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G = C8.12M4(2)  order 128 = 27

7th non-split extension by C8 of M4(2) acting via M4(2)/C8=C2

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

Aliases: C8.12M4(2), (C4×C16)⋊4C2, C4⋊C4.8C8, C4⋊C1616C2, C165C49C2, C22⋊C4.4C8, C8⋊C4.18C4, C2.6(D4○C16), C23.10(C2×C8), C8.100(C4○D4), C22⋊C16.10C2, (C2×C16).67C22, (C4×C8).322C22, C42.245(C2×C4), (C2×C8).631C23, C4.69(C2×M4(2)), C42⋊C2.23C4, (C2×M4(2)).34C4, C22.51(C22×C8), C4.78(C42⋊C2), C82M4(2).21C2, (C22×C8).419C22, C2.12(C42.12C4), (C2×C4).28(C2×C8), (C2×C8).196(C2×C4), (C22×C4).289(C2×C4), (C2×C4).616(C22×C4), SmallGroup(128,896)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C8.12M4(2)
C1C2C4C8C2×C8C4×C8C82M4(2) — C8.12M4(2)
C1C22 — C8.12M4(2)
C1C2×C8 — C8.12M4(2)
C1C2C2C2C2C4C4C2×C8 — C8.12M4(2)

Generators and relations for C8.12M4(2)
 G = < a,b,c | a8=c2=1, b8=a4, bab-1=cac=a5, cbc=a6b5 >

Subgroups: 92 in 69 conjugacy classes, 48 normal (26 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×5], C22, C22 [×3], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×2], C23, C16 [×4], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], C22×C4, C4×C8 [×2], C8⋊C4 [×2], C2×C16 [×4], C42⋊C2, C22×C8, C2×M4(2), C4×C16, C165C4, C22⋊C16 [×2], C4⋊C16 [×2], C82M4(2), C8.12M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, C2×C8 [×6], M4(2) [×2], C22×C4, C4○D4 [×2], C42⋊C2, C22×C8, C2×M4(2), C42.12C4, D4○C16 [×2], C8.12M4(2)

Smallest permutation representation of C8.12M4(2)
On 64 points
Generators in S64
(1 53 48 23 9 61 40 31)(2 62 33 32 10 54 41 24)(3 55 34 25 11 63 42 17)(4 64 35 18 12 56 43 26)(5 57 36 27 13 49 44 19)(6 50 37 20 14 58 45 28)(7 59 38 29 15 51 46 21)(8 52 39 22 16 60 47 30)
(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 45)(4 47)(6 33)(8 35)(10 37)(12 39)(14 41)(16 43)(17 25)(18 60)(19 27)(20 62)(21 29)(22 64)(23 31)(24 50)(26 52)(28 54)(30 56)(32 58)(49 57)(51 59)(53 61)(55 63)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I4J4K8A···8H8I8J8K8L8M8N8O8P16A···16P16Q···16X
order12222444444444448···88888888816···1616···16
size11114111122224441···1222244442···24···4

56 irreducible representations

dim11111111111222
type++++++
imageC1C2C2C2C2C2C4C4C4C8C8M4(2)C4○D4D4○C16
kernelC8.12M4(2)C4×C16C165C4C22⋊C16C4⋊C16C82M4(2)C8⋊C4C42⋊C2C2×M4(2)C22⋊C4C4⋊C4C8C8C2
# reps111221422884416

Matrix representation of C8.12M4(2) in GL4(𝔽17) generated by

13000
01300
0020
001615
,
0100
1000
0053
00612
,
1000
01600
0014
00016
G:=sub<GL(4,GF(17))| [13,0,0,0,0,13,0,0,0,0,2,16,0,0,0,15],[0,1,0,0,1,0,0,0,0,0,5,6,0,0,3,12],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,4,16] >;

C8.12M4(2) in GAP, Magma, Sage, TeX

C_8._{12}M_4(2)
% in TeX

G:=Group("C8.12M4(2)");
// GroupNames label

G:=SmallGroup(128,896);
// by ID

G=gap.SmallGroup(128,896);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,723,58,102,124]);
// Polycyclic

G:=Group<a,b,c|a^8=c^2=1,b^8=a^4,b*a*b^-1=c*a*c=a^5,c*b*c=a^6*b^5>;
// generators/relations

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