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G = C24.220C23  order 128 = 27

60th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.220C23, C23.249C24, C22.812+ (1+4), C22.602- (1+4), C4.99(C4×D4), C42⋊28(C2×C4), C4⋊C4.397D4, C4.4D4⋊24C4, C2.7(Q8⋊5D4), C2.11(D4⋊5D4), C23.22(C22×C4), (C23×C4).58C22, C23.8Q8⋊18C2, C22.140(C23×C4), (C2×C42).440C22, C22.120(C22×D4), (C22×C4).1254C23, C24.C22⋊22C2, (C22×D4).491C22, (C22×Q8).407C22, C23.67C23⋊22C2, C24.3C22.29C2, C2.C42.65C22, C2.7(C22.50C24), C2.4(C22.53C24), C2.35(C23.33C23), (C4×C4⋊C4)⋊47C2, (C2×C4×Q8)⋊10C2, C2.43(C2×C4×D4), (C2×C4×D4).40C2, (C2×Q8)⋊26(C2×C4), C2.37(C4×C4○D4), (C4×C22⋊C4)⋊44C2, C22⋊C4⋊17(C2×C4), (C2×C4).894(C2×D4), (C2×D4).170(C2×C4), (C2×C4).47(C22×C4), (C2×C4).892(C4○D4), (C2×C4⋊C4).979C22, C4⋊C4○4(C2.C42), (C2×C4.4D4).17C2, C22.134(C2×C4○D4), (C2×C22⋊C4).38C22, SmallGroup(128,1099)

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

Derived series
C1 — C22 — C24.220C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C4⋊C4 — C24.220C23
Lower central
C1 — C22 — C24.220C23
Upper central
C1 — C23 — C24.220C23
Jennings
C1 — C23 — C24.220C23

Subgroups: 540 in 316 conjugacy classes, 152 normal (42 characteristic)
C1, C2 [×7], C2 [×4], C4 [×4], C4 [×20], C22 [×7], C22 [×20], C2×C4 [×22], C2×C4 [×36], D4 [×8], Q8 [×8], C23, C23 [×4], C23 [×12], C42 [×4], C42 [×10], C22⋊C4 [×16], C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×10], C22×C4 [×7], C22×C4 [×6], C22×C4 [×10], C2×D4 [×4], C2×D4 [×4], C2×Q8 [×4], C2×Q8 [×4], C24 [×2], C2.C42 [×2], C2.C42 [×6], C2×C42 [×3], C2×C42 [×4], C2×C22⋊C4 [×10], C2×C4⋊C4 [×4], C2×C4⋊C4 [×2], C4×D4 [×4], C4×Q8 [×4], C4.4D4 [×8], C23×C4 [×2], C22×D4, C22×Q8, C4×C22⋊C4 [×2], C4×C4⋊C4 [×2], C23.8Q8 [×2], C24.C22 [×4], C24.3C22, C23.67C23, C2×C4×D4, C2×C4×Q8, C2×C4.4D4, C24.220C23

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, C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4 [×3], 2+ (1+4), 2- (1+4), C2×C4×D4, C4×C4○D4, C23.33C23, D4⋊5D4, Q8⋊5D4, C22.50C24, C22.53C24, C24.220C23

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

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

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

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

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

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

Matrix representation ►G ⊆ GL5(𝔽5)

10000
04000
00100
00001
00010
,
10000
01000
00100
00040
00004
,
10000
04000
00400
00010
00001
,
40000
04000
00400
00010
00001
,
30000
03000
00300
00010
00004
,
10000
00400
04000
00003
00030
,
40000
04000
00400
00001
00040

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

50 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4X4Y···4AL
order12···222224···44···4
size11···144442···24···4

50 irreducible representations

dim111111111112244
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C4D4C4○D42+ (1+4)2- (1+4)
kernelC24.220C23C4×C22⋊C4C4×C4⋊C4C23.8Q8C24.C22C24.3C22C23.67C23C2×C4×D4C2×C4×Q8C2×C4.4D4C4.4D4C4⋊C4C2×C4C22C22
# reps12224111111641211

In GAP, Magma, Sage, TeX 

C_2^4._{220}C_2^3
 
% in TeX
 
G:=Group("C2^4.220C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1099);
 
// by ID
 
G=gap.SmallGroup(128,1099);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,268,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=d,f^2=g^2=b,e*a*e^-1=g*a*g^-1=a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,f*e*f^-1=g*e*g^-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>;
 
// generators/relations
 

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