Copied to
clipboard

G = C24.225C23  order 128 = 27

65th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.225C23, C23.258C24, C22.892+ (1+4), C22.642- (1+4), C22.D4⋊11C4, C23.27(C22×C4), C23.8Q8⋊19C2, C23.7Q8⋊28C2, (C23×C4).313C22, (C2×C42).446C22, (C22×C4).486C23, C22.149(C23×C4), C24.C22⋊26C2, C23.23D4.12C2, (C22×D4).113C22, C23.65C23⋊30C2, C23.63C23⋊24C2, C2.38(C22.11C24), C2.2(C22.54C24), C24.3C22.31C2, C2.C42.66C22, C2.1(C22.56C24), C2.1(C22.57C24), C2.38(C23.33C23), C4⋊C4⋊19(C2×C4), C22⋊C4⋊20(C2×C4), (C22×C4)⋊38(C2×C4), (C2×D4).134(C2×C4), (C2×C4).55(C22×C4), (C2×C4⋊C4).194C22, (C2×C22⋊C4).40C22, (C2×C22.D4).9C2, SmallGroup(128,1108)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C22 — C24.225C23
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C2×C22.D4 — C24.225C23
Lower central
C1 — C22 — C24.225C23
Upper central
C1 — C23 — C24.225C23
Jennings
C1 — C23 — C24.225C23

Subgroups: 492 in 256 conjugacy classes, 132 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×18], C22 [×3], C22 [×4], C22 [×20], C2×C4 [×10], C2×C4 [×42], D4 [×4], C23, C23 [×4], C23 [×12], C42 [×3], C22⋊C4 [×12], C22⋊C4 [×7], C4⋊C4 [×8], C4⋊C4 [×7], C22×C4, C22×C4 [×16], C22×C4 [×5], C2×D4 [×4], C2×D4 [×2], C24 [×2], C2.C42 [×10], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×8], C2×C4⋊C4, C2×C4⋊C4 [×8], C22.D4 [×8], C23×C4 [×2], C22×D4, C23.7Q8 [×2], C23.8Q8 [×2], C23.23D4, C23.63C23 [×2], C24.C22 [×4], C23.65C23 [×2], C24.3C22, C2×C22.D4, C24.225C23

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C23×C4, 2+ (1+4) [×4], 2- (1+4) [×2], C22.11C24, C23.33C23 [×2], C22.54C24, C22.56C24 [×2], C22.57C24, C24.225C23

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=g2=1, e2=d, f2=b, eae-1=gag=ab=ba, ac=ca, ad=da, faf-1=abc, 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, geg=bce, fg=gf >

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

(1 9)(2 10)(3 11)(4 12)(5 38)(6 39)(7 40)(8 37)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 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 63)(34 64)(35 61)(36 62)

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

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

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

Matrix representation ►G ⊆ GL9(𝔽5)

100000000
000100000
041130000
010000000
000040000
000000010
000000001
000001000
000000100
,
100000000
040000000
004000000
000400000
000040000
000004000
000000400
000000040
000000004
,
100000000
040000000
004000000
000400000
000040000
000001000
000000100
000000010
000000001
,
400000000
040000000
004000000
000400000
000040000
000001000
000000100
000000010
000000001
,
300000000
020000000
003000000
000300000
003320000
000000002
000000020
000000300
000003000
,
400000000
001000000
040000000
041130000
040140000
000000100
000004000
000000004
000000010
,
400000000
010000000
001000000
000400000
041040000
000001000
000000100
000000040
000000004

G:=sub<GL(9,GF(5))| [1,0,0,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,3,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[3,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,3,0,3,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0],[4,0,0,0,0,0,0,0,0,0,0,4,4,4,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,3,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0],[4,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4] >;
 

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4Z
order12···222224···4
size11···144444···4

38 irreducible representations

dim111111111144
type++++++++++-
imageC1C2C2C2C2C2C2C2C2C42+ (1+4)2- (1+4)
kernelC24.225C23C23.7Q8C23.8Q8C23.23D4C23.63C23C24.C22C23.65C23C24.3C22C2×C22.D4C22.D4C22C22
# reps1221242111642

In GAP, Magma, Sage, TeX 

C_2^4._{225}C_2^3
 
% in TeX
 
G:=Group("C2^4.225C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1108);
 
// by ID
 
G=gap.SmallGroup(128,1108);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,555,100,1571,346]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=g^2=1,e^2=d,f^2=b,e*a*e^-1=g*a*g=a*b=b*a,a*c=c*a,a*d=d*a,f*a*f^-1=a*b*c,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,g*e*g=b*c*e,f*g=g*f>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀