22.49C24 [×2], C24.221C23

Smallest permutation representation of C24.221C23
On 64 points
Generators in S64
(1 18)(2 43)(3 20)(4 41)(5 39)(6 21)(7 37)(8 23)(9 33)(10 29)(11 35)(12 31)(13 64)(14 58)(15 62)(16 60)(17 27)(19 25)(22 56)(24 54)(26 44)(28 42)(30 50)(32 52)(34 49)(36 51)(38 53)(40 55)(45 61)(46 59)(47 63)(48 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 49 28 10)(2 24 25 39)(3 51 26 12)(4 22 27 37)(5 15 54 46)(6 44 55 20)(7 13 56 48)(8 42 53 18)(9 64 52 57)(11 62 50 59)(14 29 45 34)(16 31 47 36)(17 33 41 32)(19 35 43 30)(21 60 40 63)(23 58 38 61)
(1 34 61 8)(2 35 62 5)(3 36 63 6)(4 33 64 7)(9 48 37 17)(10 45 38 18)(11 46 39 19)(12 47 40 20)(13 22 41 52)(14 23 42 49)(15 24 43 50)(16 21 44 51)(25 30 59 54)(26 31 60 55)(27 32 57 56)(28 29 58 53)

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

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

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

50 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4X4Y···4AL
order12···222224···44···4
size11···144442···24···4

50 irreducible representations

dim1111111124
type++++++++
imageC1C2C2C2C2C2C2C4C4○D42+ 1+4
kernelC24.221C23C424C4C4×C22⋊C4C23.23D4C24.C22C23.67C23C2×C4.4D4C4.4D4C2×C4C22
# reps124242116162

Matrix representation of C24.221C23 in GL5(𝔽5)

10000
02400
03300
00012
00004
,
10000
04000
00400
00010
00001
,
10000
01000
00100
00040
00004
,
40000
04000
00400
00040
00004
,
30000
02400
00300
00031
00002
,
40000
02000
00200
00031
00022
,
10000
01200
00400
00020
00002

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

C24.221C23 in GAP, Magma, Sage, TeX

C_2^4._{221}C_2^3
% in TeX

G:=Group("C2^4.221C2^3");
// GroupNames label

G:=SmallGroup(128,1104);
// by ID

G=gap.SmallGroup(128,1104);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,568,758,100,346,136]);
// Polycyclic

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