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G = C24.338C23  order 128 = 27

178th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.338C23, C23.474C24, C22.2572+ (1+4), C22⋊C4⋊19Q8, C42⋊8C4⋊45C2, C23.26(C2×Q8), C2.38(D4⋊3Q8), (C2×C42).70C22, C23⋊Q8.10C2, (C22×C4).844C23, (C23×C4).407C22, C23.Q8.15C2, C23.8Q8.33C2, C23.7Q8.54C2, C22.113(C22×Q8), (C22×Q8).142C22, C23.63C23⋊91C2, C23.81C23⋊43C2, C23.78C23⋊19C2, C23.67C23⋊66C2, C2.27(C22.32C24), C24.C22.33C2, C2.55(C22.45C24), C2.C42.210C22, C2.29(C22.49C24), C2.30(C23.37C23), C2.85(C23.36C23), (C4×C4⋊C4)⋊100C2, (C2×C4).256(C2×Q8), (C4×C22⋊C4).65C2, (C2×C4).394(C4○D4), (C2×C4⋊C4).321C22, C22.350(C2×C4○D4), (C2×C22⋊C4).510C22, SmallGroup(128,1306)

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

Derived series
C1 — C23 — C24.338C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C22⋊C4 — C24.338C23
Lower central
C1 — C23 — C24.338C23
Upper central
C1 — C23 — C24.338C23
Jennings
C1 — C23 — C24.338C23

Subgroups: 404 in 218 conjugacy classes, 100 normal (82 characteristic)
C1, C2 [×7], C2 [×2], C4 [×20], C22 [×7], C22 [×10], C2×C4 [×12], C2×C4 [×40], Q8 [×4], C23, C23 [×2], C23 [×6], C42 [×6], C22⋊C4 [×4], C22⋊C4 [×7], C4⋊C4 [×14], C22×C4 [×14], C22×C4 [×5], C2×Q8 [×4], C24, C2.C42 [×14], C2×C42 [×4], C2×C22⋊C4 [×6], C2×C4⋊C4 [×9], C23×C4, C22×Q8, C4×C22⋊C4, C4×C4⋊C4, C23.7Q8, C42⋊8C4, C23.8Q8, C23.63C23 [×3], C24.C22 [×2], C23.67C23, C23⋊Q8, C23.78C23, C23.Q8, C23.81C23, C24.338C23

Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×8], C24, C22×Q8, C2×C4○D4 [×4], 2+ (1+4) [×2], C23.36C23, C23.37C23, C22.32C24, C22.45C24, D4⋊3Q8 [×2], C22.49C24, C24.338C23

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
040000
001000
000400
000010
000004
,
400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
010000
400000
002000
000200
000001
000010
,
300000
020000
000400
001000
000010
000001
,
400000
010000
001000
000100
000030
000003

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

38 conjugacy classes

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

38 irreducible representations

dim1111111111111224
type+++++++++++++-+
imageC1C2C2C2C2C2C2C2C2C2C2C2C2Q8C4○D42+ (1+4)
kernelC24.338C23C4×C22⋊C4C4×C4⋊C4C23.7Q8C42⋊8C4C23.8Q8C23.63C23C24.C22C23.67C23C23⋊Q8C23.78C23C23.Q8C23.81C23C22⋊C4C2×C4C22
# reps11111132111114162

In GAP, Magma, Sage, TeX 

C_2^4._{338}C_2^3
 
% in TeX
 
G:=Group("C2^4.338C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1306);
 
// by ID
 
G=gap.SmallGroup(128,1306);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,568,758,723,100,675,136]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=f^2=d*b=b*d,g^2=c,a*b=b*a,a*c=c*a,f*a*f^-1=a*d=d*a,e*a*e^-1=a*b*c,a*g=g*a,b*c=c*b,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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