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G = C24.255C23  order 128 = 27

95th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.255C23, C23.321C24, C22.962- (1+4), (C2×Q8).219D4, C42⋊8C4⋊22C2, C2.12(Q8⋊5D4), C23.23(C4○D4), (C22×C4).55C23, (C23×C4).336C22, (C2×C42).469C22, C23.11D4.3C2, C22.201(C22×D4), C23.7Q8.34C2, C23.83C23⋊4C2, C4.48(C22.D4), (C22×Q8).416C22, C23.63C23⋊29C2, C23.67C23⋊31C2, C23.65C23⋊39C2, C24.C22.11C2, C2.C42.84C22, C2.8(C22.35C24), C2.20(C23.36C23), C2.10(C22.50C24), C2.16(C22.46C24), (C2×C4×Q8)⋊13C2, (C2×C4).311(C2×D4), (C4×C22⋊C4).35C2, (C2×C22⋊Q8).22C2, (C2×C4).854(C4○D4), (C2×C4⋊C4).846C22, C22.200(C2×C4○D4), C2.18(C2×C22.D4), (C2×C22⋊C4).113C22, SmallGroup(128,1153)

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

Derived series
C1 — C23 — C24.255C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C4×C22⋊C4 — C24.255C23
Lower central
C1 — C23 — C24.255C23
Upper central
C1 — C23 — C24.255C23
Jennings
C1 — C23 — C24.255C23

Subgroups: 420 in 242 conjugacy classes, 104 normal (42 characteristic)
C1, C2 [×7], C2 [×2], C4 [×4], C4 [×16], C22 [×7], C22 [×10], C2×C4 [×12], C2×C4 [×40], Q8 [×8], C23, C23 [×2], C23 [×6], C42 [×8], C22⋊C4 [×12], C4⋊C4 [×18], C22×C4 [×6], C22×C4 [×8], C22×C4 [×6], C2×Q8 [×4], C2×Q8 [×4], C24, C2.C42 [×2], C2.C42 [×12], C2×C42 [×2], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×4], C2×C4⋊C4 [×3], C2×C4⋊C4 [×6], C4×Q8 [×4], C22⋊Q8 [×4], C23×C4, C22×Q8, C4×C22⋊C4, C23.7Q8, C42⋊8C4, C23.63C23 [×2], C24.C22 [×2], C23.65C23, C23.67C23, C23.11D4 [×2], C23.83C23 [×2], C2×C4×Q8, C2×C22⋊Q8, C24.255C23

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×8], C24, C22.D4 [×4], C22×D4, C2×C4○D4 [×4], 2- (1+4) [×2], C2×C22.D4, C23.36C23, C22.35C24, Q8⋊5D4 [×2], C22.46C24, C22.50C24, C24.255C23

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=g2=b, f2=c, ab=ba, eae-1=ac=ca, faf-1=ad=da, ag=ga, bc=cb, bd=db, 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 >

Smallest permutation representation
►On 64 points
Generators in S64
(2 10)(4 12)(5 24)(6 49)(7 22)(8 51)(14 42)(16 44)(17 36)(18 62)(19 34)(20 64)(21 40)(23 38)(26 56)(28 54)(30 60)(32 58)(33 46)(35 48)(37 50)(39 52)(45 61)(47 63)

(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 9)(2 10)(3 11)(4 12)(5 39)(6 40)(7 37)(8 38)(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 55)(26 56)(27 53)(28 54)(29 59)(30 60)(31 57)(32 58)(33 62)(34 63)(35 64)(36 61)

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
340000
001000
001400
000010
000044
,
400000
040000
001000
000100
000010
000001
,
400000
040000
004000
000400
000010
000001
,
100000
010000
004000
000400
000040
000004
,
330000
020000
002100
002300
000040
000004
,
300000
420000
001300
001400
000043
000001
,
200000
130000
004000
000400
000010
000001

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

38 conjugacy classes

class 1 2A···2G2H2I4A···4H4I···4X4Y4Z4AA4AB
order12···2224···44···44444
size11···1442···24···48888

38 irreducible representations

dim1111111111112224
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42- (1+4)
kernelC24.255C23C4×C22⋊C4C23.7Q8C42⋊8C4C23.63C23C24.C22C23.65C23C23.67C23C23.11D4C23.83C23C2×C4×Q8C2×C22⋊Q8C2×Q8C2×C4C23C22
# reps11112211221141242

In GAP, Magma, Sage, TeX 

C_2^4._{255}C_2^3
 
% in TeX
 
G:=Group("C2^4.255C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1153);
 
// by ID
 
G=gap.SmallGroup(128,1153);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,232,758,723,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=g^2=b,f^2=c,a*b=b*a,e*a*e^-1=a*c=c*a,f*a*f^-1=a*d=d*a,a*g=g*a,b*c=c*b,b*d=d*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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