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G = C24.548C23  order 128 = 27

29th non-split extension by C24 of C23 acting via C23/C22=C2

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

Aliases: C24.548C23, C23.211C24, C22.332- (1+4), C22.492+ (1+4), (C22×C4)⋊4Q8, C22⋊Q8⋊17C4, C22.5(C4×Q8), C23.94(C2×Q8), (C23×C4).49C22, C2.2(C23⋊2Q8), C23.8Q8.4C2, C23.226(C4○D4), C22.33(C22×Q8), (C2×C42).418C22, (C22×C4).476C23, C22.102(C23×C4), C23.126(C22×C4), C23.7Q8.27C2, (C22×Q8).87C22, C23.63C23⋊7C2, C2.5(C22.32C24), C23.67C23⋊15C2, C23.65C23⋊15C2, C2.17(C22.11C24), C2.C42.47C22, C2.3(C23.41C23), C2.5(C22.33C24), C2.16(C23.33C23), C4⋊C4⋊11(C2×C4), C2.11(C2×C4×Q8), (C2×Q8)⋊13(C2×C4), (C2×C4).161(C2×Q8), (C4×C22⋊C4).21C2, C22⋊C4.31(C2×C4), (C2×C4).31(C22×C4), C22.96(C2×C4○D4), (C2×C22⋊Q8).16C2, (C2×C4⋊C4).181C22, (C22×C4).304(C2×C4), (C2×C22⋊C4).429C22, (C2×C2.C42).19C2, SmallGroup(128,1061)

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

Derived series
C1 — C22 — C24.548C23
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C2×C2.C42 — C24.548C23
Lower central
C1 — C22 — C24.548C23
Upper central
C1 — C23 — C24.548C23
Jennings
C1 — C23 — C24.548C23

Subgroups: 460 in 264 conjugacy classes, 148 normal (30 characteristic)
C1, C2 [×7], C2 [×4], C4 [×22], C22 [×7], C22 [×4], C22 [×12], C2×C4 [×16], C2×C4 [×46], Q8 [×4], C23, C23 [×6], C23 [×4], C42 [×4], C22⋊C4 [×8], C22⋊C4 [×4], C4⋊C4 [×12], C4⋊C4 [×6], C22×C4 [×2], C22×C4 [×20], C22×C4 [×8], C2×Q8 [×4], C2×Q8 [×2], C24, C2.C42 [×14], C2×C42 [×4], C2×C22⋊C4 [×4], C2×C4⋊C4 [×3], C2×C4⋊C4 [×6], C22⋊Q8 [×8], C23×C4 [×3], C22×Q8, C2×C2.C42, C4×C22⋊C4 [×2], C23.7Q8, C23.8Q8 [×2], C23.63C23 [×4], C23.65C23 [×2], C23.67C23 [×2], C2×C22⋊Q8, C24.548C23

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], Q8 [×4], C23 [×15], C22×C4 [×14], C2×Q8 [×6], C4○D4 [×2], C24, C4×Q8 [×4], C23×C4, C22×Q8, C2×C4○D4, 2+ (1+4) [×3], 2- (1+4), C2×C4×Q8, C22.11C24, C23.33C23, C22.32C24, C22.33C24, C23⋊2Q8, C23.41C23, C24.548C23

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

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

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

Matrix representation ►G ⊆ GL8(𝔽5)

40000000
04000000
00100000
00010000
00004000
00000400
00000010
00000001
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
10000000
01000000
00400000
00040000
00004000
00000400
00000040
00000004
,
01000000
10000000
00030000
00300000
00000010
00000001
00004000
00000400
,
20000000
02000000
00400000
00040000
00003000
00000200
00000020
00000003
,
01000000
40000000
00010000
00400000
00000100
00001000
00000001
00000010

G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3],[0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
 

44 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4AF
order12···222224···44···4
size11···122222···24···4

44 irreducible representations

dim11111111112244
type+++++++++-+-
imageC1C2C2C2C2C2C2C2C2C4Q8C4○D42+ (1+4)2- (1+4)
kernelC24.548C23C2×C2.C42C4×C22⋊C4C23.7Q8C23.8Q8C23.63C23C23.65C23C23.67C23C2×C22⋊Q8C22⋊Q8C22×C4C23C22C22
# reps112124221164431

In GAP, Magma, Sage, TeX 

C_2^4._{548}C_2^3
 
% in TeX
 
G:=Group("C2^4.548C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1061);
 
// by ID
 
G=gap.SmallGroup(128,1061);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,680,758,219,184,675]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=d,f^2=d*c=c*d,g^2=c,e*a*e^-1=a*b=b*a,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,g*e*g^-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>;
 
// generators/relations
 

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