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G = C24.365C23  order 128 = 27

205th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.365C23, C23.524C24, C22.2202- (1+4), C22.3012+ (1+4), C23.195(C2×D4), (C22×C4).402D4, C23.4Q8⋊27C2, C23.7Q8⋊78C2, C23.11D4⋊58C2, C23.10D4⋊59C2, (C23×C4).426C22, (C2×C42).603C22, (C22×C4).134C23, C22.349(C22×D4), C24.3C22⋊66C2, C4.96(C22.D4), (C22×D4).195C22, C2.37(C22.29C24), C23.65C23⋊102C2, C2.C42.250C22, C2.25(C22.31C24), C2.45(C22.36C24), C2.25(C22.34C24), (C2×C4).383(C2×D4), (C2×C4⋊D4).40C2, (C2×C42⋊C2)⋊38C2, (C2×C4).658(C4○D4), (C2×C4⋊C4).355C22, C22.396(C2×C4○D4), C2.42(C2×C22.D4), (C2×C22⋊C4).215C22, SmallGroup(128,1356)

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

Derived series
C1 — C23 — C24.365C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C22×D4 — C24.3C22 — C24.365C23
Lower central
C1 — C23 — C24.365C23
Upper central
C1 — C23 — C24.365C23
Jennings
C1 — C23 — C24.365C23

Subgroups: 580 in 274 conjugacy classes, 100 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×12], C22 [×3], C22 [×4], C22 [×24], C2×C4 [×8], C2×C4 [×40], D4 [×12], C23, C23 [×2], C23 [×20], C42 [×4], C22⋊C4 [×18], C4⋊C4 [×14], C22×C4 [×2], C22×C4 [×14], C22×C4 [×4], C2×D4 [×14], C24, C24 [×2], C2.C42 [×6], C2×C42 [×2], C2×C22⋊C4 [×14], C2×C4⋊C4 [×3], C2×C4⋊C4 [×6], C42⋊C2 [×4], C4⋊D4 [×4], C23×C4, C22×D4, C22×D4 [×2], C23.7Q8, C23.65C23 [×2], C24.3C22 [×2], C23.10D4 [×4], C23.11D4 [×2], C23.4Q8 [×2], C2×C42⋊C2, C2×C4⋊D4, C24.365C23

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22.D4 [×4], C22×D4, C2×C4○D4 [×2], 2+ (1+4) [×3], 2- (1+4), C2×C22.D4, C22.29C24, C22.31C24, C22.34C24 [×2], C22.36C24 [×2], C24.365C23

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

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

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽5)

01000000
10000000
00400000
00040000
00003100
00002200
00000024
00000033
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00100000
00010000
00004000
00000400
00000040
00000004
,
10000000
01000000
00400000
00040000
00004000
00000400
00000040
00000004
,
30000000
03000000
00010000
00100000
00000010
00000001
00004000
00000400
,
10000000
04000000
00100000
00040000
00001000
00004400
00000040
00000011
,
10000000
01000000
00100000
00010000
00004300
00001100
00000043
00000011

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E···4N4O···4T
order12···2222244444···44···4
size11···1448822224···48···8

32 irreducible representations

dim1111111112244
type+++++++++++-
imageC1C2C2C2C2C2C2C2C2D4C4○D42+ (1+4)2- (1+4)
kernelC24.365C23C23.7Q8C23.65C23C24.3C22C23.10D4C23.11D4C23.4Q8C2×C42⋊C2C2×C4⋊D4C22×C4C2×C4C22C22
# reps1122422114831

In GAP, Magma, Sage, TeX 

C_2^4._{365}C_2^3
 
% in TeX
 
G:=Group("C2^4.365C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1356);
 
// by ID
 
G=gap.SmallGroup(128,1356);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,232,758,723,185,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=c,g^2=b,e*a*e^-1=a*b=b*a,f*a*f=a*c=c*a,a*d=d*a,a*g=g*a,b*c=c*b,b*d=d*b,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,f*e*f=d*e=e*d,d*f=f*d,d*g=g*d,e*g=g*e>;
 
// generators/relations
 

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