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G = C24.553C23  order 128 = 27

34th non-split extension by C24 of C23 acting via C23/C22=C2

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

Aliases: C24.553C23, C23.226C24, C22.452- (1+4), C22.66(C4×D4), C4⋊2(C42⋊C2), C2.2(D4⋊6D4), C22⋊C4.183D4, C23.414(C2×D4), C23.289(C4○D4), (C23×C4).300C22, (C2×C42).427C22, C22.117(C23×C4), C23.216(C22×C4), (C22×C4).755C23, C22.104(C22×D4), C23.7Q8.29C2, C23.34D4.11C2, C23.65C23⋊18C2, C23.63C23⋊13C2, C2.C42.472C22, C2.2(C22.46C24), C2.14(C23.32C23), (C4×C4⋊C4)⋊32C2, (C2×C4⋊C4)⋊36C4, C2.30(C2×C4×D4), C22⋊C4○3(C4⋊C4), C4⋊C4.206(C2×C4), (C2×C4).885(C2×D4), (C4×C22⋊C4).25C2, (C22×C4⋊C4).27C2, (C2×C4).789(C4○D4), (C2×C4⋊C4).973C22, (C22×C4).309(C2×C4), (C2×C4).231(C22×C4), C2.27(C2×C42⋊C2), C22.111(C2×C4○D4), (C2×C42⋊C2).31C2, (C2×C22⋊C4).436C22, SmallGroup(128,1076)

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

Derived series
C1 — C22 — C24.553C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C22×C4⋊C4 — C24.553C23
Lower central
C1 — C22 — C24.553C23
Upper central
C1 — C23 — C24.553C23
Jennings
C1 — C23 — C24.553C23

Subgroups: 460 in 302 conjugacy classes, 156 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×20], C22 [×3], C22 [×8], C22 [×12], C2×C4 [×20], C2×C4 [×56], C23, C23 [×6], C23 [×4], C42 [×12], C22⋊C4 [×4], C22⋊C4 [×6], C4⋊C4 [×16], C4⋊C4 [×12], C22×C4 [×2], C22×C4 [×24], C22×C4 [×12], C24, C2.C42 [×12], C2×C42 [×6], C2×C22⋊C4 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×16], C42⋊C2 [×8], C23×C4, C23×C4 [×2], C4×C22⋊C4, C4×C4⋊C4 [×2], C23.7Q8, C23.34D4 [×2], C23.63C23 [×4], C23.65C23 [×2], C22×C4⋊C4, C2×C42⋊C2 [×2], C24.553C23

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×6], C24, C42⋊C2 [×4], C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4 [×3], 2- (1+4) [×2], C2×C42⋊C2, C2×C4×D4, C23.32C23, D4⋊6D4 [×2], C22.46C24 [×2], C24.553C23

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
040000
004000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
400000
040000
004000
000400
000010
000001
,
100000
010000
004000
000400
000010
000001
,
010000
100000
000100
004000
000014
000004
,
200000
020000
003000
000300
000023
000043
,
100000
010000
004000
000400
000032
000002

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

50 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4T4U···4AL
order12···222224···44···4
size11···122222···24···4

50 irreducible representations

dim11111111112224
type++++++++++-
imageC1C2C2C2C2C2C2C2C2C4D4C4○D4C4○D42- (1+4)
kernelC24.553C23C4×C22⋊C4C4×C4⋊C4C23.7Q8C23.34D4C23.63C23C23.65C23C22×C4⋊C4C2×C42⋊C2C2×C4⋊C4C22⋊C4C2×C4C23C22
# reps112124212164842

In GAP, Magma, Sage, TeX 

C_2^4._{553}C_2^3
 
% in TeX
 
G:=Group("C2^4.553C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1076);
 
// by ID
 
G=gap.SmallGroup(128,1076);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=d,f^2=c,g^2=b,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,a*f=f*a,a*g=g*a,b*c=c*b,b*d=d*b,f*e*f^-1=b*e=e*b,g*f*g^-1=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,e*g=g*e>;
 
// generators/relations
 

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