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G = C24.388C23  order 128 = 27

228th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.388C23, C23.580C24, C22.2632- (1+4), C22.3542+ (1+4), C2.44(D42), C22⋊C4.9D4, C42⋊9C4⋊32C2, C23.206(C2×D4), C2.57(D4⋊6D4), C23.8Q8⋊97C2, C23.11D4⋊76C2, (C22×C4).867C23, (C23×C4).448C22, (C2×C42).637C22, C22.389(C22×D4), C23.10D4.39C2, (C22×D4).219C22, (C22×Q8).176C22, C23.78C23⋊39C2, C24.C22⋊120C2, C23.81C23⋊77C2, C2.C42.289C22, C2.9(C22.57C24), C2.57(C22.33C24), C2.54(C23.38C23), (C2×C4).88(C2×D4), (C2×C22⋊Q8)⋊35C2, (C2×C4).417(C4○D4), (C2×C4⋊C4).396C22, C22.443(C2×C4○D4), (C2×C22⋊C4).249C22, (C2×C22.D4).23C2, SmallGroup(128,1412)

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

Derived series
C1 — C23 — C24.388C23
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C2×C22.D4 — C24.388C23
Lower central
C1 — C23 — C24.388C23
Upper central
C1 — C23 — C24.388C23
Jennings
C1 — C23 — C24.388C23

Subgroups: 548 in 284 conjugacy classes, 104 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], Q8 [×4], C23, C23 [×4], C23 [×12], C42 [×2], C22⋊C4 [×8], C22⋊C4 [×16], C4⋊C4 [×22], C22×C4 [×3], C22×C4 [×10], C22×C4 [×10], C2×D4 [×5], C2×Q8 [×5], C24 [×2], C2.C42 [×2], C2.C42 [×6], C2×C42, C2×C22⋊C4 [×10], C2×C4⋊C4 [×2], C2×C4⋊C4 [×10], C22⋊Q8 [×8], C22.D4 [×8], C23×C4 [×2], C22×D4, C22×Q8, C42⋊9C4, C23.8Q8 [×2], C24.C22 [×2], C23.10D4, C23.78C23, C23.11D4 [×2], C23.81C23 [×2], C2×C22⋊Q8 [×2], C2×C22.D4 [×2], C24.388C23

Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C22×D4 [×2], C2×C4○D4, 2+ (1+4), 2- (1+4) [×3], C23.38C23 [×2], C22.33C24, D42, D4⋊6D4 [×2], C22.57C24, C24.388C23

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=c, g2=b, faf=ab=ba, ac=ca, ad=da, eae-1=abc, ag=ga, bc=cb, bd=db, fef=be=eb, bf=fb, bg=gb, cd=dc, geg-1=ce=ec, cf=fc, cg=gc, de=ed, gfg-1=df=fd, dg=gd >

Smallest permutation representation
►On 64 points
Generators in S64
(2 38)(4 40)(5 17)(6 8)(7 19)(10 44)(12 42)(13 45)(14 16)(15 47)(18 20)(22 52)(24 50)(26 56)(28 54)(29 57)(30 32)(31 59)(33 35)(34 63)(36 61)(46 48)(58 60)(62 64)

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
040000
001000
003400
000010
000001
,
100000
010000
004000
000400
000010
000001
,
400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
010000
400000
002200
001300
000010
000001
,
100000
010000
004400
000100
000001
000010
,
400000
010000
003000
000300
000010
000004

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4N4O···4T
order12···222224···44···4
size11···144444···48···8

32 irreducible representations

dim11111111112244
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2D4C4○D42+ (1+4)2- (1+4)
kernelC24.388C23C42⋊9C4C23.8Q8C24.C22C23.10D4C23.78C23C23.11D4C23.81C23C2×C22⋊Q8C2×C22.D4C22⋊C4C2×C4C22C22
# reps11221122228413

In GAP, Magma, Sage, TeX 

C_2^4._{388}C_2^3
 
% in TeX
 
G:=Group("C2^4.388C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1412);
 
// by ID
 
G=gap.SmallGroup(128,1412);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,758,723,100,1571,346]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=c,g^2=b,f*a*f=a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b*c,a*g=g*a,b*c=c*b,b*d=d*b,f*e*f=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,g*e*g^-1=c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,g*f*g^-1=d*f=f*d,d*g=g*d>;
 
// generators/relations
 

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