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G = C24.303C23  order 128 = 27

143rd non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.303C23, C23.392C24, C22.1922+ (1+4), C22.1442- (1+4), C2.19(D4×Q8), C4⋊C4.230D4, C22⋊C4⋊15Q8, C23.17(C2×Q8), C2.63(D4⋊5D4), C23⋊Q8.6C2, C2.19(D4⋊3Q8), C4.44(C4.4D4), C22.85(C22×Q8), (C2×C42).520C22, (C22×C4).827C23, (C23×C4).377C22, C22.272(C22×D4), C23.7Q8.43C2, (C22×Q8).117C22, C23.83C23⋊20C2, C23.67C23⋊50C2, C24.C22.19C2, C2.C42.144C22, C2.19(C23.37C23), C2.21(C22.50C24), C2.29(C22.36C24), (C4×C4⋊C4)⋊70C2, (C2×C4⋊Q8)⋊11C2, (C2×C4).63(C2×D4), (C2×C4).38(C2×Q8), (C4×C22⋊C4).46C2, C2.18(C2×C4.4D4), (C2×C22⋊Q8).30C2, (C2×C4).123(C4○D4), (C2×C4⋊C4).262C22, C22.269(C2×C4○D4), (C2×C22⋊C4).157C22, SmallGroup(128,1224)

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

Derived series
C1 — C23 — C24.303C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C4×C22⋊C4 — C24.303C23
Lower central
C1 — C23 — C24.303C23
Upper central
C1 — C23 — C24.303C23
Jennings
C1 — C23 — C24.303C23

Subgroups: 468 in 254 conjugacy classes, 112 normal (42 characteristic)
C1, C2 [×7], C2 [×2], C4 [×4], C4 [×18], C22 [×7], C22 [×10], C2×C4 [×16], C2×C4 [×38], Q8 [×12], C23, C23 [×2], C23 [×6], C42 [×6], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×12], C22×C4 [×6], C22×C4 [×8], C22×C4 [×6], C2×Q8 [×16], C24, C2.C42 [×2], C2.C42 [×12], C2×C42 [×2], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×4], C2×C4⋊C4 [×3], C2×C4⋊C4 [×4], C22⋊Q8 [×4], C4⋊Q8 [×4], C23×C4, C22×Q8, C22×Q8 [×2], C4×C22⋊C4, C4×C4⋊C4, C23.7Q8, C24.C22 [×2], C23.67C23 [×2], C23.67C23 [×2], C23⋊Q8 [×2], C23.83C23 [×2], C2×C22⋊Q8, C2×C4⋊Q8, C24.303C23

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×6], C24, C4.4D4 [×4], C22×D4, C22×Q8, C2×C4○D4 [×3], 2+ (1+4), 2- (1+4), C2×C4.4D4, C23.37C23, C22.36C24, D4⋊5D4, D4×Q8, D4⋊3Q8, C22.50C24, C24.303C23

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=db=bd, f2=cb=bc, g2=b, ab=ba, eae-1=ac=ca, faf-1=ad=da, ag=ga, 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
(2 27)(4 25)(5 39)(6 44)(7 37)(8 42)(9 54)(11 56)(14 57)(16 59)(17 33)(18 45)(19 35)(20 47)(21 41)(22 38)(23 43)(24 40)(30 49)(32 51)(34 63)(36 61)(46 64)(48 62)

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

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
040000
001000
000100
000010
000014
,
400000
040000
004000
000400
000010
000001
,
400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
010000
400000
002000
000300
000030
000003
,
100000
040000
000100
004000
000013
000004
,
200000
030000
000100
004000
000010
000001

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

38 conjugacy classes

class 1 2A···2G2H2I4A···4H4I···4X4Y4Z4AA4AB
order12···2224···44···44444
size11···1442···24···48888

38 irreducible representations

dim111111111122244
type++++++++++-++-
imageC1C2C2C2C2C2C2C2C2C2Q8D4C4○D42+ (1+4)2- (1+4)
kernelC24.303C23C4×C22⋊C4C4×C4⋊C4C23.7Q8C24.C22C23.67C23C23⋊Q8C23.83C23C2×C22⋊Q8C2×C4⋊Q8C22⋊C4C4⋊C4C2×C4C22C22
# reps1111242211441211

In GAP, Magma, Sage, TeX 

C_2^4._{303}C_2^3
 
% in TeX
 
G:=Group("C2^4.303C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1224);
 
// by ID
 
G=gap.SmallGroup(128,1224);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,344,758,723,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=d*b=b*d,f^2=c*b=b*c,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,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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