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G = C24.251C23  order 128 = 27

91st non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.251C23, C23.316C24, C22.932- (1+4), C22.1302+ (1+4), C4⋊C4⋊35D4, C2.9(D42), (C2×D4)⋊30D4, C4⋊4(C4⋊D4), (C22×C4)⋊24D4, C23.36(C2×D4), C22⋊2(C4⋊D4), C2.13(D4⋊6D4), C23.Q8⋊3C2, C2.11(Q8⋊5D4), (C22×C4).51C23, C23.7Q8⋊36C2, C23.326(C4○D4), C23.10D4⋊11C2, C23.23D4⋊28C2, (C23×C4).334C22, (C2×C42).465C22, C22.196(C22×D4), C24.3C22⋊29C2, (C22×D4).120C22, C23.65C23⋊36C2, C2.C42.80C22, C2.7(C22.31C24), C2.13(C22.47C24), (C2×C4×D4)⋊22C2, (C2×C4⋊D4)⋊6C2, (C2×C4).46(C2×D4), (C22×C4⋊C4)⋊18C2, C2.20(C2×C4⋊D4), (C2×C4).805(C4○D4), (C2×C4⋊C4).845C22, C22.195(C2×C4○D4), (C2×C22⋊C4).111C22, SmallGroup(128,1148)

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

Derived series
C1 — C23 — C24.251C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C4×D4 — C24.251C23
Lower central
C1 — C23 — C24.251C23
Upper central
C1 — C23 — C24.251C23
Jennings
C1 — C23 — C24.251C23

Subgroups: 804 in 394 conjugacy classes, 124 normal (42 characteristic)
C1, C2 [×7], C2 [×8], C4 [×4], C4 [×14], C22 [×7], C22 [×4], C22 [×36], C2×C4 [×14], C2×C4 [×46], D4 [×28], C23, C23 [×8], C23 [×24], C42 [×2], C22⋊C4 [×22], C4⋊C4 [×4], C4⋊C4 [×14], C22×C4 [×5], C22×C4 [×10], C22×C4 [×20], C2×D4 [×4], C2×D4 [×32], C24 [×2], C24 [×2], C2.C42 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C22⋊C4 [×10], C2×C4⋊C4 [×4], C2×C4⋊C4 [×4], C2×C4⋊C4 [×4], C4×D4 [×4], C4⋊D4 [×16], C23×C4 [×2], C23×C4 [×2], C22×D4 [×2], C22×D4 [×4], C23.7Q8, C23.23D4 [×2], C23.65C23, C24.3C22, C23.10D4 [×2], C23.Q8 [×2], C22×C4⋊C4, C2×C4×D4, C2×C4⋊D4 [×2], C2×C4⋊D4 [×2], C24.251C23

Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C4○D4 [×4], C24, C4⋊D4 [×8], C22×D4 [×3], C2×C4○D4 [×2], 2+ (1+4), 2- (1+4), C2×C4⋊D4 [×2], C22.31C24, D42, D4⋊6D4, Q8⋊5D4, C22.47C24, C24.251C23

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=c, f2=g2=b, 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
(1 18)(2 17)(3 20)(4 19)(5 46)(6 45)(7 48)(8 47)(9 15)(10 14)(11 13)(12 16)(21 30)(22 29)(23 32)(24 31)(25 38)(26 37)(27 40)(28 39)(33 41)(34 44)(35 43)(36 42)(49 57)(50 60)(51 59)(52 58)(53 61)(54 64)(55 63)(56 62)

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
040000
000100
001000
000003
000020
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000040
000004
,
400000
040000
004000
000400
000010
000001
,
100000
010000
000400
001000
000030
000002
,
010000
100000
000200
003000
000001
000040
,
100000
010000
001000
000100
000001
000040

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A4B4C4D4E···4T4U4V
order12···22222222244444···444
size11···12222448822224···488

38 irreducible representations

dim11111111112222244
type++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2D4D4D4C4○D4C4○D42+ (1+4)2- (1+4)
kernelC24.251C23C23.7Q8C23.23D4C23.65C23C24.3C22C23.10D4C23.Q8C22×C4⋊C4C2×C4×D4C2×C4⋊D4C4⋊C4C22×C4C2×D4C2×C4C23C22C22
# reps11211221144444411

In GAP, Magma, Sage, TeX 

C_2^4._{251}C_2^3
 
% in TeX
 
G:=Group("C2^4.251C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1148);
 
// by ID
 
G=gap.SmallGroup(128,1148);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,120,758,723,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=c,f^2=g^2=b,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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