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G = C24.355C23  order 128 = 27

195th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.355C23, C23.507C24, C22.2872+ (1+4), (C22×C4)⋊5Q8, C23.64(C2×Q8), C23⋊Q8.14C2, C2.6(C23⋊2Q8), (C23×C4).134C22, (C22×C4).553C23, (C2×C42).594C22, C23.4Q8.13C2, C22.128(C22×Q8), C23.34D4.21C2, (C22×Q8).148C22, C23.67C23⋊69C2, C23.63C23⋊108C2, C2.71(C22.45C24), C2.C42.236C22, C2.39(C23.37C23), C2.32(C22.53C24), (C2×C4).166(C2×Q8), (C4×C22⋊C4).71C2, (C2×C4).165(C4○D4), (C2×C4⋊C4).346C22, C22.383(C2×C4○D4), (C2×C22⋊C4).516C22, SmallGroup(128,1339)

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

Derived series
C1 — C23 — C24.355C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C22⋊C4 — C24.355C23
Lower central
C1 — C23 — C24.355C23
Upper central
C1 — C23 — C24.355C23
Jennings
C1 — C23 — C24.355C23

Subgroups: 420 in 222 conjugacy classes, 100 normal (12 characteristic)
C1, C2, C2 [×6], C2 [×2], C4 [×20], C22 [×3], C22 [×4], C22 [×10], C2×C4 [×12], C2×C4 [×40], Q8 [×8], C23, C23 [×2], C23 [×6], C42 [×4], C22⋊C4 [×10], C4⋊C4 [×10], C22×C4 [×18], C22×C4 [×2], C2×Q8 [×10], C24, C2.C42 [×16], C2×C42 [×4], C2×C22⋊C4 [×2], C2×C22⋊C4 [×4], C2×C4⋊C4 [×6], C23×C4, C22×Q8 [×2], C4×C22⋊C4 [×2], C23.34D4, C23.63C23 [×4], C23.67C23 [×4], C23⋊Q8 [×2], C23.4Q8 [×2], C24.355C23

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], C23.37C23 [×2], C23⋊2Q8, C22.45C24 [×2], C22.53C24 [×2], C24.355C23

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

Smallest permutation representation
►On 64 points
Generators in S64
(2 40)(4 38)(5 36)(6 62)(7 34)(8 64)(10 42)(12 44)(13 57)(14 30)(15 59)(16 32)(17 61)(18 33)(19 63)(20 35)(22 50)(24 52)(26 54)(28 56)(29 45)(31 47)(46 58)(48 60)

(1 39)(2 40)(3 37)(4 38)(5 17)(6 18)(7 19)(8 20)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 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 62)(34 63)(35 64)(36 61)

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
040000
001000
000400
000010
000004
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
400000
040000
004000
000400
000010
000001
,
100000
010000
003000
000200
000001
000010
,
010000
100000
000100
004000
000010
000001
,
300000
030000
003000
000300
000020
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],[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],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,3] >;
 

38 conjugacy classes

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

38 irreducible representations

dim1111111224
type+++++++-+
imageC1C2C2C2C2C2C2Q8C4○D42+ (1+4)
kernelC24.355C23C4×C22⋊C4C23.34D4C23.63C23C23.67C23C23⋊Q8C23.4Q8C22×C4C2×C4C22
# reps12144224162

In GAP, Magma, Sage, TeX 

C_2^4._{355}C_2^3
 
% in TeX
 
G:=Group("C2^4.355C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1339);
 
// by ID
 
G=gap.SmallGroup(128,1339);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,758,723,184,675,304]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=f^2=c,g^2=d*b=b*d,e*a*e^-1=a*b=b*a,a*c=c*a,f*a*f^-1=a*d=d*a,a*g=g*a,b*c=c*b,g*e*g^-1=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f^-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,f*g=g*f>;
 
// generators/relations
 

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